The Computer Science of TeX and LaTex
Content
THE COMPUTER SCIENCE OF TEX AND LATEX;
COMPUTER SCIENCE COURSE 594, FALL 2004
VICTOR EIJKHOUT
DEPARTMENT OF COMPUTER SCIENCE,
UNIVERSITY OF TENNESSEE, KNOXVILLE TN 37996
DRAFT LECTURE NOTES
About this book.
These are the lecture notes of a course I taught in the fall of 2004. This was the first time
I taught the course, and the first time this course was taught, period. These lecture notes,
therefore, are probably full of inaccuracies, mild fibs, and gross errors. Ok, make that ‘are
definitely full of &c’, because I know of several errors that time has prevented me from
addressing.
However, I would be interested in hearing any comments and suggestions you have, or
errors you would like to point out.
Victor Eijkhout
[email protected]
Knoxville, TN, december 2004.
Enjoy!
2.7
2.8
Contents
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
2
2.1
2.2
2.3
2.4
2.5
2.6
2
About this book 1
TEX and LATEX 5
LATEX 6
Document markup 6
The absolute basics of
LATEX 8
The TEX conceptual
model of
typesetting 10
Text elements 11
Tables and figures 19
Math 19
References 21
Some TEXnical
issues 23
Customizing LATEX 23
Extensions to LATEX 27
TEX programming 30
TEX visuals 31
Projects for this
chapter 32
Parsing 33
Parsing theory 34
Levels of parsing 34
Very short
introduction 34
Lexical analysis 37
Finite state automata and
regular languages 37
Lexical analysis with
FSAs 41
Syntax parsing 43
Context-free
languages 43
Parsing context-free
languages 45
2.9
2.10
2.11
2.12
2.13
2.14
2.15
2.16
2.17
2.18
2.19
2.20
2.21
2.22
2.23
2.24
2.25
2.26
2.27
3
3.1
3.2
3.3
3.4
3.5
3.6
Lex 56
Introduction 56
Structure of a lex
file 56
Definitions section 57
Rules section 57
Regular expressions 60
Remarks 60
Examples 61
Yacc 65
Introduction 65
Structure of a yacc
file 65
Motivating example 65
Definitions section 67
Lex Yacc
interaction 67
Rules section 69
Operators; precedence
and associativity 69
Further remarks 70
Examples 73
Hashing 79
Introduction 79
Hash functions 79
Collisions 82
Other applications of
hashing 86
Discussion 87
Projects for this
chapter 88
Breaking things into
pieces 89
Dynamic
Programming 90
Some examples 90
Discussion 96
TEX paragraph
breaking 98
The elements of a
paragraph 98
TEX’s line breaking
algorithm 102
NP completeness 110
Introduction 110
Basics 111
CONTENTS
3.7
3.8
3.9
3.10
3.11
4
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5
5.1
6
Complexity
classes 112
NP-completeness 114
Page breaking 118
Introduction 118
TEX’s page breaking
algorithm 118
Theory of page
breaking 120
Projects for this
chapter 128
Fonts 129
Bezier curves 130
Introduction to curve
approximation 130
Parametric curves 135
Practical use 143
Curve plotting with
gnuplot 145
Introduction 145
Plotting 146
Raster graphics 147
Vector graphics and
raster graphics 147
Basic raster
graphics 147
Rasterizing type 150
Anti-aliasing 154
Projects for this
chapter 157
TEX’s macro language
– unfinished
chapter 159
Lambda calculus in
TEX 160
Logic with TEX 160
Character
encoding 171
Victor Eijkhout
3
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
7
7.1
7.2
7.3
7.4
7.5
Input file encoding 172
History and
context 172
Unicode 175
More about character
sets and encodings 178
Character issues in TEX /
LATEX 180
Font encoding 183
Basic terminology 183
Æsthetics 185
Font technologies 186
Font handling in TEX
and LATEX 189
Input and output
encoding in LATEX 191
The fontenc
package 191
Projects for this
chapter 192
Software
engineering 193
Literate
programming 194
The Web system 194
Knuth’s philosophy of
program
development 194
Software
engineering 195
Extremely brief history
of TEX 195
TEX’s
development 195
Markup 198
History 198
Projects for this
chapter 200
4
CONTENTS
TEX – LATEX – CS 594
Chapter 1
TEX and LATEX
In this chapter we will learn
•
The use of LATEX for document preparation,
•
LATEX style file programming,
•
TEX programming.
Handouts and further reading for this chapter
For LATEX use the ‘Not so short introduction to LATEX’ by Oetiker et al. For further reading
and future reference, it is highly recommended that you get ‘Guide to LATEX’ by Kopka and
Daly [15]. The original reference is the book by Lamport [16]. While it is a fine book, it has
not kept up with developments around LATEX, such as contributed graphics and other packages. A book that does discuss extensions to LATEX in great detail is the ‘LATEX Companion’
by Mittelbach et al. [17].
For the TEX system itself, consult ‘TEX by Topic’. The original reference is the book by
Knuth [12], and the ultimate reference is the published source [11].
5
6
CHAPTER 1. TEX AND LATEX
LATEX.
1.1
Document markup
If you are used to ‘wysiwyg’ (what you see is what you get) text processors, LATEX may
seem like a strange beast, primitive, and probably out-dated. While it is true that there is a
long history behind TEX and LATEX, and the ideas are indeed based on much more primitive
technology than what we have these days, these ideas have regained surprising validity in
recent times.
1.1.1
A little bit of history
Document markup dates back to the earliest days of computer typesetting. In those days,
terminals were strictly character-based: they could only render mono-spaced built-in fonts.
Graphics terminals were very expensive. (Some terminals could switch to a graphical character set, to get at least a semblance of graphics.) As a result, compositors had to key in
text on a terminal – or using punched cards in even earlier days – and only saw the result
when it would come out of the printer.
Any control of the layout, therefore, also had to be through character sequences. To set
text in bold face, you may have had to surround it with <B> .. the text .. </B>.
Doesn’t that look like something you still encounter every day?
Such ‘control sequences’ had a second use: they could serve a template function, expanding
to often used bits of text. For instance, you could imagine $ADAM$ expanding to ‘From
our correspondent in Amsterdam:’.
LATEX works exactly the same. There are command control sequences; for instance, you get
bold type by specifying \bf, et cetera. There are also control sequences that expand to bits
of text: you have to type \LaTeX to get the characters ‘LATEX’ plus the control codes for
all that shifting up and down and changes in font size.
\TeX => T\kern -.1667em\lower .5ex\hbox {E}\kern -.125emX
\LaTeX => L\kern -.36em {\sbox \z@ T\vbox to\ht \z@ {\hbox
{\check@mathfonts \fontsize \sf@size \z@ \math@fontsfalse
\selectfont A} \vss }}\kern -.15em\TeX
1.1.2
Macro packages
The old typesetting systems were limited in their control sequences: they had a fixed repertoire of commands related to typesetting, and there usually was some mechanism to defining ‘macros’ with replacement text. Formally, a macro is a piece of the input that gets
replaced by its definition text, which can be a combination of literal text and more macros
or typesetting commands.
TEX – LATEX – CS 594
1.1. DOCUMENT MARKUP
7
An important feature of many composition programs is the ability to
designate by suitable input instructions the use of specified formats. Previously stored sequences of commands or text replace the instructions,
and the expanded input is then processed. In more sophisticated systems, formats may summon other formats, including themselves [”System/360 Text Processor Pagination/360, Application Description Manual,” Form No. GE20-0328, IBM Corp., White Plains, New York.].
That was the situation with commercial systems by manufacturers of typesetting equipment such as Linotype. Systems developed by (and for!) computer scientists, such Scribe or
nroff/troff, were much more customizable. In fact, they sometimes would have the equivalent of a complete programming language on board. This makes it possible to take the
basic language, and design a new language of commands on top of it. Such a repertoire of
commands is called a macro package.
In our case, TEX is the basic package with the strange macro programming language, and
LATEX is the macro package1 . LATEX was designed for typesetting scientific articles and
books: it offers a number of styles, each with slightly different commands (for instance,
there are no chapters in the article style) and slightly different layout (books need a title
page, articles merely a title on the first page of the text). Styles can also easily be customized. For different purposes (art books with fancy designs) it is often better to write
new macros in TEX, rather than to bend the existing LATEX styles.
However, if you use an existing LATEX style, the whole of the underlying TEX programming
language is still available, so many extensions to LATEX have been written. The best place
to find them is through CTAN http://wwww.ctan.org/.
Exercise 1. Discuss the difference between a macro and a function or
procedure in a normal programming language. In a procedural language,
looping is implemented with goto instructions. How would you do that
in a macro language? Is there a difference in efficiency?
1.1.3
Logical markup
Macro packages are initially motivated as a labour-saving device: a macro abbreviates a
commonly used sequence of commands. However, they have another important use: a well
designed macro package lets you use commands that indicate the structure of a document
rather than the formatting. This means that you would write \section{Introduction}
and not worry about the layout. The layout would be determined by a statement elsewhere
as to what macros to load2 . In fact, you could take the same input and format it two different
ways. This is convenient in cases such as an article being reprinted as part of a collection,
or a book being written before the final design is commissioned.
In a well written document, there will be few explicit typesetting commands. Almost all
macros should be of the type that indicates the structure, and any typesetting is taken care
1.
In this tutorial I will say ‘TEX’ when a statement applies to the basic system, and ‘LATEX’ if it only applies
to the macro package.
2.
Compare this to the use of CSS versus HTML in web pages.
Victor Eijkhout
8
CHAPTER 1. TEX AND LATEX
of in the definition of these. Further control of the layout of the document should be done
through global parameter settings in the preamble.
1.2
The absolute basics of LATEX
Here is the absolute minimum you need to know to use LATEX.
1.2.1
Different kinds of characters
A TEX input file mostly contains characters that you want to typeset. That is, TEX passes
them on from input to output without any action other than placement on the page and font
choice. Now, in your text there can be commands of different sorts. So TEX has to know
how to recognize commands. It does that by making a number of characters special. In this
section you will learn which characters have special meaning.
•
•
•
•
•
•
3.
Anything that starts with a backslash is a command or ‘control sequence’. A control sequence consists of the backslash and the following sequence of letters – no
digits, no underscores allowed either – or one single non-letter character.
Spaces at the beginning and end of a line are ignored. Multiple spaces count as
one space.
Spaces are also ignored after control sequences, so writing \LaTeX is fun
comes out as ‘LATEXis fun’. To force a space there, write \LaTeX{} is fun
or \LaTeX\ is fun. Spaces are not ignored after control symbols such as \$,
but they are again after the ‘control space’ \ 3 .
A single newline or return in the input file has no meaning, other than giving a
space in the input. You can use newlines to improve legibility of the input. Two
newlines (leading to one empty line) or more cause a paragraph to end. You can
also attain this paragraph end by the \par command.
Braces {,} are mostly used for delimiting the arguments of a control sequence.
The other use is for grouping. Above you saw an example of the use of an empty
group; similarly \TeX{}ing is fun comes out as ‘TEXing is fun’.
Letters, digits, and most punctuation can be typed normally. However, a bunch of
characters mean something special to LATEX: %$&ˆ_#˜{}. Here are their functions:
% comment: anything to the end of line is ignored.
$,_,ˆ inline math (toggle), subscript, superscript. See section 1.6.
& column separator in tables.
˜ nonbreaking space. (This is called an ‘active character’)
{} Macro arguments and grouping.
In order to type these characters, you need to precede them with a backslash, for
instance \% to get ‘%’. This is called a ‘control symbol’. Exception: use $\backslash$
to get ‘\’.
The funny bucket character here is how we visualize the space character.
TEX – LATEX – CS 594
1.2. THE ABSOLUTE BASICS OF LATEX
•
9
Some letters do not exist in all styles. As the most commonly encountered example, angle brackets <> do not exist in the roman text font (note that they are
in the typewriter style here, in roman you would get ‘¡¿’), so you need to write,
somewhat laboriously $\langle$S$\rangle$ to get ‘hSi’4 .
Exercise 2. You read in a document ‘This happens only in 90rest of
the time it works fine.’ What happened here? There are articles in print
where the word ‘From’ has an upside down question mark in front of it.
Try to think of an explanation.
1.2.2
LATEX document structure
Every LATEX document has the following structure:
\documentclass[ <class options> ]{ <class name> }
<preamble>
\begin{document}
<text>
\end{document}
Typical document classes are article, report, book, and letter. As you may expect, that last one has rather different commands from the others. The class options are
optional; examples would be a4paper, twoside, or 11pt.
The preamble is where additional packages get loaded, for instance
\usepackage{times}
switches the whole document to the Times Roman typeface. This is also the place to define
new commands yourself (section 1.9.2).
1.2.2.1
Title
To list title and author, a document would start with
\title{My story}
\author{B.C. Dull}
\date{2004} %leave this line out to get today’s date
\maketitle
After the title of an article and such, there is often an abstract. This can be specified with
\begin{abstract}
... The abstract text ...
\end{abstract}
The stretch of input from \begin to \end is called an ‘environment’; see section 1.4.1.2.
1.2.2.2
Sectioning
The document text is usually segmented by calls
4.
That’s what macros are good for.
Victor Eijkhout
10
CHAPTER 1. TEX AND LATEX
\section{This}
\subsection{That}
\subsection{The other}
\paragraph{one}
\subparagraph{two}
\chapter{Ho}
\part{Hum}
which all get numbered automatically. Chapters only exist in the report and book styles.
Paragraphs and subparagraphs are not numbered. To prevent sections et cetera from being
numbered, use \section*{...} and such5 .
1.2.2.3
Frontmatter, backmatter
You can use commands \frontmatter, \mainmatter, \backmatter – in book
class only – to switch page numbering from roman to arabic, and, for the back matter,
section numbering from numbers to letters.
1.2.3
Running LATEX
With the last two sections you should know enough to write a LATEX document. Now how
do you get to see the final output? This takes basically two steps: formatting and viewing.
You need to know that TEX’s original output format is slightly unusual. It is called ‘DVI’
for DeVice Independent. There are viewers for this format, but usually you need another
step to print it.
Traditionally, you would run an executable called latex (or tex), which gives you a dvi
file, which you then view with a previewer such as xtex or xdvi. To print this file, you
would use
dvips -Pcmz foo.dvi -o foo.ps
to generate a ps file. This can be printed, or converted to pdf.
There are version of the latex executable that output to other formats, for instance
pdflatex (there is also a pdftex) goes straight to pdf, which you can view with the
Adobe Acrobat Reader, or xpdf. The big advantage of this approach is that you can get
hyperlinks in your pdf file; see section 1.10.3.
Exercise 3. Set up a document that will have the answers to your
homework exercises of the whole course.
1.3
The TEX conceptual model of typesetting
In TEX, the question ‘on what page does this character appear’ is hard to answer. That is
because TEX typesets all material for a page, sort of on a long scroll, before cutting a page
5.
This also makes the title not go into the table of contents. See section 1.7.2 on how to remedy that.
TEX – LATEX – CS 594
1.4. TEXT ELEMENTS
11
off that scroll. That means that when a piece of text is set, you do not know if it falls before
or after a certain page break.
A similar story holds for paragraph breaking, but the question on what line something
occurs is not usually interesting.
1.4
Text elements
Here are the main elements that make up a LATEX document.
1.4.1
Large scale text structure
We already discussed sectioning commands in section 1.2.2.2. Here are more major text
elements in a LATEX document.
1.4.1.1
Input files
Use \include{<file>} to input a file beginning on a new page, and \input for just
plain input. With
\includeonly{file1,file2}
you can save processing time – provided the files are \included to begin with.
The .tex extension can usually be left off; because of the way TEX works, be careful with
funny characters in the file name.
On Unix installations, input files can be in other directories, specified by the TEXINPUTS
environment variable.
1.4.1.2
Environments
If a certain section of text needs to be treated in a different way from the surrounding text,
it can be segmented off by
\begin{<environment name>}
... text ...
\end{<environment name>}
An environment defines a group, so you can have local definitions and changes of parameters.
Some predefined environments are
flushleft (flushright) for text that is left (right) aligned, but not right (left).
center for text that is centered.
quote, quotation for text that needs to be set apart, by indenting both margins. The quote
environment is for single paragraphs, the quotation for multiple.
abstract for the abstract that starts an article, report, or book. In the report and book style
it is set on a separate page. In the article style it is set in a smaller type size, and
with indented margins.
verbatim see section 1.4.1.3.
Victor Eijkhout
12
1.4.1.3
CHAPTER 1. TEX AND LATEX
Verbatim text
As we have observed already, TEX has a number of special characters, which can be printed
by prefixing them with a backslash, but that is a hassle. Good thing that there is a mechanism for printing input completely verbatim. For inline text, use \verb+&###\text+ to
get ‘&###\text’. The essential command here is \verb. Unlike with other commands
that have arguments, the argument is not delimited by braces, but by two occurrences of
a character that does not appear in the verbatim text. A plus sign is a popular choice. The
\verb* variant makes spaces visible: \verb*+{ }+ gives ‘{ }’.
For longer verbatim text there is a verbatim environment. The verbatim* version
prints each space as a symbol. To input whole files verbatim, use \verbatiminput{file},
which is defined in the verbatim package.
For TEXnical reasons, verbatim text can not appear in some locations such as footnotes or
command definitions.
Exercise 4. Why does the \verb command not have its argument in
braces?
1.4.1.4
Lists
Lists in LATEX are a special case of an environment; they are specified by
\begin{<list type>}
\item ...
\item ...
\end{<list type>}
The three main list types are unnumbered lists, itemize, numbered lists, enumerate,
and definition or description lists, description.
In the case of a description list, it is mandatory to give the item label:
\begin{description}
\item[Do] A deer, a female deer.
\item[Re] A drop of golden sun.
...
\end{description}
You can give item labels in the other list types too.
Putting a list inside a list item will change the style of the item labels and numbers in a way
that is dictated by the document class.
You can put a \label command after an item to be able to refer to the item number.
\begin{enumerate}
\item\label{first:item} One
\item Two comes after \ref{first:item}
\end{enumerate}
Output:
TEX – LATEX – CS 594
1.4. TEXT ELEMENTS
13
1.
One
2.
Two comes after 1
This only makes sense with enumerate environments.
1.4.1.5
Tabbing
The tabbing environment is useful for setting pieces of text, such as source code, that use
a small number of ‘tab stops’. Tab stops (a term deriving from old mechanical typewriters)
are locations on the line that one can ‘tab to’, no matter how much material is currently on
the line.
Example:
\begin{tabbing}
The first line sets this: \=point;\\
the second jumps\>there
\end{tabbing}
Output:
The first line sets this: point;
the second jumps
there
The \= command in the first line defines a tab stop; in every subsequent line a \> command
will jump to that position, if it has not been reached yet. There can be multiple tab stops,
not necessarily defined in the same line, and tab stops can be redefined.
A more interesting case is where the tab stop is used before the line that defines it. For this
case there is the \kill command, which prevents a line from being displayed. Example:
\begin{tabbing}
while \=\kill
do\>\{\\
\>$i_1\leftarrow{}$\=1\\
\>$\ldots$\>2\\
\>\}\\
while (1)
\end{tabbing}
Output:
do {
i1 ← 1
... 2
}
while (1)
1.4.1.6
Tabular material
The tabular environment generates a table. Tables are often placed independently of the
text, at the top or bottom of the page; see section 1.5 for details. The table itself is generated
by
Victor Eijkhout
14
CHAPTER 1. TEX AND LATEX
\begin{tabular}{<alignment>}
... material ...
\end{tabular}
Each line of the table has items separated by characters, and \\ at the end of each line but
the last.
In its simplest form, the alignment directions are a combination of the letters l,r,c:
\begin{tabular}{rl}
"Philly" Joe & Jones\\ Dizzie & Gillespie\\ Art&Tatum
\end{tabular}
Output:
”Philly” Joe
Dizzie
Art
Jones
Gillespie
Tatum
Vertical rules are inserted by placing a | character in the alignment specification; horizontal
lines you get from \hline.
\begin{tabular}{|r|rl|}
\hline
instrument&name&\\ \hline
drums: &"Philly" Joe & Jones\\
trumpet:& Dizzie & Gillespie\\
piano: &Art&Tatum\\ \hline
\end{tabular}
Output:
instrument
drums:
trumpet:
piano:
name
”Philly” Joe
Dizzie
Art
Jones
Gillespie
Tatum
Some more tricks:
•
•
In headings you often want to span one item over several columns. Use
\begin{tabular}{|r|rl|}
\hline
instrument&\multicolumn{2}{|c|}{name}\\ \hline
drums: &"Philly" Joe & Jones\\
trumpet:& Dizzie & Gillespie\\
piano: &Art&Tatum\\ \hline
\end{tabular}
Output:
instrument
name
drums: ”Philly” Joe Jones
trumpet:
Dizzie Gillespie
piano:
Art Tatum
LATEX inserts a standard amount of space between columns. You can override this
with @{<stuff>}:
TEX – LATEX – CS 594
1.4. TEXT ELEMENTS
•
1.4.1.7
15
\begin{tabular}{r@{.}l}
2&75\\ 800&1
\end{tabular}
2.75
gives
800.1
A column specification of p{<size>} (where <size> is something like 5.08cm
or 2in) makes the items in that column formatted as paragraphs with the width
as specified.
Footnotes
Use the command \footnote. The numbering style is determined by the document class.
The kinds of footnote to denote affiliation of the author of a paper and such (these often
use asterisk symbols and such, even if footnotes are numbered in the rest of the document)
are given by the command \thanks.
There are two common cases where want more detailed control over footnotes:
•
You want to dictate the label yourself, for instance using the same label again (this
often happens with author affiliations)
•
You want to place a footnote in a table or such; LATEX has trouble with that.
In such cases you can use \footnotemark to place the marker, and \footnotetext
for the text. You can also set or change the footnote counter explicitly with counter
functions (section 1.9.4), or use
\footnote[<label>]{<text>}
where the label is used instead, and the counter is not increased.
1.4.1.8
Text boxes
Sometimes you want a small amount of text to behave like one or more paragraphs, except
not as part of the main page. The main commands for that are
\parbox[pos]{width}{text}
\begin{minipage}[pos]{width} text \end{minipage}
The optional pos parameter specifies whether the top (t) or bottom (b) line of the box
should align with surrounding text: top-aligned box of text:
Chapter 1. \parbox[t]{2in}{\slshape Introduction. First easy
lessons. Exercises. More about things to come. Conclusions}
Output:
Chapter 1. Introduction. First easy lessons. Exercises. More about things to come.
Conclusions
The default is a vertically centered position.
The minipage environment is meant for longer pieces of text; it can also handle other
environments in the text.
The \mbox command is for text (or other objects) that need to stay on one line.
Victor Eijkhout
16
CHAPTER 1. TEX AND LATEX
1.4.2
Minor text issues
1.4.2.1
Text styles
You can switch between roman (the style for this text), italic (also called ‘cursive’), slanted
(in some typefaces, italic and slanted may be identical), and bold with the commands
\texrm, \textit, \textsl, and \textbf respectively, used as
Text is stated \textbf{boldly} or \textsl{with a slant}.
These combinations are not independent: nesting the commands can give you bold slanted
text.
The above commands are mostly for short bits of text. See section 1.4.2.2 for commands
to change font parameters in a more global manner.
If you are using italics for emphasis, consider using \emph instead, which works better,
especially if you emphasize something in text that is already italic.
1.4.2.2
Fonts and typefaces
You already saw commands such as \textrm and \textit for switching from one type
style to another. These commands hide a more complicated reality: LATEX handles its fonts
as combination of three parameters. These individual switches can be used inside a group,
or as an environment:
{\ttfamily This is typewriter text}
\begin{mdseries}
This text is set in medium weight.
\end{mdseries}
Here are the categories and possible values.
family roman, sans serif, typewriter type: \rmfamily, \sffamily, \ttfamily.
series medium and bold: \mdseries, \bfseries.
shape upright, italic, slanted, and small caps: \upshape, \itshape, \slshape, \scshape.
1.4.2.3
Comments
Anything from % to the end of line is ignored. For multiline comments, load either
\usepackage{comment}
or
\usepackage{verbatim}
and in both cases surround text with
\begin{comment}
to be ignored
\end{comment}
where the closing line has to be on a line of its own.
TEX – LATEX – CS 594
1.4. TEXT ELEMENTS
1.4.2.4
17
Hyphenation
Sometimes TEX has a hard time finding a breakpoint in a word. When you are fixing the
final layout of a document, you can help it with helico\-pter. If TEX consistently
breaks your name wrong, do
\hyphenation{Eijk-hout}
in the preamble.
This is not the mechanism for telling TEX about a new language; see section 1.10.5.
To keep text together, write \mbox{do not break}. You could also write this with
a non-breaking space as do˜not˜break. (See also section 1.4.2.5.) It is a good idea to
write
A˜note on...
increase by˜$1$.
to prevent single characters at the beginning of a line (first example), or the end of a line
(second example). The second example could even give a line with a single character on it
if it were to occur at the end of a paragraph.
1.4.2.5
Tilde
The tilde character has special meaning as a nonbreaking space; see section 1.4.2.4. To get
a tilde accent, use \˜. To get a literal tilde, do \˜{}, $\sim$, or \char‘\˜. If you need
a tilde in URLs, consider using the url or hyperref package; see section 1.10.3.
1.4.2.6
Accents
In basic TEX, accents are formed by putting a control symbol of that accent in front of the
letter:
Sch\"on b\ˆet\’e
for ‘Sch¨on bˆet´e’. If you have an occasional foreign word in English text this works fine.
However, if your terminal allows you to input accented characters, you can use them in
LATEX with the inputenc package.
Standard TEX (or LATEX) does not understand Unicode encodings such as UTF-8.
1.4.2.7
Line/page breaking
In general, you should leave line and page breaking to TEX, at most adjusting parameters.
However,
should
you
really
need
it,
you can use the commands \linebreak[<num>] and \pagebreak[<num>], where
the number is 1,2,3,4, with 4 the highest
urgency. There is also \nolinebreak and \nopagebreak with a similar urgency parameter.
In this last paragraph there was a \linebreak after ‘need it’. You notice that TEX still
tried to fill out to the right margin, with ugly consequences. After ‘highest’ there was
a \newline, which really breaks then and there. Similarly, there is \newpage.
Victor Eijkhout
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CHAPTER 1. TEX AND LATEX
There is also \nolinebreak and \nopagebreak, both with optional numerical parameter, to discourage breaking.
1.4.2.8
Manual spacing
Most of the time you should leave spacing decisions to LATEX, but for custom designs it is
good to know the commands.
\hspace{1cm} \hspace*{1in} \hspace{\fill}
\vspace{1cm} \vspace*{1in} \vspace{\fill}
•
The *-variants give space that does not disappear at the beginning or end of a line
(for horizontal) or page (vertical).
A space of size \fill is infinite: this means it will stretch to take up however
much space is needed to the end of the line or page.
•
1.4.2.9
Drawing lines
Let us get one thing out of the way: underlining is a typewriter way of emphasizing text. It
looks bad in typeset text, and using italics or slanted text is a much better way. Use \emph.
Lines can be used a typographical decorations, for instance drawn between the regular text
and the footnotes on a page, or as part of chapter headings. The command is
\rule[lift]{width}{height}
Example
1\ \rule{2cm}{\fboxrule}\ The title
Output:
1
The title
You can draw a whole box around text: \fbox{text} gives text . The thickness of the
line is \fboxrule.
1.4.2.10 Horizontal and vertical mode
TEX is in horizontal or vertical mode while it is processing6 . In horizontal mode, elements
– typically letters – are aligned next to each other; in vertical mode elements are stacked on
top of one another. Most of the time you do not have to worry about that. When TEX sees
text, it switches to horizontal mode, and LATEX environments will briefly switch to vertical
mode so that they start on a new line.
In certain cases you want to force vertical mode; for that you can use \par. You can force
things into a line with \mbox (section 1.4.1.8). In rare cases, \leavevmode.
6.
The story is actually more complicated; for the whole truth see the notes about TEX.
TEX – LATEX – CS 594
1.5. TABLES AND FIGURES
1.5
19
Tables and figures
Tables and figures are objects that typically do not appear in the middle of the text. At the
very least they induce a paragraph break, and often they are placed at the top or bottom of
a page. Also, some publishers’ styles demand that a document have a list of tables and a
list of figures. LATEX deals with this by having environments
\begin{<table or figure>}[placement]
... table or figure material ...
\caption{Caption text}\label{tabfig:label}
\end{<table or figure>}
In this,
•
The ‘placement’ specifier is a combination of the letters htbp for ‘here’, ‘top’,
‘bottom’, and ‘page’, telling LATEX where to place the material, if possible. Suppose a placement of [ht] is given, then the material is placed ‘right here’, unless
there is not enough space on the page, in which case it will be placed on top of
the page (this one or the next).
•
Table material is given by a tabular environment; see section 1.4.1.6.
•
Figure material needs some extra mechanism, typically provided by another package; see section 1.10.4.
•
The caption goes into the list of tables/figures.
•
The label will refer to the number, which is automatically generated.
The list of tables/figures is generated by the command \listoftables or \listoffigures.
1.6
Math
TEX was designed by a computer scientist to typeset some books with lots of mathematics.
As a result, TEX, and with it LATEX’s, math capabilities are far better than those of other
typesetters.
1.6.1
Math mode
You can not just write formulas in the middle of the text. You have to surround them with
$<formula>$ or \(<stuff>\) for inline formulas, or
\begin{displaymath} ... \end{displaymath}
\[ ... \]
for unnumbered and
\begin{equation} ... \end{equation}
for numbered displayed equations respectively. You can refer to an equation number by
including a \label statement.
In math mode, all sorts of rules for text typesetting are changed. For instance, all letters
are considered variables, and set italic: $a$ gives ‘a’. Roman text is either for names
of functions, for which there are control sequences – \sin(x) gives ‘sin(x)’ – or for
connecting text, which has to be marked as such:
Victor Eijkhout
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CHAPTER 1. TEX AND LATEX
\forall x \in \mathbf{R}
\quad \mathrm{(sufficiently large)} \quad: \qquad x>5
Output:
∀x ∈ R
(sufficientlylarge)
:
x>5
(1.1)
A formula is limited to one line; if you want to break it, or if you need several formulas
vertically after one another, you have to do it yourself. The eqnarray environment is
useful here. It acts as a three-column alignment.
\begin{eqnarray}
\sin x&=&x-\frac{xˆ3}{3!}+\frac{xˆ5}{5!}- \nonumber \\
&&{}-\frac{xˆ7}{7!}+\cdots
\end{eqnarray}
Output:
x5
x3
+
−
3!
5!
x7
+ ···
(1.2)
−
7!
Note the use of \nonumber here; with the eqnarray* all lines would be unnumbered
by default.
sin x = x −
In AMS LATEX there is an align environment which looks better than eqnarray.
1.6.2
Super and subscripts
In math mode, the character ˆ denotes a superscript, and _ denotes a subscript: x_iˆ2 is x2i .
(Outside of math mode these characters give an error.) Sub and superscripts of more than
one character have to be grouped.
1.6.3
Grouping
2
Grouping, in math mode as outside, is done with braces: x_{i-1}ˆ{nˆ2} looks like xni−1 .
1.6.4
Display math vs inline
Math looks different when used inline in a paragraph from that used as display math. This
is mostly clear for operators with ‘limits’:
∞
X
P∞
text mode: i=1 displaymode :
i=1
1.6.5
Delimiters, matrices
Delimiters are ()[]\{\}. You can prefix them with \big, \Big and such, but TEX can
resize them automatically:
TEX – LATEX – CS 594
1.7. REFERENCES
21
\left( \frac{1}{1-xˆ2} \right)
\left\{ \begin{array}{ccc}
\mathrm{(a)}&\Rightarrow&x>0\\
\mathrm{(b)}&\Rightarrow&x=0\\
\mathrm{(c)}&\Rightarrow&x<0
\end{array} \right.
Output:
(a) ⇒ x > 0
(b) ⇒ x = 0
(c) ⇒ x < 0
Note that with \right. you get a omitted right delimiter.
1
1 − x2
(1.3)
In the above example you also saw the array environment, which can be used for anything
tabular in math mode, in particular matrices. Here is a good example of a matrix. Note the
different kinds of dots:
A = \left( \begin{array}{cccccc}
a_{11}&0&&\ldots&0&a_{1n}\\
&a_{22}&0&\ldots&0&a_{2n}\\
&&\ddots&\ddots&\vdots&\vdots\\
&&&a_{n-2n-2}&0&a_{n-2n}\\
&\emptyset&&&a_{n-1n-1}&a_{n-1n}\\
&&&&&a_{nn}
\end{array} \right)
Output:
A=
1.6.6
a11
0
a22
0
..
.
...
...
..
.
0
0
..
.
a1n
a2n
..
.
an−2n−2
0
an−2n
an−1n
ann
∅
an−1n−1
(1.4)
There is more
See a good book for the whole repertoire of symbols. If what LATEX has is not enough, you
can also get AMS LATEX, which has even more fonts and tricky constructs.
1.7
References
1.7.1
Referring to document parts
One of the hard things in traditional typesetting is to keep references such as ‘see also
section 3’ in sync with the text. This is very easy in LATEX. You write
\section{Results}\label{section:results}
Victor Eijkhout
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CHAPTER 1. TEX AND LATEX
after which you can use this as
see also section˜\ref{section:results}
on page˜\pageref{section:results}.
The \label command can appear after headings, or in general every time some counter
has been increased, whether that’s a section heading or a formula number.
LATEX implements this trick by writing the information to an auxiliary file – it has extension
.aux – and reading it in next run. This means that a LATEX document usually has to be
typeset twice for all references to be both defined and correct. You get a reminder after the
first run if a second one is needed, or if there are missing or duplicately defined labels.
Exercise 5. A document with references usually takes two passes to
get right. Explain why a table of contents can increase this number to
three.
1.7.2
Table of contents
Something that typically goes into the front or back matter is the table of contents. This
gets inserted automatically by the command \tableofcontents. No other actions required. You can add your own material to the contents with \addcontentsline or
\addtocontents.
1.7.3
Bibliography references
Another kind of the reference is that to external bibliographies. This needs a bit more work.
•
•
•
•
You write \cite{Knuth:1978} where you want the citation.
At the end of your document you write
\bibliographystyle{plain}
\bibliography{cs}
to get the bibliography included.
The bibliography references have to be defined in a file cs.bib.
After running LATEX once, you need to invoke bibtex <yourfile>, which
creates another auxiliary file, this time with .bbl extension, and run LATEX once
or twice more.
The bibliography files have a syntax of their own, but you can figure that out from looking
at some examples.
1.7.4
Index
Finally, a document can have an index. For this you need to have a statement \usepackage{makeidx}
in the preamble, \printindex wherever you want the index, and commands \index{<some term>}
throughout your document. Additionally, as with bibtex, you need to run the program
makeindex to generate the external .ind file.
Further indexing commands: \index{name!sub} for subentry; \index{foo@\textit{foo})
for sorting under ‘foo’ but formatted differently.
TEX – LATEX – CS 594
1.8. SOME TEXNICAL ISSUES
1.8
Some TEXnical issues
1.8.1
Commands inside other commands
23
For deep technical reasons you can get completely incomprehensible error messages by
writing things like
\section{My first section \footnote{and not my last}}
Remedy that by writing
\section{My first section \protect\footnote{and not my last}}
1.8.2
Grouping
Most modern programming languages have a block structure of some sort, where variables
can be declared inside a block, and are no longer known after the block ends. TEX has a
stronger mechanism, where assignments to a variable made inside a block are reverted at
the end of that block.
In LATEX you notice that only \newcommand and \newenvironment declarations are
local; \newcounters are global, as are \setcounter assignments. However, \savebox
assignments are local.
1.9
Customizing LATEX
LATEX offers a number of tools (on top of the possibility of doing straight TEX programming)
for customizing your document. The easiest customization is to change already defined
parameters. However, you can also define new commands and environments of your own.
In fact, several of the customization we will see in this section are not part of standard
LATEX, but have been written by other users. If they do not come with your installation, you
can download them from the Central TEX Archive Network; see section 1.10.1.
1.9.1
Page layout
1.9.1.1
Layout parameters
Page layout is controlled by parameters such as \textheight, \textwidth, \topmargin
(distance to the running head, not to the first text line), and \odd/evensidemargin
(distance to the ‘spine’ of the document). These are set with commands like
\setlength{\textwidth}{10in}
\addtolength{\oddsidemargin}{-1cm}
Some lengths are ‘rubber length’
\setlength{\parskip}{10pt plus 3pt minus 2pp}
Victor Eijkhout
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1.9.1.2
CHAPTER 1. TEX AND LATEX
Page styles
Use the commands
\pagestyle{<style>}
and
\thispagestyle{<style>}
to change the style of all pages or one page. Available styles are empty (no page numbers), plain (the default), and headings (page numbers and running headers). See also
section 1.10.2 for many more options.
For two-sided printing, use the twoside option for the document class.
Exercise 6. Take a look at the headers and footers in Oetiker’s ‘Not
so short introduction’ and ‘TEX by Topic’ (the LATEX and TEX part of
the handout). Can you find a reason to prefer one over the other from a
point of usability? In both books, what is the rationale behind the header
on the odd pages? See in particular page 35 of the former and 77 of the
latter. Do you agree with this design?
1.9.1.3
Running page headers
The headings page style (section 1.9.1.2) uses running heads that can change through
the document. For instance it would have chapter titles in the left page head and section
titles in the right head. You can achieve this effect yourself by using the myheadings
page style, and using the
\markright{<right head>}
\markboth{<left>}{<right>}
You have access to these texts as \rightmark and \leftmark; this is needed in the
fancyhdr style.
1.9.1.4
Multicolumn text
Load
\usepackage{multicol}
and write
\begin{multicol}{3}
text in three column mode
\end{multicol}
1.9.2
New commands
You can define your own commands in LATEX. As example of a a simple command, consider
an often used piece of text
\newcommand{\IncrByOne}{increased by˜$1$}
The replacement text can have parameters:
TEX – LATEX – CS 594
1.9. CUSTOMIZING LATEX
25
\newcommand{\IncrDecrBy}[2]{#1creased by˜$#2$}
In this definition, the number of arguments is listed after the command name: [2], and
occurrences of the arguments in the replacement text are indicated by #1, #2 etc. Example
calls: \IncrDecrBy{in}{5}, \IncrDecrBy{de}{2}.
The declaration of a new command can specify an optional argument. If we define
\newcommand{\IncrDecrBy}[2][in]{#1creased by˜$#2$}
the [in] specification means that the first argument is optional (only the first argument
can ever be optional) with a default value of in. Example calls:
\newcommand{\IncrDecrBy}[2][in]{#1creased by˜$#2$}
\IncrDecrBy[de]{1}, \IncrDecrBy{5}.
Output:
decreased by 1, increased by 5.
To redefine an existing command, use \renewcommand.
1.9.3
New environments
It is possible to define environments, by specifying the commands to be used at their start
and end:
\newenvironment{example}%
{\begin{quote}\textbf{Example.}}%
{\end{quote}}
which, used as \begin{example}...\end{example} gives a quote environment
that starts with the word ‘Example’ in bold face. While defining that environment does not
save a lot of typing, it is a good idea nevertheless from a point of view of logical markup.
Using the example environment throughout ensures a uniform layout, and makes design
changes easy if you ever change your mind.
Special case: defining mathematical statements with
\newtheorem{majorfact}{Theorem}
\newtheorem{minorfact}[majorfact]{Lemma}
\begin{minorfact}Small fact\end{minorfact}
\begin{majorfact}Big fact\end{majorfact}
giving
Lemma 1 Small fact
Theorem 2 Big fact
The optional argument in the definition of lemma makes it use the theorem counter.
Exercise 7. Why does this not work:
\newenvironment{examplecode}%
{\textbf{Example code.}\begin{verbatim}}{\end{verbatimm}}
Victor Eijkhout
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CHAPTER 1. TEX AND LATEX
Exercise 8. Write macros for homework typesetting; make one master
document that will contain all your homework throughout this course.
1. Define an environment exercise so that
\begin{exercise}
My answer is...
\end{exercise}
gives
Problem 5. My answer is. . .
The counter should be incremented automatically. List your solution in your answer, and find a way that the listing is guaranteed to
be the code you actually use.
2. Write a macro \Homework that will go to a new page, and output
Answers to the exercises for chapter 3
at the top of the page. The exercise environment should now
take the question as argument:
\begin{exercise}{Here you paraphrase the question that was asked}
My answer is...
\end{exercise}
and this outputs
Problem 1.8 Here you paraphrase the question
that was asked
My answer is. . .
(Hint: read the section on text boxes. Also be sure to use \par to
get LATEX to go to a new line.) Allow for the question to be more
than one line long. Unfortunately you can not get verbatim text in
the question. Find a way around that.
1.9.4
Counters
LATEX has a number of counters defined, for sections and such. You can define your own
counters too. Here are the main commands:
create A new counter is created with
\newcounter{<name>}[<other counter>]
where the name does not have a backslash. The optional other counter indicates a counter (such as chapter) that resets the new counter every time
it is increased. (To do this reset for an already existing counter, check out the
chngcntr package.)
change values A counter can be explicitly set or changed as
\setcounter{<name>}{<value>}
\addtocounter{<name>}{<value>}
The command \refstepcounter also make the new value the target for a \label
command.
use To get a counter value numerically, use \value. To print the value, use
\arabic{<name>}, \roman{<name>}, \Roman{<name>}
et cetera.
TEX – LATEX – CS 594
1.10. EXTENSIONS TO LATEX
1.9.5
27
Lengths
Parameters such as \textwidth (section 1.9.1.1) are called ‘lengths’. You can define
your own with
\newlength{\mylength}
\setlength{\mylength}{5in}
These lengths can be used in horizontal or vertical space commands (section 1.4.2.8) for
your own designs.
1.9.6
The syntax of \new... commands
Have you noticed by now that the name you define starts with a backslash in \newcommand
and \newlength, but not in \newenvironment or \newcounter? Confusing.
1.10
Extensions to LATEX
LATEX may be a package on top of TEX, but that doesn’t mean that the programming power
of TEX is no longer available. Thus, many people have written small or large extensions to
be loaded in addition to LATEX. We will discuss a couple of popular ones here, but first we’ll
see how you can find them.
1.10.1
How to find a package, how to use it
Packages are typically loaded in the file preamble with
\usepackage{pack1,pack2,...}
(These course notes load about a dozen packages.)
Many popular packages are already part of the standard LATEX distribution, but you will
have to search to find where they are stored on your computer. Make a document that uses
a common package, say fancyhdr, and see in the log output on the screen or in the log
file where the file is loaded from. A typical location is /usr/share/texmf/.... With
a bit of searching you can also find7 the documentation, which can be a dvi, ps, or pdf
file.
If you have heard of a package and it is not on your system, go to the ‘Comprehensive TEX
Archive Network’ (CTAN for short) and download it from there: http://www.ctan.
org/.
1.10.2
Fancy page headers and footers
The fancyhdr8 package provides customized headers and footers. The simple interface
is
\lhead{<text>} \chead{<text>} \rhead{<text>}
7.
8.
For instance using the Unix command ‘find’.
This supersedes the fancyheadings package.
Victor Eijkhout
28
CHAPTER 1. TEX AND LATEX
for specifying text to get left, center, and right in the header. Likewise \lfoot and such
for the footer.
This is assuming that all pages are the same. If you format for two-sided printing (section 1.9.1.2), you can specify different text for odd and even pages:
\fancyhead[LE,RO]{<text>}
for text that is Left on Even and Right on Odd pages. Typically, you specify text for
[LE,RO] and [RE,LO], for instance
\fancyhead[EL,OR]{\textsl{\rightmark}}
(see section 1.9.1.3).
1.10.3
Pdf file generation
Making beautiful pdf documents, complete with hyperlinks and table of contents, from
your LATEX files is simplicity itself. Insert
\usepackage[pdftex]{hyperref}
in the preamble, and format with pdflatex. That’s it. Do see section 1.10.4.2 about
including pictures.
1.10.4
Graphics
Because of TEX’s ancient origins – and its desire to be machine-independent – graphics is not standard, and frankly a bit of a hassle. The basic TEX system knows nothing
about graphics, it just keeps some space open. An extension mechanism (‘specials’) then
puts information in the output file that the printer driver can use to place graphics. With
pdflatex this situation has become a bit less messy: now any graphics go straight into
the pdf output file.
1.10.4.1 The picture environment
There is a way to generate graphics from inside LATEX, using some graphics fonts rather
than a full drawing mode. While primitive and limited, the picture environment has two
advantages:
•
•
It is easier to get the fonts for labels to be the same as the text font.
Since it involves explicit drawing instructions, you can automatically draw bar
charts and such.
1.10.4.2 Including external graphics
Most of the time, you will have graphics to include that come from some drawing package.
Using the graphicx package, you write
\includegraphics[key=value,...]{<file name>}
TEX – LATEX – CS 594
1.10. EXTENSIONS TO LATEX
29
where the file name can refer to any format, but if you use pdflatex, Postscript can not be
used; if your picture is in Postscript, you can convert it with ps2pdf.
Commands such as \includegraphics, as well as similar commands in other packages, leave space in your document for the graphic. Now you have to be careful: you can
not leave space for a 3 inch picture, an inch from the bottom of the page. Here are two
approaches for placing a picture:
•
Just place it where you want it, and if it falls on a page break, deal with it later by
moving it.
•
Put the figure in a floating figure object (section 1.5) and let LATEX sort out the
placement.
You can also have text wrap around a figure, by using the wrapfig package.
There is a package color for colour output.
1.10.5
Other languages than English
The fact that TEX and LATEX were written by Americans becomes obvious in a couple of
places.
•
Various typographical conventions are geared towards American English.
•
Words like ‘Chapter’ are the default in the style files9 .
To address this and make LATEX easier to use with other languages, there is a package
babel.
9.
They used to be hard-wired, so the situation is improved.
Victor Eijkhout
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CHAPTER 1. TEX AND LATEX
TEX programming.
No separate handout for this chapter; see the book ‘TEX by
Topic’.
Exercise 9. Write a macro \intt (‘in typewriter type’) such that
\intt{foo} and \intt{foo_bar} are output as foo and foo_bar,
in typewriter type.
Exercise 10. Write a macro that constructs another macro: \tees\three3
should be equivalent to \def\three{TTT}, \tees\five5 equivalent to \def\five{TTTTT} et cetera. In other words, the first argument of \tees is the name of the macro you are defining, the second
is the number of letters ‘T’ the defined macro expands to. To make sure
that your solution really expands to that string of ‘T’s, and not some
code that generates it when the macro is called, do \show\five and
check the screen output.
Exercise 11. TEX natively has addition, multiplication, and division
arithmetic. Write a square root routine in TEX. Hint: Use Newton’s method.
Exercise 12. Make this work:
\def\LeftDelim{(}\def\RightDelim{)}
\DefineWithDelims{foo}{My argument is ‘#1’.}
\def\LeftDelim{<}\def\RightDelim{>}
\DefineWithDelims{bar}{But my argument is ‘#1’.}
\foo(one)\par
\bar<two>
Output:
My argument is ‘one’.
But my argument is ‘two’.
In other words, \DefineWithDelims defines a macro – in this case
\foo – and this macro has one argument, delimited by custom delimiters. The delimiters can be specified for each macro separately.
Hint: \DefineWithDelims is actually a macro with only one argument. Consider this code snippet:
\Define{foo}{ ... #1 ...}
\def\Define#1{
\expandafter\def\csname #1\endcsname##1}
TEX – LATEX – CS 594
1.10. EXTENSIONS TO LATEX
31
TEX visuals.
Exercise 13. Use the \everypar command so that the first paragraph after a heading (define your own heading command) will have a
bullet ($\bullet$) in the left margin.
Exercise 14. Set TEX up so that every paragraph starts in mediaeval
‘initial’ style: the first letter of the paragraph is set in a large type size,
and takes the first two or three lines. Use the following auxiliary macro:
\def\Hang#1{\hbox to 0pt
{\raise 1.2ex \vbox to 0pt
{\hbox{#1}\vss}\hss}}
% small test:
A \Hang{$\bullet$} B \Hang{\Huge B} C. \bigskip
Output:
A •B C.
B
Also, set \parindent=0pt. The result should look like this. Input:
This is an old-fashioned mediaeval paragraph that has lots
of text and...
Also, the second paragraph is an old-fashioned mediaeval
paragraph that...
with output:
his is an old-fashioned mediaeval paragraph that
has lots of text and a very long first sentence. The
second sentence is also long, and only serves the purpose to make this more than 2 or so lines long. For
good measure we throw in a third line which should
make this four lines long, if not five with a little luck.
T
the second paragraph is an old-fashioned meA lso,
diaeval paragraph that has lots of text and a very
long first sentence. The second sentence is also long,
and only serves the purpose to make this more than 2
or so lines long. For good measure we throw in a third
line which should make this four lines long, if not five
with a little luck.
Victor Eijkhout
32
CHAPTER 1. TEX AND LATEX
Projects for this chapter.
Project 1.1. TEX has a syntax that can to a large extent be altered, dynamically or statically. This has had the effect that macro packages typically use a syntax that is
somewhere between ad hoc and plain unsystematic. Explore how it would be possible to set up a macro package with object-oriented syntax and design. It would
probably be a good idea to read [3, 4, 5], and to look at macro package such as
Lollipop and ConTeXt.
Project 1.2. The web site http://wwww.cookingforengineers.com uses a tablelike layout for notating recipes. Design an easy syntax for inputting these diagrams, and write a LATEX package that implements them.
TEX – LATEX – CS 594
Chapter 2
Parsing
The programming language part of TEX is rather unusual. In this chapter we will learn the
basics of language theory and parsing, and apply this to parsing TEX and LATEX. Although
TEXcan not be treated like other programming languages, it is interesting to see how far we
can get with existing tools.
Handouts and further reading for this chapter
The theory of languages and automata is discussed in any number of books, such as the
Hopcroft and Ulman one. For a discussion that is more specific to compilers, see the compilers book by Aho and Ulman or Aho, Seti, and Ulman.
The tutorials on lex and yacc should suffice you for most applications. The O’Reilly book
by Levine, Mason, and Brown is probably the best reference on lex and yacc. A copy of it
is on reserve in the library, QA76.76.U84M37 .
The definitive reference on hashing is Knuth’s volume 3 of The Art of Computer Programming [14], section 6.4. This is on reserve, QA76.5.K57 .
33
34
CHAPTER 2. PARSING
Parsing theory.
2.1
Levels of parsing
A compiler, or other translation software, has two main tasks: checking the input for validity, and if it is valid, understanding its meaning and transforming it into an executable
that realizes this meaning. We will not go into the generation of the executable code here,
but focus on the validity check and the analysis of the meaning, both of which are parsing
tasks.
A parser needs to look at the input on all sorts of levels:
•
•
•
•
Are all characters valid – no 8-bit ascii?
Are names, or identifiers, well-formed? In most programming languages a1 is a
valid name, but 1a is not. By contrast, in TEX a name can only have letters, while
in certain Lisp dialects !!important_name!! is allowed.
Are expressions well-formed? An arithmetic expression like 5/*6- does not
make sense, nor does CALL )FOO( in Fortran.
If the input is well-formed, are constraints satisfied such as that every name that
is used is defined first?
These different levels are best handled by several different software components. In this
chapter we will look at the two initial stages of most translators1 .
1.
2.
2.2
First of all there is the lexical analysis. Here a file of characters is turned into a
stream of tokens. The software that performs this task is called a tokenizer, and
it can be formalized. The theoretical construct on which the tokenizer is based is
called a ‘Finite State Automaton’.
Next, we need to check if the tokens produced by the tokenizer come in a legal
sequence. For instance, opening and closing parentheses need to come in matched
pairs. This stage is called the syntactical analysis, and the software doing this is
called a parser.
Very short introduction
A language is a set of words (strings) over an alphabet, that satisfies certain properties. It
is also possible to define a language as the output of a certain type of grammar, or as the
strings accepted by a certain type of automaton. We then need to prove the equivalences of
the various formulations. In this section we briefly introduce the relevant concepts.
1.
I will use the terms ‘translating’ and ‘translater’ as informal concepts that cover both compilers and interpreters and all sorts of mixed forms. This is not the place to get philosophical about the differences.
TEX – LATEX – CS 594
2.2. VERY SHORT INTRODUCTION
2.2.1
35
Languages
A language is a set of words that are constructed from an alphabet. The alphabet is finite
in size, and words are finite in length, but languages can have an infinite number of words.
The alphabet is often not specified explicitly.
Languages are often described with set notation and regular expressions, for example ‘L =
{an b∗ cn |n > 0}’, which says that the language is all strings of equal number of as and cs
with an arbitrary number of bs in between.
Regular expressions are built up from the following ingredients:
α|β either the expression α or β
αβ the expression α followed by the expression β
α∗ zero or more occurrences of α
α+ one or more occurrences of α
α? zero or one occurrences of α
We will see more complicated expressions in the lex utility.
2.2.2
Automata
A description of a language is not very constructive. To know how to generate a language
we need a grammar. A grammar is a set of rules or productions α → β that state that,
in deriving a word in the language, the intermediate string α can be replaced by β. These
strings can be a combination of
•
•
•
•
A start symbol S,
‘Terminal’ symbols, which are letters from the alphabet; these are traditionally
rendered with lowercase letters.
‘Non-terminal’ symbols, which are not in the alphabet, and which have to be
replaced at some point in the derivation; these are traditionally rendered with
uppercase letters.
The empty symbol .
Languages can be categorized according to the types of rules in their grammar:
type 0 These are called ‘recursive languages’, and their grammar rules can be of any form:
both the left and right side can have any combination of terminals, non-terminals,
and .
type 1 ‘Context-sensitive languages’ are limited in that can not appear in the left side of
a production. A typical type 1 rule would look like
αAβ → γ
which states that A, in the context of αAβ, is replaced by γ. Hence the name of
this class of languages.
type 2 ‘Context-free languages’ are limited in that the left side of a production can only
consist of single non-terminal, as in A → γ. This means that replacement of the
non-terminal is done regardless of context; hence the name.
type 3 ‘Regular languages’ can additionally have only a single non-terminal in each righthand side.
Victor Eijkhout
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CHAPTER 2. PARSING
In the context of grammars, we use the notation α ⇒ β to indicate that the string β as
derived from α by a single application of a grammar rule; α ⇒∗ β indicates multiple
rules. For example, αAβ ⇒ αBγ indicates that the rhs string was derived from the lhs by
replacing Aβ with Bγ.
2.2.3
Automata
Corresponding to these four types of formal languages, there are four types of ‘automata’:
formal machines that can recognize these languages. All these machines have a starting
state, they go from one state to another depending on the input symbols they encounter, and
if they reach the end state, the string is accepted as being in the language. The difference
between the different types of automata lies in the amount of memory they have to store
information. Very briefly the classes of automaton are:
for type 3 Finite State Automata. These machines have no memory. They can only make
transitions.
for type 2 Pushdown Automata. These machines have a stack where they can store information; only the top of the stack can be inspected.
for type 1 Linear Bounded Automata. These have random-access memory, the size of
which is equal to (a linear function of) the size of the input.
for type 0 Turing machines. These have an unbounded tape for storing intermediate calculations.
TEX – LATEX – CS 594
2.3. FINITE STATE AUTOMATA AND REGULAR LANGUAGES
37
Lexical analysis.
The lexical analysis phase of program translation takes in a stream of characters and outputs
a stream of tokens.
A token is a way of recognizing that certain characters belong together, and form an object
that we can classify somehow. In some cases all that is necessary is knowing the class, for
instance if the class has only one member. However, in general a token is a pair consisting
of its type and its value. For instance, in 1/234 the lexical analysis recognizes that 234 is
a number, with the value 234. In an assignment abc = 456, the characters abc are recognized as a variable. In this case the value is not the numeric value, but rather something
like the index of where this variable is stored in an internal table.
Lexical analysis is relatively simple; it is performed by software that uses the theory of
Finite State Automata and Regular Languages; see section 2.3.
Remark. It might be tempting to consider the input stream to consist of lines, each of
which consist of characters, but this does not always make sense. Programming languages
such as Fortran do look at the source, one line at a time; C does not. TEX is even more
complicated: the interpretation of the line end is programmable.2
2.3
Finite state automata and regular languages
Regular languages are the strings accepted by a particularly simple kind of automaton.
However, we initially define these languages – non-constructively – by so-called ‘regular
expressions’.
2.3.1
Definition of regular languages
A regular language over some alphabet can be described by a ‘regular expression’.
•
•
•
•
denotes the empty language: the language with no words in it.
If a is a letter in the alphabet, then a denotes the language {a}.
If α and β are expressions denoting regular languages A and B, then
–
αβ or α · β denotes the language {xy|x ∈ A, y ∈ B}.
–
α|β denotes the language A ∪ B.
–
α∗ denotes the language ∪n≥0 An .
Parentheses can be used to indicate grouping: (α) simply denotes the language A.
Any regular expression built up this way describes a regular language.
2.
Ok, if we want to be precise, TEX does look at the input source on a line-by-line basis. There is something
of a preprocessor before the lexical analysis which throws away the machine-dependent line end, and replaces it
with the TEX-defined one.
Victor Eijkhout
38
2.3.2
CHAPTER 2. PARSING
Non-deterministic automata
A Finite State Automaton is an abstract machine that recognizes (‘accepts’) words from a
language:
•
•
•
The automaton is initially in a beginning state;
every letter or ‘symbol’ from the input word causes unambiguously a transition
to the same or to a next state; if no transition is defined for a given combination
of current state and input symbol, then the word is not in the language;
a word is accepted if the last symbol causes a transition to a state that is marked
as an accepting state.
Formally, we can define a FSA as the combination of
•
•
•
A set S of states, with a starting state S0 and a set of final states.
A finite input alphabet I.
A transition diagram I × S → S that specifies how the combination of a state and
an input symbol effects a transition to a new state.
This kind of automaton is deterministic in the sense that every transition from one state
to the next is deterministically made by accepting an input symbol. However, in the context of lexical analysis, the so-called ‘non-deterministic FSA’ is more convenient. A nondeterministic FSA (also NFA) differs in two ways from the deterministic type:
•
•
An NFA can make spontaneous transitions from one state to another. If an automaton has such a transition, we can say that this is caused by the symbol , and
this is called an -transition.
An NFA can be ambiguous in that there can be more than one possible transition
for a given state and input symbol.
Exercise 15. Show that the second condition in the definition of an
NFA can be reduced to the first. Is a reduction the other way possible?
2.3.3
The NFA of a given language
We now construct a nondeterministic automaton that accepts a regular language.
•
The automaton that accepts the expression has a single transition from the starting state to the accepting state.
0
•
The automaton that accepts the expression a has a single transition from the starting state to the accepting state.
a
0
•
1
1
If A and B are automata accepting the languages A and B with expressions α
and β, then
–
the language AB is accepted by the automaton that has the states and transition of both automata combined, with the initial state of A as the new initial
state, the accepting state of B as the new accepting state, and an -transition
TEX – LATEX – CS 594
2.3. FINITE STATE AUTOMATA AND REGULAR LANGUAGES
39
from the accepting state of A to the initial state of B;
A
1
0
1
B
the language A ∪ B is accepted by an automaton with a new starting state
that has -transitions to the initial states of A and B;
A
1
0
0
–
s
s
1
B
the expression α∗ is accepted by A modified such that the initial state is
also the accepting state, or equivalently by adding an -transition from the
starting to the accepting state, and one the other way around.
0
–
2.3.4
Examples and characterization
Any language that can be described by the above constructs of repetition, grouping, concatenation, and choice, is a regular language. It is only slightly harder to take a transition
diagram and write up the regular expression for the language that it accepts.
An informal way of characterizing regular languages is to say that FSAs ‘do not have
memory’. That means that any language where parts of words are related, such as {an bm |
m ≥ n}, can not be recognized by a FSA. Proof: suppose there is a recognizing FSA.
When it first accepts a b, it can come from only a fixed number of states, so that limits the
information it can carry with it.
We can give a slightly more rigorous proof if we first characterize regular languages:
Theorem 1 Let L be a regular language, then there is an n so that all strings α in L longer
than n can be written as α = uvw, such that for any k uv k w is also in the language.
Using this theorem it is easy to see that the above language can not be regular.
This theorem is proved by observing that in order to accept a sufficiently long string the
same state must have been encountered twice. The symbols accepted in between these
encounters can then be repeated arbitrarily many times.
2.3.5
Deterministic automata
Non-deterministic automata, as defined above, are easy to define. However, from a practical
point of view they do not look very constructive: a string in the language is accepted by
the automaton if there is any sequence of transitions that accepts it. Fortunately, for every
NFSA, there is a DFSA that accepts the same language.
Victor Eijkhout
40
CHAPTER 2. PARSING
Sometimes it is easy to derive the DFSA. Consider the language a∗ |b∗ and the automaton
a
1
0
b
2
The following automaton is derived by splitting off one a and one b:
a
1
a
0
b
b
2
This next example leads up to what happens in the lexical analysis of a compiler:
B
E–G–I
...
5
1
2
N
0
7
A–Z
a–z
6
The resulting DFA is a bit more messy:
E
G
B
1
2
¬E
0
3
...
text
¬G
¬B
6
A–Z
8
7
A–Z
...
A–Z
(and we can collapse states 6 . . . to one.)
Sketch of the proof: the states of the DFSA are sets of states of the NFSA. The states
we are actually interested in are defined inductively, and they satisfy the property that
TEX – LATEX – CS 594
2.4. LEXICAL ANALYSIS WITH FSAS
41
they are closed under -transitions of the original NFSA. The starting state contains the
original starting state plus everything reachable with -transitions from it. Given a state
of the DFSA, we then define more states by considering all transitions from the states
contained in this state: if there is a transition based on a symbol x, the next state has all
states reachable from this state by accepting x, plus any subsequent -transitions.
Since the number of subsets of a finite set of states is finite, this will define a finite number
of states for the DFSA, and it is not hard to see that an accepting sequence in the one
automaton corresponds to an accepting sequence in the other.
2.3.6
Equivalences
Above, we saw how the NFA of a regular language is constructed. Does every NFA correspond to a regular language, and if so, how can that be derived? We make a detour by first
talking about the equivalence of automata and grammars.
Let X be a string in the language L of a DFA, and suppose that after t transitions state i
is reached. That means we can split X = Xi (t)Yi . This is merely one of the strings that
is in state i at time t; let us call the set of all these strings Li (t). Let us call the set of all
strings that, given a state i, bring the automaton to an accepting state Ri . This set is clearly
m
not dependent on t. Defining Li = ∪∞
t=0 Li (t), we have that L = ∪i=1 Li Ri where m is the
number of states.
This inspires us to tackle the derivation of a grammar by describing the production of
the remainder strings Ri . Suppose the automaton is in state i; we will derive the productions Ni → . . .. If state i is an accepting state, there will be a production Ni → ; for all
other transitions by a symbol x to a state Ni0 we add a production Ni → xNi0 . It is easy
to see the equivalence of strings accepted by the DFA and derivations of the grammar thus
constructed.
Going the other way, constructing an automaton from a grammar runs into a snag. If there
are productions Ni → aNi0 and Ni → aNi00 , we can of necessity only construct an NFA.
However, we know that these are equivalent to DFAs.
We note that the grammars used and constructed in this – informal – proof are rightrecursive, so they generate precisely the regular languages.
Exercise 16. Show how this proof can be modified to use left-recursive
grammars, that is, grammars that have productions of the form Ni →
Ni0 a.
2.4
Lexical analysis with FSAs
A FSA will recognize a sequence of language elements. However, it’s not enough to simply
say ‘yes, this was a legal sequence of elements’: we need to pass information on to the next
stage of the translation. This can be done by having some executable code attached to
the accepting state; if that state is reached, the code snippet tells the next stage what kind
Victor Eijkhout
42
CHAPTER 2. PARSING
of element has been recognized, and its value. This value can be a numerical value for
numbers recognized, but more generally it will be an index into some table or other.
Formally, we can extend the definition of a FSA (section 2.3.2) by the addition of an output
alphabet O and an output table I × S → O. This models the output of a symbol, possibly ,
at each transition.
Exercise 17. One could also define the output with a mapping S → O.
Show that the definitions are equivalent.
An FSA is not enough to recognize a whole language, but it can recognize elements from a
language. For instance, we can build multiple FSAs for each of the keywords of a language
(‘begin’ or ‘void’), or for things like numbers and identifiers. We can then make one big
FSA for all the language elements by combining the multiple small FSAs into one that has
•
a starting state with -transitions to the start states of the element automata, and
•
from each of the accepting states an -transition back to the start state.
B
EGIN
s
s. . . . . . . . . . . . s
s E s N s D s
0–9
s
s
s
a-z, 0-9
s
a-z
s
Exercise 18. Write a DFA that can parse Fortran arithmetic expressions. In Fortran, exponentiation is written like 2**n. It is also not allowed to have two operators in a row, so 2 × −3 is notated 2*(-3).
There is a problem with the -transition from the final state to the initial state in the above
NFA. This transition should only be taken if no other transitions can be taken, in other
words, if the maximal string is recognized. For instance, most programming languages allow quote characters inside a quoted string by doubling them: ‘"And then he said ""Boo!"""’.
The final state is reached three times in the course of this string; only the last time should
the jump back be taken.
However, sometimes finding the maximum matched string is not the right strategy. For
instance, in most languages, 4.E3 is a floating point number, but matching the E after the
decimal point is not necessarily right. In Fortran, the statement IF (4.EQ.VAR) ...
would then be misinterpreted. What is needed here is one token ‘look-ahead’: the parser
needs to see what character follows the E.
At this point it would be a good idea to learn the Unix tool lex.
TEX – LATEX – CS 594
2.5. CONTEXT-FREE LANGUAGES
43
Syntax parsing.
Programming languages have for decades been described using formal grammars. One
popular way of notating those grammars is Backus Naur Form, but most formalisms are
pretty much interchangable. The essential point is that the grammars are almost invariably
of the context-free type. That is, they have rules like
hfunction calli −→ hfunction namei ( hoptargsi )
hoptargsi −→ empty | hargsi
hargsi −→ word | word , hargsi
The second and third rule in this example can be generated by a regular grammar, but the
first rule is different: when the opening parenthesis is matched, the parser has to wait an
unlimited time for the closing parenthesis. This rule is of context-free typeCHECKTHIS.
It is important to keep some distinctions straight:
•
A grammar has a set of rules, each indicating possible replacements during a
derivation of a string in the language. Each rule looks like A → α.
•
A derivation is a specific sequence of applications of rules; we denote each step
in a derivation as α ⇒ β, where β can be derived from α by application of some
rule. The derivation of some string α is a sequence of step such that S ⇒ · · · ⇒ α;
we abbreviate this as S ⇒∗ α.
•
Ultimately, we are interested in the reverse of a derivation: we have a string that
we suspect is in the language, and we want to reconstruct whether and how it
could be derived. This reconstruction process is called ‘parsing’, and the result
often takes the form of a ‘parse tree’.
We will first give some properties of context-free languages, then in section 2.6 we will
discuss the practical parsing of context-free languages.
2.5
Context-free languages
Context-free languages can be defined as the output of a particular kind of grammar (the left
side can only consist of a single nonterminal), or as the set of string accepted by a certain
kind of automaton. For languages of this type, we use a Pushdown Automaton (PDA) to
recognize them. A PDA is a finite-state automaton, with some scratch memory that takes
the form of a stack: one can only push items on it, and inspect or remove the top item. Here
we will not give an equivalence proof.
An example of a language that is context-free but not regular is {an bn }. To parse this, the
automaton pushes as on the stack, then pops them when it finds a b in the input, and the
string is accepted if the stack is empty when the input string is fully read.
2.5.1
Pumping lemma
As with regular languages (section 2.3.4), there is a way to characterize the strings of a
context-free language.
Victor Eijkhout
44
CHAPTER 2. PARSING
Theorem 2 Let L be a context-free language, then there is an n so that all strings α in L
longer than n can be written as α = uvwxy, such that for any k the string uv k wxk y is also
in the language.
The proof is as before: the derivation of a sufficiently long string must have used the same
production twice.
S
HH
H
u
A
HH
v A x
H
y
w
2.5.2
Deterministic and non-deterministic PDAs
As with Finite State Automata, there are deterministic and non-deterministic pushdown
automata. However, in this case they are not equivalent. As before, any DPA is also a NPA,
so any language accepted by a DPA is also accepted by a NPA. The question is whether
there are languages that are accepted by a NPA, and that are not accepted by a DPA.
A similar example to the language {an bn } above is the language over an alphabet of at
least two symbols L = {ααR }, where αR stands for the reverse of α. To recognize this
language, the automaton pushes the string α on the stack, and pops it to match the reverse
part. However, the problem is knowing when to start popping the stack.
Let the alphabet have at least three letters, then the language Lc = {αcαR |c 6∈ α} can
deterministically be recognized. However, in absence of the middle symbol, the automaton
needs an -transition to know when to start popping the stack.
2.5.3
Normal form
Context-free grammars have rules of the form A → α with A a single nonterminal and α
any combination of terminals and nonterminals. However, for purposes of parsing it is
convenient to have the rules in a ‘normal form’. For context-free grammars that is the form
A → aα where a is a terminal symbol.
One proof that grammars can always be rewritten this way uses ‘expression equations’. If x
and y stand for sets of expressions, then x + y, xy, and x∗ stand for union, concatenation,
and repetition respectively.
Consider an example of expression equations. The scalar equation x = a + xb states that
x contains the expressions in a. But then it also contains ab, abb, et cetera. One can verify
that x = ab∗ .
The equation in this example had a regular language as solution; the expression x = a +
bxc does not have a regular solution.
TEX – LATEX – CS 594
2.6. PARSING CONTEXT-FREE LANGUAGES
45
Now let x be a vector of all non-terminals in the grammar of a context-free language, and
let f be the vector of righthandsides of rules in the grammar that are of normal form. We
can then write the grammar as
xt = xt A + f t
where the multiplication with A describes all rules not of normal form.
Example:
S
X
Y
→ aSb|XY |c
→ Y Xc|b
→ XS
⇒
φ
[S, X, Y ] = [S, X, Y ] Y
φ
φ φ
φ S + [aSb + c, b, φ]
Xc φ
The solution to this equation is
xt = f t A∗
which describes rules on normal form. However, we need to find a more explicit expression
for A∗ .
Noting that A∗ = λ + AA∗ we get
xt = f t + f t AA∗ = f t + f t B
(2.1)
∗
where B = AA . This is a grammar on normal form. It remains to work out the rules
for B. We have
B = AA∗ = A + AAA∗ = A + AB
These rules need not be of normal form. However, any elements of A that start with a
nonterminal, can only start with nonterminals in x. Hence we can substitute a rule from
equation (2.1).
2.6
Parsing context-free languages
The problem of parsing is this:
Given a grammar G and a string α, determine whether the string is in
the language of G, and through what sequence of rule applications it
was derived.
We will discuss the LL and LR type parser, which correspond to a top-down and bottom-up
way of parsing respectively, then go into the problem of ambiguity
2.6.1
Top-down parsing: LL
One easy parsing strategy starts from the fact that the expression has to come from the start
symbol. Consider the expression 2*5+3, which is produced by the grammar
Expr −→ number Tail
Tail −→ | + number Tail | * number Tail
In the following analysis the stack has its bottom at the right
Victor Eijkhout
46
CHAPTER 2. PARSING
initial queue:
2∗5+3
start symbol on stack:
Expr
replace
number Tail
match
∗5+3
Tail
replace
* number Tail
match
5+3
number Tail
match
+3
Tail
replace
+ number Tail
match
3
number Tail
match
Tail
match
The derivation that we constructed here is
E ⇒ nT ⇒ n ∗ nT ⇒ n ∗ n + nT ⇒ n ∗ n + n
that is, we are replacing symbols from the left. Therefore this kind of parsing is called
LL parsing: read from left to right, replace from left to right. Because we only need to look
at the first symbol in the queue to do the replacement, without need for further ‘look ahead’
tokens, this is LL(1) parsing.
But this grammar was a bit strange. Normally we would write
Expr −→ number | number + Expr | number * Expr
If our parser can now see the first two symbols in the queue, it can form
initial queue:
2∗5+3
start symbol on stack:
Expr
replace
number * Expr
match
5+3
Tail
replace
number + Expr
match
3
Expr
replace
3
number
match
This is called LL(2) parsing: we need one token look ahead.
2.6.1.1
Problems with LL parsing
If our grammar had been written
Expr −→ number | Expr + number | Expr * number
an LL(k) parser, no matter the value of k, would have gone into an infinite loop.
In another way too, there are many constructs that can not be parsed with an LL(k) parser
for any k. For instance if both A<B and A<B> are legal expressions, where B can be of
arbitrary length, then no finite amount of look-ahead will allow this to be parsed.
2.6.1.2 LL and recursive descent
The advantages of LL(k) parsers are their simplicity. To see which rule applies at a given
point is a recursive-descent search, which is easily implemented. The code for finding
which rule to apply can broadly be sketched as follows:
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2.6. PARSING CONTEXT-FREE LANGUAGES
47
define FindIn(Sym,NonTerm)
for all expansions of NonTerm:
if leftmost symbol == Sym
then found
else if leftmost symbol is nonterminal
then FindIn(Sym,that leftmost symbol)
This implies that a grammar is LL-parsable if distinct rules for some non-terminal can not
lead to different terminals. In other words, by looking at a terminal, it should be clear what
production was used.
The LR parsers we will study next are more powerful, but much more complicated to
program. The above problems with LL(k) are largely non-existent in languages where
statements start with unique keywords.
2.6.2
Bottom-up parsing: shift-reduce
In this section we will look at the ‘bottom-up’ parsing strategy, where terminal symbols are
gradually replaced by non-terminals.
One easily implemented bottom-up parsing strategy is called ‘shift-reduce parsing’. The
basic idea here is to move symbols from the input queue to a stack, and every time the
symbols on top of the stack form a right hand size of a production, reduce them to the left
hand side.
For example, consider the grammar
E −→ number | E + E | E * E
and the expression 2 ∗ 5 + 3. We proceed by moving symbols from the left side of the queue
to the top of the stack, which is now to the right.
stack
initial state:
shift
reduce
shift
shift
reduce
reduce
shift, shift, reduce
reduce
2
E
E*
E*5
E*E
E
E+E
E
queue
2∗5+3
*5+3
*5+3
5+3
+3
+3
+3
(Can you tell that we have ignored something important here?)
The derivation we have reconstructed here is
E ⇒E+E ⇒E+3⇒E∗E+3⇒E∗5+3⇒2∗5+3
which proceeds by each time replacing the right-most nonterminal. This is therefore called
a ‘rightmost derivation’. Analogously we can define a ‘leftmost derivation’ as one that
proceeds by replacing the leftmost nonterminal.
Victor Eijkhout
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CHAPTER 2. PARSING
For a formal definition of shift-reduce parsing, we should also define an ‘accept’ and ‘error’
action.
2.6.3
Handles
Finding the derivation of a legal string is not trivial. Sometimes we have a choice between
shifting and reducing, and reducing ‘as soon as possible’ may not be the right solution.
Consider the grammar
S −→ aAcBe
A −→ bA | b
B −→ d
and the string abbcde. This string can be derived (writing the derivation backwards for a
change) as
abbcde ⇐ abAcde ⇐ aAcde ⇐ aAcBe ⇐ S.
However, if we had started
abbcde ⇐ aAbcde ⇐ aAAcde ⇐?
we would be stuck because no further reductions would be applicable.
The problem then is to know where to start replacing terminal symbols and, later in the
derivation, non-terminals. The shift-reduce strategy of the previous section is here seen to
lead to problems, so some extra power is needed. We introduce the concept of ‘handle’
as a formal definition of ‘the right production and place to start reducing’. The following
definition is totally unhelpful:
If S ⇒∗ αAw ⇒ αβw is a right-most derivation, then A → β at the
position after α is a handle of αAw.
Clearly, if we can identify handles, we can derive a parse tree for a given string. However,
the story so far has not been constructive. Next we will look at ways of actually finding
handles.
2.6.4
Operator-precedence grammars
It is easy to find handles if a grammar is of an ‘operator grammar’ form. Loosely, by this
we mean that expressions in the language look like expression-operator-expression. More
strictly, we look at grammars where there are never two adjacent nonterminals, and where
no right hand side is . We also assume that precedence relations between operators and
terminals are known.
Let us look again at arithmetic expressions; we will introduce relations op1 l op2 if the
first operator has lower precedence, and op1 m op2 if it has higher precedence. If the two
operators are the same, we use predence to force associativity rules. For instance, right
associativity corresponds to definitions such as + m +.
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2.6. PARSING CONTEXT-FREE LANGUAGES
49
For the + and ∗ operators we then have the following table:
number + ×
number
m m
l
m l
+
×
l
m m
Now we can find a handle by scanning left-to-right for the first m character, then scanning
back for the matching l. After reducing handles thus found, we have a string of operators
and nonterminals. Ignoring the nonterminals, we insert again the comparisons; this allows
us to find handles again.
For example, 5 + 2 ∗ 3 becomes l5 m + l 2 m ∗ l 3m; replacing handles this becomes
E + E ∗ E. Without the nonterminals, the precedence structure is l + l ∗ m, in which we
find lE ∗ Em as the handle. Reducing this leaves us with E + E, and we find that we have
parsed the string correctly.
This description sounds as if the whole expression is repeatedly scanned to insert precedence relations and find/reduce handle. This is not true, since we only need to scan as far
as the right edge of the first handle. Thus, a shift/reduce strategy will still work for operator
grammars.
2.6.5
LR parsers
We will now consider LR parsers in more detail. These are the parsers that scan the input
from the left, and construct a rightmost derivation, as in the examples we have seen in
section 2.6.2. Most constructs in programming languages can be parsed in an LR fashion.
An LR parser has the following components
•
•
•
A stack and an input queue as in the shift-reduce examples you have already seen
in section 2.6.2. The difference is that we now also push state symbols on the
stack.
Actions ‘shift’, ‘reduce’, ‘accept’, ‘error’, again as before.
An Action and Goto function that work as follows:
–
Suppose the current input symbol is a and the state on top of the stack is s.
–
If Action(a, s) is ‘shift’, then a and a new state s0 = Goto(a, s) are
pushed on the stack.
–
If Action(a, s) is ‘reduce A → β’ where |β| = r, then 2r symbols are
popped from the stack, a new state s0 = Goto(a, s00 ) is computed based on
the newly exposed state on the top of the stack, and A and s0 are pushed.
The input symbol a stays in the queue.
An LR parser that looks at the first k tokens in the queue is called an LR(k) parser. We will
not discuss this issue of look-ahead any further.
It is clear that LR parser are more powerful than a simple shift-reduce parser. The latter
has to reduce when the top of the stack is the right hand side of a production; an LR parser
additionally has states that indicate whether and when the top of the stack is a handle.
Victor Eijkhout
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CHAPTER 2. PARSING
2.6.5.1
A simple example of LR parsing
It is instructive to see how LR parsers can deal with cases for which simple shift/reduce
parsing is insufficient. Consider again the grammar
E −→ E + E | E * E
and the input string 1 + 2 ∗ 3 + 4. Give the + operator precedence 1, and the * operator
precedence 2. In addition to moving tokens onto the stack, we also push the highest precedence seen so far. In the beginning we declare precedence 0, and pushing a non-operator
does not change the precedence.
Shift/reduce conflicts are now resolved with this rule: if we encounter at the front of the
queue a lower precedence than the value on top of the stack, we reduce the elements on top
of the stack.
1 S0
1 S0
1 S0
1 S0
1 S0
1 S0
1 S0
1 S0
1 S0
11
+ S1
+ S1
+ S1
+ S1
+ S1
+ S1
+ S1
+ S1
2 S1
2 S1 * S2
2 S1 * S2 3 S2
6 S1
6 S1 + S1
6 S1 + S1 4 S1
10 S1
1 + 2 ∗ 3 + 4 push symbol; highest precedence is 0
+2 ∗ 3 + 4 highest precedence now becomes 1
2∗3+4
∗3 + 4
highest precedence becoming 2
3+4
+4
reduce because P(+) < 2
+4
the highest exposed precedence is 1
4
at the end of the queue we reduce
Even though this example is phrased very informally, we see the key points:
•
•
only the top of the stack and the front of the queue are inspected;
we have a finite set of rules governing shift/reduce behaviour.
As we shall see, this mechanism can also identify handles.
2.6.5.2
States of an LR parser
An LR parser is constructed automatically from the grammar. Its states are somewhat complicated, and to explain them we need a couple of auxiliary constructs.
item An ‘item’ is a grammar rule with a location indicated. From the rule A → B C we
get the items A → • B C, A → B • C, A → B C• . The interpretation of an item
will be that the symbols left of the dot are on the stack, while the right ones are
still in the queue. This way, an item describes a stage of the parsing process.
closure The closure of an item is defined as the smallest set that
•
Contains that item;
•
If the closure contains an item A → α • B β with B a nonterminal symbol,
then it contains all items B → • γ. This is a recursive notion: if γ starts with
a non-terminal, the closure would also contain the items from the rules of γ.
The states of our LR parser will now be closures of items of the grammar. We motivate
this by an example.
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2.6. PARSING CONTEXT-FREE LANGUAGES
51
Consider now an item A → β1• β2 in the case that we have recognized αβ1 so far. The item
is called valid for that string, if a rightmost derivation S ⇒∗ αAw ⇒ αβ1 β2 w exists. If
β2 = , then A → β1 is a handle and we can reduce. On the other hand, if β2 6= , we have
not encountered the full handle yet, so we shift β2 .
As an example, take the grammar
E −→ E+T | T
T −→ T*F | F
F −→ (E) | id
and consider the partially parsed string E+T*. The (rightmost) derivation
E ⇒E+T ⇒E+T ∗F
shows that T → T*• F is a valid item,
E ⇒ E + T ⇒ E + T ∗ F ⇒ E + T ∗ (E)
gives F → • (E) as a valid item, and
E ⇒ E + T ⇒ E + T ∗ F ⇒ E + T ∗ id
gives F → • id as a valid item.
2.6.5.3
States and transitions
We now construct the actual states of our parser.
•
•
•
•
We add a new start symbol S’, and a production S0 → S.
The starting state is the closure of S0 → • S.
The transition function d(s, X) of a state s and a symbol X is defined as the closure
of
{A → α X• β|A → α • X β is in s}
The ‘follow’ of a symbol A is the set of all terminal symbols that can follow its
possible expansions. This set is easy to derive from a grammar.
Here is an example
We construct the states and transition for the grammar
S −→ (S)S |
which consists of all strings of properly matched left and right parentheses.
Solution: we add the production S0 → • S. We now find the states
1. {S0 → • S, S → • (S)S, S → • }
2. {S0 → S• }
3. {S → (• S)S, S → • (S)S, S → • }
4. {S → (S• )S}
5. {S → (S)• S, S → • (S)S, S → • }
6. {S → (S)S• }
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CHAPTER 2. PARSING
with transitions
d(0, S) = 1
d(0,0 (0 ) = 2
d(2, S) = 3
d(2,0 (0 ) = 2
d(3,0 )0 ) = 4
d(4, S) = 5
d(4,0 (0 ) = 2
The only thing missing in our parser is the function that describes the stack handling. The
parsing stack consists of states and grammar symbols (alternating). Initially, push the start
state onto the stack. The current state is always the state on the top of the stack. Also, add
a special endmarker symbol to the end of the input string.
Loop:
(1) if the current state contains S0 → S•
accept the string
(2) else if the current state contains any other final item A → α•
pop all the tokens in α from the stack, along with the corresponding states;
let s be the state left on top of the stack: push A, push d(s,A)
(3) else if the current state contains any item A → α • x β,
where x is the next input token
let s be the state on top of the stack: push x, push d(s,x)
else report failure
Explanation:
1.
2.
3.
If we have recognized the initial production, the bottom-up parse process was
successful.
If we have a string of terminals on the stack, that is the right hand side of a
production, replace by the left hand side non-terminal.
If we have a string of terminals on the stack that is the start of a right hand side,
we push the current input symbol.
Exercise 19. Give the states and transitions for the grammar
S −→ x
S −→ (L)
L −→ S
L −→ L S
Apply the above parsing algorithm to the string (x,x,(x)).
The parsers derived by the above algorithm can only handle cases where there is no ambiguity in condition (3). The class of grammars recognized by this type of parser is called LR(0)
and it is not very interesting. We get the more interesting class of SLR(1) by adding to condition (2) the clause that the following symbol is in the follow of A. Another similar class,
which is the one recognized by yacc, is LALR(1).
TEX – LATEX – CS 594
2.6. PARSING CONTEXT-FREE LANGUAGES
2.6.6
53
Ambiguity and conflicts
The problem of finding out how a string was derived is often important. For instance, with
a grammar
hexpri −→ hnumberi | hexpri + hexpri | hexpri × hexpri
the expression 2 + 5 ∗ 3 is ambiguous: it can mean either (2 + 5) ∗ 3 or 2 + (5 ∗ 3).
+
H
H
H
2
*
+
3
H
H
H
H
3
5
2
5
An LR parser would report a ‘shift/reduce conflict’ here: after 2 + 5 has been reduced to
<expr> + <expr>, do we reduce that further to <expr>, or do we shift the minus,
since <expr> - is the start of a legitimate reducible sequence?
*
H
Another example of ambiguity is the ‘dangling else’ problem. Consider the grammar
hstatementi −→ if hclausei then hstatementi | if hclausei
then hstatementi else hstatementi
and the string
if c1 then if c2 then s1 else s2
This can be parsed two ways:
S
P
P
@
@ PPP
PP
@
Then
S
Else
If
H
H
If Then
S
H
HH
HH
H
If
Then
S
H
H
H
If
Then Else
Does the else clause belong to the first if or the second?
Let us investigate the first example. We can solve the ambiguity problem in two ways:
•
Reformulate the grammar as
hexpri −→ hmulexi | hmulexi + hmulexi
hmulexi −→ htermi | htermi × htermi
htermi −→ number
so that the parser can unambiguously reconstruct the derivation,
Victor Eijkhout
54
CHAPTER 2. PARSING
expr
HH
HH
H
mulex
+
mulex
H
HH
term
term * term
•
2
5
3
or
Teach the parser about precedence of operators. This second option may be easier
to implement if the number of operators is large: the first option would require a
large number of rules, with probably a slower parser.
Exercise 20. Rewrite the grammar of the second example to eliminate
the dangling else problem.
Since we are not used to thinking of keywords such as then in terms of precedence, it is
a better solution to eliminate the dangling else problem by introducing a fi keyword to
close the conditional. Often, however, ambiguity is not so easy to eliminate.
Exercise 21. In case of a shift-reduce conflict, yacc shifts. Write an example that proves this. Show what this strategy implies for the dangling
else problem.
Another type of conflict is the ‘reduce/reduce conflict’. Consider this grammar:
A −→ B c d | E c f
B −→ x y
E −→ x y
and the input string that starts x y c.
•
•
•
An LR(1) parser will shift x y, but can not decide whether to reduce that to B
or E on the basis of the look-ahead token c.
An LR(2) parser can see the subsequent d or f and make the right decision.
An LL parser would also be confused, but already at the x. Up to three tokens (x
y c) is unsufficient, but an LL(4) parser can again see the subsequent d or f.
The following grammar would confuse any LR(n) or LL(n) parser with a fixed amount of
look-ahead:
A
B
E
C
−→
−→
−→
−→
BCd | ECf
xy
xy
c | Cc
which generates x y cn {d|f }.
As usual, the simplest solution is to rewrite the grammar to remove the confusion e.g.:
A −→ BorE c d | BorE c f
BorE −→ x y
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2.6. PARSING CONTEXT-FREE LANGUAGES
55
or assuming we left-factorise the grammar for an LL(n) parser:
A −→ BorE c tail
tail −→ d | f
BorE −→ x y
Another example of a construct that is not LR parsable, consider languages such as Fortran,
where function calls and array indexing both look like A(B,C):
hexpressioni −→ hfunction calli | harray elementi
hfunction calli −→ name ( hparameter listi )
harray elementi −→ name ( hexpression listi )
hparameter listi −→ name | name , hparameter listi
hexpression listi −→ name | name, hexpression listi
After we push B on the stack, it is not clear whether to reduce it to the head of a parameter list or of an expression list, and no amount of lookahead will help. This problem can
be solved by letting the lexical analyzer have access to the symbol table, so that it can
distinguish between function names and array names.
Victor Eijkhout
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CHAPTER 2. PARSING
Lex.
2.7
Introduction
The unix utility lex parses a file of characters. It uses regular expression matching; typically
it is used to ‘tokenize’ the contents of the file. In that context, it is often used together
with the yacc utility. However, there are many other applications possible. By itself, lex is
powerful enough to build interesting programs with, as you will see in a few examples.
2.8
Structure of a lex file
A lex file looks like
...definitions...
%%
...rules...
%%
...code...
Here is a simple example:
%{
int charcount=0,linecount=0;
%}
%%
. charcount++;
\n {linecount++; charcount++;}
%%
int main()
{
yylex();
printf("There were %d characters in %d lines\n",
charcount,linecount);
return 0;
}
In this example, all three sections are present:
definitions All code between %{ and %} is copied to the beginning of the resulting C file.
rules A number of combinations of pattern and action: if the action is more than a single
command it needs to be in braces.
code This can be very elaborate, but the main ingredient is the call to yylex, the lexical
analyser. If the code segment is left out, a default main is used which only calls
yylex.
TEX – LATEX – CS 594
2.9. DEFINITIONS SECTION
2.8.1
57
Running lex
If you store your lex code in a file count.l, you can build an executable from it by
lex -t count.l > count.c
cc -c -o count.o count.c
cc -o counter count.o -ll
You see that the lex file is first turned into a normal C file, which is then compiled and
linked.
If you use the make utility (highly recommended!) you can save a few steps because make
knows about lex:
counter: count.o
cc -o counter count.o -ll
2.9
Definitions section
There are three things that can go in the definitions section:
C code Any indented code between %{ and %} is copied to the C file. This is typically
used for defining file variables, and for prototypes of routines that are defined in
the code segment.
definitions A definition is very much like a #define cpp directive. For example
letter [a-zA-Z]
digit [0-9]
punct [,.:;!?]
nonblank [ˆ \t]
These definitions can be used in the rules section: one could start a rule
{letter}+ {...
state definitions If a rule depends on context, it’s possible to introduce states and incorporate those in the rules. A state definition looks like %s STATE, and by default
a state INITIAL is already given. See section 2.10.2 for more info.
2.10
Rules section
The rules section has a number of pattern-action pairs. The patterns are regular expressions
(see section 2.11, and the actions are either a single C command, or a sequence enclosed in
braces.
If more than one rule matches the input, the longer match is taken. If two matches are the
same length, the earlier one in the list is taken.
It is possible to associate one action with more than one pattern:
[0-9]+
[0-9]+\.[0-9]*
\.[0.9]+
Victor Eijkhout
process_integer();
|
process_real();
58
2.10.1
CHAPTER 2. PARSING
Matched text
When a rule matches part of the input, the matched text is available to the programmer as
a variable char* yytext of length int yyleng.
To extend the example from the introduction to be able to count words, we would write
%{
int charcount=0,linecount=0,wordcount=0;
%}
letter [ˆ \t\n]
%%
{letter}+ {wordcount++; charcount+=yyleng;}
. charcount++;
\n {linecount++; charcount++;}
Exercise 22. Write an integer postfix calculator in lex: expression such
as 1 2 + and 1 2 3 4/*- should be evaluated to 3 and -.5 respectively. White space only serves to separate number, but is otherwise
optional; the line end denotes the end of an expression. You will probably need the C function int atoi(char*) which converts strings to
ints.
2.10.2
Context
If the application of a rule depends on context, there are a couple of ways of dealing with
this. We distinguish between ‘left context’ and ‘right context’, basically letting a rule depend on what comes before or after the matched token.
See section 2.13.1 for an elaborate example of the use of context.
2.10.2.1 Left context
Sometimes, using regular expression as we have seen so far is not powerful enough. For
example:
%%
"/*".*"*/"
.
\n
;
|
ECHO;
works to filter out comments in
This line /* has a */ comment
but not in
This /* line has */ a /* comment */
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2.10. RULES SECTION
59
What we want is, after the /* string to change the behaviour of lex to throw away all
characters until */ is encountered. In other words, we want lex to switch between two
states, and there is indeed a state mechanism available.
We can consider states to implement implement a dependence on the left context of a rule,
since it changes the behaviour depending on what came earlier. To use a state, a rule is
prefixed as
<STATE>(some pattern) {...
meaning that the rule will only be evaluated if the specified state holds. Switching between
states is done in the action part of the rule:
<STATE>(some pattern) {some action; BEGIN OTHERSTATE;}
where all state names have been defined with %s SOMESTATE statements, as described in
section 2.9. The initial state of lex is INITIAL.
Here is the solution to the comment filtering problem:
%x COMM
%%
.
\n
"/*"
<COMM>"*/"
<COMM>.
<COMM>\n
|
ECHO;
BEGIN COMM;
BEGIN INITIAL;
|
;
%%
We see that the state is defined with %x COMM rather than as indicated above with %s. This
is called an ‘exclusive state’. If an exclusive state is active, rules without state prefix will
not be matched if there is a match in a rule with the prefix of the current state.
2.10.2.2 Right context
It is also possible to let a rule depend on what follows the matched text. For instance
abc/de {some action}
means ‘match abc but only when followed by de. This is different from matching on
abcde because the de tokens are still in the input stream, and they will be submitted to
matching next.
It is in fact possible to match on the longer string; in that case the command
abcde {yyless(3); .....}
pushes back everything after the first 3 characters. The difference with the slash approach
is that now the right context tokens are actually in yytext so they can be inspected.
Victor Eijkhout
60
2.11
CHAPTER 2. PARSING
Regular expressions
Many Unix utilities have regular expressions of some sort, but unfortunately they don’t all
have the same power. Here are the basics:
. Match any character except newlines.
\n A newline character.
\t A tab character.
ˆ The beginning of the line.
$ The end of the line.
<expr>* Zero or more occurrences of the expression.
<expr>+ One or more occurrences of the expression.
(<expr1>|<expr2>) One expression of another.
[<set>] A set of characters or ranges, such as [,.:;] or [a-zA-Z].
[ˆ<set>] The complement of the set, for instance [ˆ \t].
Exercise 23. It is possible to have ] and - in a character range. The
] character has to be first, and - has to be either first or last. Why?
Exercise 24. Write regular expressions that match from the beginning
of the line to the first letter ‘a’; to the last letter ‘a’. Also expressions
that match from the first and last ‘a’ to the end of the line. Include representative input and output in your answer.
2.12
Remarks
2.12.1
User code section
If the lex program is to be used on its own, this section will contain a main program. If
you leave this section empty you will get the default main:
int main()
{
yylex();
return 0;
}
where yylex is the parser that is built from the rules.
2.12.2
Input and output to lex
Normally, the executable produced from the lex file will read from standard in and write to
standard out. However, its exact behaviour is that it has two variables
FILE *yyin,*yyout;
that are by default set that way. You can open your own files and assign the file pointer to
these variables.
TEX – LATEX – CS 594
2.13. EXAMPLES
2.12.3
61
Lex and Yacc
The integration of lex and yacc will be discussed in the yacctutorial; here are just a few
general comments.
2.12.3.1 Definition section
In the section of literal C code, you will most likely have an include statement:
#include "mylexyaccprog.h"
as well as prototypes of yacc routines such as yyerror that you may be using. In some
yacc implementations declarations like
extern int yylval;
are put in the .h file that the yacc program generates. If this is not the case, you need to
include that here too if you use yylval.
2.12.3.2 Rules section
If you lexprogrammer is supplying a tokenizer, the yacc program will repeatedly call the
yylex routine. The rules will probably function by calling return everytime they have
constructed a token.
2.12.3.3 User code section
If the lex program is used coupled to a yacc program, you obviously do not want a main
program: that one will be in the yacc code. In that case, leave this section empty; thanks to
some cleverness you will not get the default main if the compiled lex and yacc programs
are linked together.
2.13
Examples
2.13.1
Text spacing cleanup
(This section illustrates the use of contexts; see section 2.10.2.)
Suppose we want to clean up sloppy spacing and punctuation in typed text. For example,
in this text:
This
text (all of it )has occasional lapses , in
punctuation(sometimes pretty bad) ,( sometimes not so).
(Ha! ) Is this : fun?Or what!
We have
•
Multiple consecutive blank lines: those should be compacted.
•
Multiple consecutive spaces, also to be compacted.
•
Space before punctuation and after opening parentheses, and
Victor Eijkhout
62
•
CHAPTER 2. PARSING
Missing spaces before opening and after closing parentheses.
That last item is a good illustration of where context comes in: a closing paren followed by
punctuation is allowed, but followed by a letter it is an error to be corrected.
We can solve this problem without using context, but the lex code would be longer and
more complicated. To see this, consider that we need to deal with spacing before and after
a parenthesis. Suppose that there are m cases of material before, and n of material after,
to be handled. A lex code without context would then likely have m × n rules. However,
using context, we can reduce this to m + m.
2.13.1.1 Right context solution
Let us first try a solution that uses ‘right context’: it basically describes all cases and corrects the spacing.
punct [,.;:!?]
text [a-zA-Z]
%%
")"" "+/{punct}
")"/{text}
{text}+" "+/")"
{printf(")");}
{printf(") ");}
{while (yytext[yyleng-1]==’ ’) yyleng--; ECHO;}
({punct}|{text}+)/"(" {ECHO; printf(" ");}
"("" "+/{text}
{while (yytext[yyleng-1]==’ ’) yyleng--; ECHO;}
{text}+" "+/{punct}
{while (yytext[yyleng-1]==’ ’) yyleng--; ECHO;}
ˆ" "+
" "+
.
\n/\n\n
\n
;
{printf(" ");}
{ECHO;}
;
{ECHO;}
In the cases where we match superfluous white space, we manipulate yyleng to remove
the spaces.
2.13.1.2 Left context solution
Using left context, we implement a finite state automaton in lex, and specify how to treat
spacing in the various state transitions. Somewhat surprisingly, we discard spaces entirely,
and reinsert them when appropriate.
We recognise that there are four categories, corresponding to having just encountered an
open or close parenthesis, text, or punctuation. The rules for punctuation and closing parentheses are easy, since we discard spaces: these symbols are inserted regardless the state. For
text and opening parentheses we need to write rules for the various states.
TEX – LATEX – CS 594
2.13. EXAMPLES
punct [,.;:!?]
text [a-zA-Z]
%s
%s
%s
%s
OPEN
CLOSE
TEXT
PUNCT
%%
" "+ ;
<INITIAL>"(" {ECHO; BEGIN OPEN;}
<TEXT>"(" {printf(" "); ECHO; BEGIN OPEN;}
<PUNCT>"(" {printf(" "); ECHO; BEGIN OPEN;}
")" {ECHO ; BEGIN CLOSE;}
<INITIAL>{text}+ {ECHO; BEGIN TEXT;}
<OPEN>{text}+ {ECHO; BEGIN TEXT;}
<CLOSE>{text}+ {printf(" "); ECHO; BEGIN TEXT;}
<TEXT>{text}+ {printf(" "); ECHO; BEGIN TEXT;}
<PUNCT>{text}+ {printf(" "); ECHO; BEGIN TEXT;}
{punct}+ {ECHO; BEGIN PUNCT;}
\n {ECHO; BEGIN INITIAL;}
%%
Exercise 25. Write a lex parser that analyzes text the way the TEX
input processor does with the normal category code values. It should
print its output with
•
<space> denoting any space that is not ignored or skipped, and
•
<cs: command> for recognizing a control sequence \command;
•
open and close braces should also be marked as <{>, <}>.
Here is some sample input:
this is {a line} of text.
handle \control sequences \andsuch
with \arg{uments}.
Aha!
this line has %a comment
x
y%
Victor Eijkhout
63
64
CHAPTER 2. PARSING
z
\comm%
and
TEX – LATEX – CS 594
2.14. INTRODUCTION
65
Yacc.
2.14
Introduction
The unix utility yacc (Yet Another Compiler Compiler) parses a stream of token, typically
generated by lex, according to a user-specified grammar.
2.15
Structure of a yacc file
A yacc file looks much like a lex file:
...definitions...
%%
...rules...
%%
...code...
definitions As with lex, all code between %{ and %} is copied to the beginning of the
resulting C file. There can also be various definitions; see section 2.17.
rules As with lex, a number of combinations of pattern and action. The patterns are now
those of a context-free grammar, rather than of a regular grammar as was the case
with lex.
code This can be very elaborate, but the main ingredient is the call to yyparse, the
grammatical parse.
2.16
Motivating example
It is harder to give a small example of yacc programming than it was for lex. Here is a
program that counts the number of different words in a text. (We could have written this
particular example in lex too.)
First consider the lex program that matches words:
%{
#include "words.h"
int find_word(char*);
extern int yylval;
%}
%%
[a-zA-Z]+ {yylval = find_word(yytext);
return WORD;}
.
;
Victor Eijkhout
66
\n
CHAPTER 2. PARSING
;
%%
The lexer now no longer has a main program, but instead returns a WORD return code. It
also calls a routine find_word, which inserts the matched word in a list if it is not already
there.
The routine find_word is defined in the yacc code:
%{
#include <stdlib.h>
#include <string.h>
int yylex(void);
#include "words.h"
int nwords=0;
#define MAXWORDS 100
char *words[MAXWORDS];
%}
%token WORD
%%
text : ;
| text WORD ; {
if ($2<0) printf("new word\n");
else printf("matched word %d\n",$2);
}
%%
int find_word(char *w)
{
int i;
for (i=0; i<nwords; i++)
if (strcmp(w,words[i])==0) {
return i;
}
words[nwords++] = strdup(w);
return -1;
}
int main(void)
{
yyparse();
printf("there were %d unique words\n",nwords);
TEX – LATEX – CS 594
2.17. DEFINITIONS SECTION
67
}
Other things to note:
•
The WORD token that was used in the lex code is defined here in the definitions
section; lex knows about it through including the words.h file.
•
The lex rule also sets a variable yylval; this puts a value on the stack top, where
yacc can find it with $1, $2, et cetera.
All of this will be explained in detail below.
2.17
Definitions section
There are three things that can go in the definitions section:
C code Any code between %{ and %} is copied to the C file. This is typically used for
defining file variables, and for prototypes of routines that are defined in the code
segment.
definitions The definitions section of a lex file was concerned with characters; in yacc this
is tokens. These token definitions are written to a .h file when yacc compiles this
file.
associativity rules These handle associativity and priority of operators; see section 2.20.
2.18
Lex Yacc interaction
Conceptually, lex parses a file of characters and outputs a stream of tokens; yacc accepts a
stream of tokens and parses it, performing actions as appropriate. In practice, they are more
tightly coupled.
If your lex program is supplying a tokenizer, the yacc program will repeatedly call the
yylex routine. The lex rules will probably function by calling return every time they
have parsed a token. We will now see the way lex returns information in such a way that
yacc can use it for parsing.
2.18.1
The shared header file of return codes
If lex is to return tokens that yacc will process, they have to agree on what tokens there are.
This is done as follows.
•
The yacc file will have token definitions
%token NUMBER
in the definitions section.
•
When the yacc file is translated with yacc -d -o, a header file <file>.h3 is
created that has definitions like
#define NUMBER 258
This file can then be included in both the lex and yacc program.
3.
If you leave out the -o option to yacc, the file is called y.tab.h.
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CHAPTER 2. PARSING
•
The lex file can then call return NUMBER, and the yacc program can match on
this token.
The return codes that are defined from %TOKEN definitions typically start at around 258,
so that single characters can simply be returned as their integer value:
/* in the lex program */
[0-9]+ {return NUMBER}
[-+*/] {return *yytext}
/* in the yacc program */
sum : NUMBER ’+’ NUMBER
The yacc code now recognizes a sum if lex returns in sequence a NUMBER token, a plus
character, and another NUMBER token.
See example 2.22.1 for a worked out code.
2.18.2
Return values
In the above, very sketchy example, lex only returned the information that there was a number, not the actual number. For this we need a further mechanism. In addition to specifying
the return code, the lex parser can return a value that is put on top of the stack, so that yacc
can access it. This symbol is returned in the variable yylval. By default, this is defined
as an int, so the lex program would have
extern int yylval;
%%
[0-9]+ {yylval=atoi(yytext); return NUMBER;}
See section 2.19.1 for how the stack values are used by yacc.
If more than just integers need to be returned, the specifications in the yacc code become
more complicated. Suppose we are writing a calculator with variables, so we want to return
double values, and integer indices in a table. The following three actions are needed.
1.
The possible return values need to be stated:
%union {int ival; double dval;}
2.
These types need to be connected to the possible return tokens:
%token <ival> INDEX
%token <dval> NUMBER
3.
The types of non-terminals need to be given:
%type <dval> expr
%type <dval> mulex
%type <dval> term
The generated .h file will now have
#define INDEX 258
#define NUMBER 259
typedef union {int ival; double dval;} YYSTYPE;
extern YYSTYPE yylval;
This is illustrated in example 2.22.2.
TEX – LATEX – CS 594
2.19. RULES SECTION
2.19
69
Rules section
The rules section contains the grammar of the language you want to parse. This looks like
name1 :
THING something OTHERTHING {action}
| othersomething THING
{other action}
name2 :
.....
This is the general form of context-free grammars, with a set of actions associated with
each matching right-hand side. It is a good convention to keep non-terminals (names that
can be expanded further) in lower case and terminals (the symbols that are finally matched)
in upper case.
The terminal symbols get matched with return codes from the lex tokenizer. They are typically defines coming from %token definitions in the yacc program or character values;
see section 2.18.1.
A simple example illustrating the ideas in this section can be found in section 2.22.1.
2.19.1
Rule actions
The example in section 2.22.1 had such rules as:
expr:
expr ’+’ mulex
| expr ’-’ mulex
| mulex
{ $$ = $1 + $3; }
{ $$ = $1 - $3; }
{ $$ = $1; }
The action belonging to the different right hand sides refer to $n quantities and to $$. The
latter refers to the stack top, so by assigning to it a new item is put on the stack top. The
former variables are assigned the values on the top of the stack: if the right hand side has
three terms, terminal or nonterminal, then $1 through $3 are assigned and the three values
are removed from the stack top.
2.20
Operators; precedence and associativity
The example in section 2.22.1 had separate rules for addition/subtraction and multiplication/division. We could simplify the grammar by writing
expr:
expr ’+’
expr ’-’
expr ’*’
expr ’/’
expr ’ˆ’
number ;
expr
expr
expr
expr
expr
;
;
;
;
;
but this would have 1+2*3 evaluate to 9. In order to indicate operator precedence, we can
have lines
Victor Eijkhout
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CHAPTER 2. PARSING
%left ’+’ ’-’
%left ’*’ ’/’
%right ’ˆ’
The sequence of lines indicates increasing operator precedence and the keyword sets the
associativity type: we want 5-1-2 to be 2, so minus is left associative; we want 2ˆ2ˆ3 to
be 256, not 64, so exponentiation is right associative.
Operators that can be both unary and binary are handled by declaring a non-associative
token, and explicitly indicating its precedence.
%left ’-’ ’+’
%nonassoc UMINUS
%
expression : expression ’+’ expression
| expression ’-’ expression
| ’-’ expression %prec UMINUS
2.21
Further remarks
2.21.1
User code section
The minimal main program is
int main()
{
yyparse();
return 0;
}
Extensions to more ambitious programs should be self-evident.
In addition to the main program, the code section will usually also contain subroutines, to
be used either in the yacc or the lex program. See for instance example 2.22.3.
Exercise 26. Try to write lex or yacc programs for the following languages:
an bm , an bn , an bn cn
Discuss the theoretical power of lex and yacc.
2.21.2
Errors and tracing
So far we have assumed that the input to yacc is syntactically correct, and yacc need only
discover its structure. However, occasionally input will be incorrect.
2.21.2.1 Tracing
If you assign yydebug=1;, yacc will produce trace output. While its states may not make
sense to you, at least you will see which tokens it matches, and which rules it can reduce.
TEX – LATEX – CS 594
2.21. FURTHER REMARKS
71
2.21.2.2 Syntax errors
Sometimes, yacc reports ‘syntax error’ and stops processing. This means that an unexpected symbol is found. A common source for this is the case that you have made a typo
in your grammar, and the symbol it is trying to match is not defined. Example: suppose we
have just matched an open token:
group : open body close
bodytext : ;
| character bodytext
If you are tracing yacc’s workings, you will probably see it matching the character,
then giving the syntax error message.
The ‘syntax error’ message is actually yacc’s default implementation of the yyerror
routine, but it can be redefined at will. For example, suppose we have a declaration
int lineno=1;
/* in yacc */
extern int lineno; /* in lex */
and every line with \n in lex increases this variable. We could then define
void yyerror(char *s)
{
printf("Parsing failed in line %d because of %s\n",
lineno,s);
}
2.21.2.3 Error recovery
Error recovery in yacc is possible through the error token. In the rule
foo : bar baz ;
| error baz printf("Hope for the best\n");
recognizing any token but bar will make yacc start skipping tokens, hoping to find baz
and recover from that point. This is not guaranteed to work.
2.21.2.4 Semantical errors
Both lex and yacc are stronger than simple finite-state or pushdown automata, for instance
if they are endowed with a symbol table. This can be used to detect semantic errors. For
instance, while you would like to write
array_slice : array_name ’[’ int_expr ’]’
you may be limited to
array_slice : ident ’[’ int_expr ’]’
{if (!is_array($1)) { ....
There are a couple of tools here:
yyerror(char*) is a default write to stderr; you can redefine it.
YYABORT is a macro that halts parsing.
Victor Eijkhout
72
2.21.3
CHAPTER 2. PARSING
Makefile rules for yacc
The make utility knows about lex and yacc, but if you want to do things yourself, here are
some good rules:
# disable normal rules
.SUFFIXES:
.SUFFIXES: .l .y .o
# lex rules
.l.o :
lex -t $*.l > $*.c
cc -c $*.c -o $*.o
# yacc rules
.y.o :
if [ ! -f $*.h ] ; then touch $*.h ; fi
yacc -d -t -o $*.c $*.y
cc -c -o $*.o $*.c ;
rm $*.c
# link lines
lexprogram : $(LEXFILE).o
cc $(LEXFILE).o -o $(LEXFILE) -ll
yaccprogram : $(YACCFILE).o $(LEXFILE).o
cc $(YACCFILE).o $(LEXFILE).o -o $(YACCFILE) -ly -ll
2.21.4
The power of yacc
Theoretically, yacc implements an LALR(1) parser, which is essentially an LR parser with
one token look-ahead. This describes a large class of useful grammars. As an example of a
grammar with two tokens look-ahead, consider
phrase −→ CART ANIMAL and cart | WORK ANIMAL and plow
CART ANIMAL −→ horse | goat
WORK ANIMAL −→ horse | ex
Now to distinguish between horse and cart and horse and plow from the word
horse takes two tokens look-ahead.
Exercise 27. Use the TEX parser you wrote in lex to parse LATEX documents. The parser should
•
Report the documentclass used;
•
Check that \begin{document} and \end{document} are
used, with no text before the begin command;
•
Know about some commands with one argument, such as \textbf,
and properly recognize that argument
•
Recognize proper matching of begin/end of an environment.
TEX – LATEX – CS 594
2.22. EXAMPLES
73
Bonus: let your parser interpret \newcommand correctly. You can limit
yourself to the the case of commands with one argument, that is
\newcommand{\foo}[1]{ ...}
2.22
Examples
2.22.1
Simple calculator
This calculator evaluates simple arithmetic expressions. The lex program matches numbers
and operators and returns them; it ignores white space, returns newlines, and gives an error
message on anything else.
%{
#include <stdlib.h>
#include <stdio.h>
#include "calc1.h"
void yyerror(char*);
extern int yylval;
%}
%%
[ \t]+ ;
[0-9]+
[-+*/]
"("
")"
\n
.
{yylval = atoi(yytext);
return INTEGER;}
{return *yytext;}
{return *yytext;}
{return *yytext;}
{return *yytext;}
{char msg[25];
sprintf(msg,"%s <%s>","invalid character",yytext);
yyerror(msg);}
Accepting the lex output, the following yacc program has rules that parse the stream of
numbers and operators, and perform the corresponding calculations.
%{
#include <stdlib.h>
#include <stdio.h>
int yylex(void);
#include "calc1.h"
%}
%token INTEGER
%%
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CHAPTER 2. PARSING
program:
line program
| line
line:
expr ’\n’
| ’n’
expr:
expr ’+’ mulex
| expr ’-’ mulex
| mulex
mulex:
mulex ’*’ term
| mulex ’/’ term
| term
term:
’(’ expr ’)’
| INTEGER
{ printf("%d\n",$1); }
{ $$ = $1 + $3; }
{ $$ = $1 - $3; }
{ $$ = $1; }
{ $$ = $1 * $3; }
{ $$ = $1 / $3; }
{ $$ = $1; }
{ $$ = $2; }
{ $$ = $1; }
%%
void yyerror(char *s)
{
fprintf(stderr,"%s\n",s);
return;
}
int main(void)
{
/*yydebug=1;*/
yyparse();
return 0;
}
Here we have realized operator precedence by having separate rules for the different priorities. The rule for plus/minus comes first, which means that its terms, the mulex expressions involving multiplication, are evaluated first.
2.22.2
Calculator with simple variables
In this example the return variables have been declared of type double. Furthermore, there
can now be single-character variables that can be assigned and used. There now are two
different return tokens: double values and integer variable indices. This necessitates the
%union statement, as well as %token statements for the various return tokens and %type
statements for the non-terminals.
TEX – LATEX – CS 594
2.22. EXAMPLES
This is all in the yacc file:
%{
#include <stdlib.h>
#include <stdio.h>
int yylex(void);
double var[26];
%}
%union { double dval; int ivar; }
%token <dval> DOUBLE
%token <ivar> NAME
%type <dval> expr
%type <dval> mulex
%type <dval> term
%%
program:
line program
| line
line:
expr ’\n’
{ printf("%g\n",$1); }
| NAME ’=’ expr ’\n’ { var[$1] = $3; }
expr:
expr ’+’ mulex
{ $$ = $1 + $3; }
| expr ’-’ mulex
{ $$ = $1 - $3; }
| mulex
{ $$ = $1; }
mulex:
mulex ’*’ term
{ $$ = $1 * $3; }
| mulex ’/’ term
{ $$ = $1 / $3; }
| term
{ $$ = $1; }
term:
’(’ expr ’)’
{ $$ = $2; }
| NAME
{ $$ = var[$1]; }
| DOUBLE
{ $$ = $1; }
%%
void yyerror(char *s)
{
fprintf(stderr,"%s\n",s);
return;
}
int main(void)
Victor Eijkhout
75
76
CHAPTER 2. PARSING
{
/*yydebug=1;*/
yyparse();
return 0;
}
The lex file is not all that different; note how return values are now assigned to a component
of yylval rather than yylval itself.
%{
#include <stdlib.h>
#include <stdio.h>
#include "calc2.h"
void yyerror(char*);
%}
%%
[ \t]+ ;
(([0-9]+(\.[0-9]*)?)|([0-9]*\.[0-9]+)) {
yylval.dval = atof(yytext);
return DOUBLE;}
[-+*/=]
{return *yytext;}
"("
{return *yytext;}
")"
{return *yytext;}
[a-z]
{yylval.ivar = *yytext;
return NAME;}
\n
{return *yytext;}
.
{char msg[25];
sprintf(msg,"%s <%s>","invalid character",yytext);
yyerror(msg);}
2.22.3
Calculator with dynamic variables
Basically the same as the previous example, but now variable names can have regular
names, and they are inserted into a names table dynamically. The yacc file defines a routine
for getting a variable index:
%{
#include <stdlib.h>
#include <stdio.h>
#include <string.h>
int yylex(void);
#define NVARS 100
char *vars[NVARS]; double vals[NVARS]; int nvars=0;
%}
TEX – LATEX – CS 594
2.22. EXAMPLES
77
%union { double dval; int ivar; }
%token <dval> DOUBLE
%token <ivar> NAME
%type <dval> expr
%type <dval> mulex
%type <dval> term
%%
program:
line program
| line
line:
expr ’\n’
| NAME ’=’ expr ’\n’
expr:
expr ’+’ mulex
{
| expr ’-’ mulex
{
| mulex
{
mulex:
mulex ’*’ term
{
| mulex ’/’ term
{
| term
{
term:
’(’ expr ’)’
{
| NAME
{
| DOUBLE
{
{ printf("%g\n",$1); }
{ vals[$1] = $3; }
$$ = $1 + $3; }
$$ = $1 - $3; }
$$ = $1; }
$$ = $1 * $3; }
$$ = $1 / $3; }
$$ = $1; }
$$ = $2; }
$$ = vals[$1]; }
$$ = $1; }
%%
int varindex(char *var)
{
int i;
for (i=0; i<nvars; i++)
if (strcmp(var,vars[i])==0) return i;
vars[nvars] = strdup(var);
return nvars++;
}
int main(void)
{
/*yydebug=1;*/
yyparse();
return 0;
}
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CHAPTER 2. PARSING
The lex file is largely unchanged, except for the rule that recognises variable names:
%{
#include <stdlib.h>
#include <stdio.h>
#include "calc3.h"
void yyerror(char*);
int varindex(char *var);
%}
%%
[ \t]+ ;
(([0-9]+(\.[0-9]*)?)|([0-9]*\.[0-9]+)) {
yylval.dval = atof(yytext);
return DOUBLE;}
[-+*/=]
{return *yytext;}
"("
{return *yytext;}
")"
{return *yytext;}
[a-z][a-z0-9]* {
yylval.ivar = varindex(yytext);
return NAME;}
\n
{return *yytext;}
.
{char msg[25];
sprintf(msg,"%s <%s>","invalid character",yytext);
yyerror(msg);}
TEX – LATEX – CS 594
2.23. INTRODUCTION
Hashing.
79
Hashing, hash functions, hash tables, come into play when a compiler, and
in particular its parser, needs to store names (of identifiers) and further information about
the object of that name.
2.23
Introduction
A compiler, and in particular its parser, needs to store variables and information about them.
The data structure for this is in effect addressed by the name of the variable, rather than by
any numerical index. This sort of storage is sometimes called ‘associative’. The design of
a data structure for this is a good example of trade-offs between efficiency and expediency.
•
•
•
•
If variables are stored in the order in which they are encountered, storage is very
fast, but searching will take time linear in the number of items stored.
If the list if kept sorted, searching for an item will take logarithmic time. However,
insertion is now more expensive, taking linear time because all elements following
have to be copied up.
A naively implemented linked list would give both insertion and retrieval time
linearly in the number of stored items. In the insertion phase we avoid the copying,
but finding the place to insert is harder, and in fact we can not use bisection here.
Items can be stored in a treewise fashion:
•
HH
HH
B
E
H
H
H
H
H
LSE ND
ART
E
H
H
GIN LL
The cost of inserting and retrieving is then linear in the length of the string, at
least for as far as is necessary to disambiguate from other strings.
These are then the issues we need to address:
•
•
•
What is the cost of inserting a new item?
What is the cost of finding and retrieving an item?
What is the cost of deleting an item?
2.24
Hash functions
A hash function is function that maps non-numeric keys into a range of integers, interpreted
as the locations in a table. It would be nice if this function was injective, to avoid mapping
two keys to the same location but surprisingly hard, as is clear from the ‘birthday paradox’:
it takes only 23 random picks from a 365-entry table to have even chances of a collision.
If we know all keys in advance, it is possible to design a function that maps them uniquely
into a table of precisely the right size, but this is unrealistic, since the number of possible
keys in a programming language is very large, indeed unbounded.
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CHAPTER 2. PARSING
Figure 2.1: A hash function without conflicts
A ‘hash function’ is then a function that maps keys in some space to a range of integers 0 . . . M − 1. A good hash function has the following properties:
•
•
•
•
The hash value is fully determined by the data being hashed. For instance, it
should not have a ‘memory’.
The hash function uses as much as possible of the input data. Program variables
often have names such as ikey, so emphasis on the first letter is a bad idea.
The hash function ”uniformly” distributes the data across the entire set of possible
hash values.
The hash function generates very different hash values for similar strings. Variables like key1, key2, et cetera should not be mapped into a cluster.
Figure 2.1 illustrates a hash function without conflicts.
Let us assume that we have found a way of mapping the names onto a large integer space,
for instance by interpreting the bit pattern of the name as an integer. A simple hash function
would be
h(K) = K mod M,
(2.2)
where M is the size of the hash table.
Certain values of M are less desirable. For instance, if M is even, say M = 2M 0 , then the
statement r = K mod M (say K = nM + r for some n) implies
K = 2K 0 ⇒ r = 2(nM 0 − K 0 )
K = 2K 0 + 1 ⇒ r = 2(nM 0 − K 0 ) + 1
so the key is even, iff the original number is. This places an undue influence on the last
digit. If M is a multiple of 3, we find for numbers stored decimally or in bytes that keys
that are a permutation of each other get mapped to numbers that differ by a multiple of 3,
since both 10n mod 3 = 1 and 28 mod 1 = 1.
TEX – LATEX – CS 594
2.24. HASH FUNCTIONS
2.24.1
81
Multiplication and division strategies
A good strategy is to take M prime, and such that rk 6= ±a mod M , where r the radix
of the number system, and a, k small. (There is a list of suitable primes on http://
planetmath.org/encyclopedia/GoodHashTablePrimes.html.)
Equation (2.2) requires you to perform a division. The same equation based on multiplication would use an integer A ≈ w/M , where w is the maxint value. Then 1/M = A/w,
which is simply A with an imaginary decimal point to its left. Observing that
K mod M = M (K/M mod 1)
we define
A
h(K) = bM
K mod 1 c.
w
As an example of the value of using a prime table size, consider hashing the Bible, which
consists of 42,829 unique words, into an open hash table with 30,241 elements (a prime
number). In the end, 76.6 percent of the slots were used and that the average chain was
1.85 words in length (with a maximum of 6). The same file run into a hash table of 30,240
elements (evenly divisible by integers 2 through 9) fills only 60.7 percent of the slots and
the average chain is 2.33 words long (maximum: 10).
2.24.2
String addition strategies
One could Derive a hash key by adding or XORing together all bytes in a string.
h = <some value>
for (i=0; i<len(var); i++)
h = h + <byte i of string>;
This runs into the problem that anagrams map into the same key, and nearby strings into
nearby keys. This could be remedied by introducing a table of random numbers:
h = <some value>
for (i=0; i<len(var); i++)
h = Rand( h XOR <byte i of string> );
Exercise 28. This algorithm only gives a one-byte key. How would
you derive longer keys? Give pseudo-code for the algorithm.
2.24.3
Examples
Here are a couple of published hash functions:
/* UNIX ELF hash
* Published hash algorithm used in the UNIX ELF format for object files
*/
unsigned long hash(char *name)
{
unsigned long h = 0, g;
while ( *name ) {
Victor Eijkhout
82
CHAPTER 2. PARSING
h = ( h << 4 ) + *name++;
if ( g = h & 0xF0000000 )
h ˆ= g >> 24;
h &= ˜g;
}
}
This hash key is then reduced to an index in the hash table by
#define HASHSIZE 997
static int M = HASHSIZE;
return h % M;
Another hash function:
/* djb2
* This algorithm was first reported by Dan Bernstein
* many years ago in comp.lang.c
*/
unsigned long hash(unsigned char *str)
{
unsigned long hash = 5381;
int c;
while (c = *str++) hash = ((hash << 5) + hash) + c;
return hash;
}
Note the use of bit shifts to implement multiplication.
2.25
Collisions
The above techniques for generating randomly spread out addresses are generally sufficient. The problem to worry about is how to handle collisions, that is, if h(k1 ) = h(k2 ) for
different keys k1 , k2 . We will investigate several techniques for dealing with this.
For all of the strategies below, any performance analysis is statistical in nature. The average
expected behaviour is often excellent, but the worst case behaviour is always very bad. In
the worst case, all hash addresses map to the same location, and search time is propertional
to the number of elements in the table.
The question is now how to find the storage locations for the elements with conflicting hash
keys. We will look at one strategy that allocates space outside the hash table (‘open hash
table’), and two that resolve the conflict by finding different locations in the table (‘closed
hash table’).
2.25.1
Separate chaining
A simple solution to hash conflicts is the create a linked list from each table entry, as shown
in figure 2.5. This way of implementing a hash table is called ‘separate chaining’ or ‘open
TEX – LATEX – CS 594
2.25. COLLISIONS
83
Figure 2.2: Separate chaining as a solution for hash conflicts
hashing’. One problem with this approach is that we need to maintain two different kinds
of storage: static in the table and dynamic for the linked lists.
The linked lists can be created by malloc (and released by free) calls. However, these
are very expensive compared to using a freespace pointer in a list. To amortize the cost, one
could allocate a block of cells, and dole them out gradually. The problem with this strategy
is dealing with deletions. If a cell is deleted, rerouting the pointers is easy, but additionally
we have to keep track of which cells in the allocated chunk are free. This makes the open
hash table somewhat complicated to implement.
Another problem is that, while this strategy is fairly efficient if the number of collisions is
low, in order to get the number of collisions low, one typically chooses the table size fairly
large, and then this solution is wasteful of storage. It is then better to store all elements in
the hash table and maintain links in the table.
Exercise 29. Discuss the value of keeping the lists ordered by key:
how does this influence the run time of insertion and retrieval? Programming languages like C have local variables. Does this change your
argument?
2.25.2
Linear probing
The easiest solution is to store a conflicting element in the location immediately after the
computed hash address.
struct { ... } node;
node Table[M]; int Free;
/* insert K */
addr = Hash(K);
if (IsEmpty(addr)) Insert(K,addr);
else {
Victor Eijkhout
84
CHAPTER 2. PARSING
Figure 2.3: Linear probing as a solution for hash conflicts
I
I
J3
J
J2
J
I
J3
J
K
J2
J
K
K
I
I
L
J3
J
J2
J
K
L
I
Figure 2.4: Coalescing of blocks in linear probing
/* see if already stored */
test:
if (Table[addr].key == K) return;
else {
addr = Table[addr].link; goto test;}
/* find free cell */
Free = addr;
do { Free--; if (Free<0) Free=M-1; }
while (!IsEmpty(Free) && Free!=addr)
if (!IsEmpty(Free)) abort;
else {
Insert(K,Free); Table[addr].link = Free;}
}
However, this is not the best solution. Suppose that the blocks of size N is occupied, then
the free pointer will search N/2 locations on average for an address that maps into this
block. While this is acceptable, if two blocks coalesce, this makes the search time double.
Note that the chance of the cell between two blocks filling up is much larger than the
chance of that exact address being generated as hash: each hash in the top block will cause
the address to be filled.
TEX – LATEX – CS 594
2.25. COLLISIONS
85
This is illustrated in figure 2.4. There is a gap of size one between h(I) and a block starting
at h(J). When a conflict h(K) = h(I) occurs, the free space pointer fills the gap. A subsequent conflict h(L) = h(I) (or h(L) = h(K)) needs the free space pointer to traverse
the whole J block to find the next location.
With α = N/M the ratio between occupied cells and total table size, the expected search
time with this algorithm is
2
1
1 1+
unsuccessful
2
1−α
T ≈
1 1+ 1
successful
2
1−α
It is clear that when α approaches 1, this time will go up unbounded.
The clumping behaviour of this algorithm makes it sensitive to the hash algorithm used.
Care has to be taken that successive keys, such as Ptr1, Ptr2. . . , do not get mapped to
successive hash values K, K + 1, . . ..
2.25.3
Chaining
The problems with linear probing can be prevented by storing conflicting elements at the
start or end of the table.
struct { ... } node;
node Table[M]; int Free = M;
/* insert K */
addr = Hash(K);
if (IsEmpty(addr)) Insert(K,addr);
else {
/* see if already stored */
test:
if (Table[addr].key == K) return;
else {
addr = Table[addr].link; goto test;}
/* find free cell */
do { Free--; }
while (!IsEmpty(Free)
if (Free<0) abort;
else {
Insert(K,Free); Table[addr].link = Free;}
}
This algorithm does the same list traversal as linear probing in the case a search is ultimately successful. However, for an unsuccessful search the Free pointer will usually be
decreased by one, and only occasionally by two or more, when it runs into already occupied positions. Since these are likely to be spread out, having to search more than two steps
will be rare.
In this algorithm, occasionally a hash address will be an address that has further links. We
say that we have lists coalescing. This increases search time slightly, but not by much, and
preventing this increases insertion time, because we would have to move cells.
Victor Eijkhout
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CHAPTER 2. PARSING
Figure 2.5: Chaining as a solution for hash conflicts
With α = N/M the fraction of used to total table entries, find that the number of entries
searched is
1 + (e2α − 1 − 2α)/4
unsuccessful
T ≈
1 + (e2α − 1 − 2α)/8α + α/4 successful
The hash algorithm of TEX is a variant of this chaining algorithm.
2.25.4
Other solutions
The solutions to the conflict problem given so far can be called ‘linear rehashing’. The
following strategies are called ‘nonlinear rehashing’.
Random probing Try (h(m) + pi ) mod s, where pi is a sequence of random numbers.
This requires either reproducible random numbers, or storing these numbers. In
order to prevent colliding keys to collide on the next try too, the random number
needs to depend on the key.
Add the hash Try (i × h(m)) mod s. This requires s to be a prime number; with this
approach clumping is prevented.
They have the advantage that occupied locations in the table remain fairly scattered. On the
other hand, they require further hash calculations. Also, because of the irregular memory
access pattern, the cost of memory operations may become significant here.
2.25.5
Deletion
A surprising aspect of closed hash table algorithms is that generally it is hard to delete
elements. Since most algorithms give coalescing lists, we can not mark a cell empty if its
key is to be removed. Instead, we mark a cell ‘deleted’, which removes the key, but leaves
the link intact. There are algorithms that can deal with deletions, but they have a fairly high
complexity.
On the other hand, deleting in an open hash table algorithm is simple. The complication
there is the freeing of the memory. If the cells are allocated in chunks, the decision to free
a chunk becomes complicated.
2.26
Other applications of hashing
The foremost application of hashing is in compilers. However, there are other uses.
TEX – LATEX – CS 594
2.27. DISCUSSION
2.26.1
87
Truncating searches
In applications such as chess programs, you want to avoid evaluating a configuration twice
if it’s arrived at two different ways. This can be done by storing evaluations in a table. This
table can be addressed by the configuration as a key itself, but these keys are long and span
a large space, so searching will probably be expensive. Instead, one can use a hash table.
If two configurations generate the same hash key, they can be, but need not be the same,
so further decision might be needed. To avoid this second stage work, a good quality hash
function is essential.
(This example comes from http://www.seanet.com/˜brucemo/topics/hashing.
htm.)
2.26.2
String searching
The question ‘does a string of length M appear anywhere in a document of length N ’ can
be answered in O(N M ) time by a sequence of string comparisons. However, we can do
considerably better, reducing the complexity to O(N + M ).
A hash function that adds characters together will give the same hash key for strings that
are anagrams of each other. This means that instead of doing a string comparison we can
compare hash keys, and only if they are equal resort to a full string comparison. To get the
complexity down, we note that if the hash function is of the form
(
)
X
h(k) =
k[i] mod K,
i
where k is a character string, then (for a text t long enough)
h(t[2 . . . n + 1]) = h(t[1 . . . n]) + t[n + 1] − t[1]
(with addition/subtraction modulo K) so we can cheaply update the hash key in O(1) time.
2.27
Discussion
In a comparison between hash tables and, for instance, tree-based storage, there is no clear
preference. Hash tables can be faster, because, until the table fills up, access is O(1). A similar complexity can be obtained for trees, but
•
memory access in trees is more chaotic, leading to worse cache or paging behaviour;
•
memory is allocated and freed dynamically; circumventing that takes considerable programming effort;
•
trees can become unbalanced, and balancing them is tricky to program, and takes
time;
•
the optimal search time can be made approximately equal, but that is again hard
to code.
Closed hash tables have the advantage of simpler storage management, and, until they fill
up, no worse performance than open tables.
Victor Eijkhout
88
CHAPTER 2. PARSING
Projects for this chapter.
Project 2.1. Use the lex and yacc programs you have written for LATEX to write a full
LATEX-to-HTML translator.
Project 2.2. A number of projects involve parsers or code generators for (parts of) TEX or
LATEX.
formulas Reinhold Heckmann and Reinhard Wilhelm. 1997. A functional description of TeX’s formula layout. Journal of Functional Programming 7(5):451485. Available online at http://rw4.cs.uni-sb.de/users/heckmann/
doc.html. For software, see ftp://ftp.cs.uni-sb.de/formulae/.
also formulas Preview-LaTeX (http://preview-latex.sourceforge.
net/) displays formulas right in the emacs edit buffer.
math on web pages see http://www.forkosh.com/mimetex.html.
LATEX to HTML HeVeA, TtH, TeX4ht and LaTeX2HTML.
front end for LATEX http://www.lyx.org/ Ages ago there was ‘griff’. Scientific Word still exists but is commercial.
reimplementation of TEX TEX in Python: http://www.pytex.org/
Investigate these parsers: what is their basic approach and theoretical power, what
are they good at, what are they bad at.
Project 2.3. Develop the full theory of the compound NFA that does lexical analysis.
•
This automaton basically parses the whole file, rather than small chunks; every once in a while it will report that it has recognized an identifier, number,
special symbol &c. This means that the definition of the output alphabet has
to be expanded. Analyze the structure of this output language.
•
As explained, the returning -transition only has to be taken if a maximal
string is recognized. Formalyze this notion.
•
The automaton may need look-ahead of one or more tokens. Formalize this
notion and give equivalence proofs.
Project 2.4. Do an experimental study of hashing. Implement open and closed hash table
algorithms, and do a search on the web for hash functions. Compare to storing
the names in a tree. Find some documents to hash: the source of some programs,
books. Construct some input data sets to trip up the various algorithms. Measure
statistics and do timings.
TEX – LATEX – CS 594
Chapter 3
Breaking things into pieces
The line breaking algorithm of TEX is interesting, in that it produces an aesthetically optimal solution in very little time.
Handouts and further reading for this chapter
If you still have the book ‘Introduction to Algorithms’ by Cormen et al., you can find a
discussion of Dynamic Programming and NP-completeness there. The books by Bellman
are the standard works in this field. Bellman’s ‘Applied Dynamic Programming’ [1] has
been put on reserve, QA264.B353 . The TEX line breaking algorithm is described in an
article by Knuth and Plass [13], reprinted in [9].
The standard work on Complexity Theory, including NP-completeness, is Garey and Johnson ‘Computers and intractibility’ [7]. There is excellent online material about this subject
on Wikipedia, for instance http://en.wikipedia.org/wiki/Complexity_classes_
P_and_NP. Issues in page breaking are discussed in Plass’ thesis [18].
89
90
CHAPTER 3. BREAKING THINGS INTO PIECES
Dynamic Programming.
Dynamic programming is an optimization tech-
nique, that is, a way of solving problems where a yield function is to be maximized, or a
cost function minimized, under certain constraints. Certain optimization problems can be
solved using calculus – unconstrained optimization being a prime example – and others by
such linear algebra techniques as the simplex method. However, such continuous methods
have a hard time dealing with integer constraints. For instance, in computing the yield of a
car factory, the number of cars has to be integer.
The type of problems that dynamic programming is suited for is the type where the problem
can be formulated as a series of decisions. For instance, in work allocation problems, there
is a separate decision for the amount of work to be allocated in each month.
We will start with some examples that show the essential features of the dynamic programming approach.
3.1
Some examples
3.1.1
Decision timing
Our first example concerns the problem of when to make a one-time decision, giving a
number of opportunities to do so. This example illustrates the concept of a series of decisions, and of starting at the final stage and working backward from that.
The surprise menu in a new restaurant works as follows. You will be shown 5 dishes in
sequence, and you can pick one that you like, but if you turn one down, you can not reconsider. Let us say that each dish scores between 0 and 1 on your attractiveness scale. How
do you maximize your choice of dish?
Call the scores you give the dishes xi , and N the number of dishes.
•
If you wait till the last dish, you have no choice.
•
The last dish can be anything, so the best you can say is that it will have an
expected attractiveness of 0.5. Therefore, if xN −1 > 0.5, you take that one, since
it is better than what you can expect in the next step.
•
Now, you will take dish N −1 in half the cases, giving you on average a .75 score,
and the other half of the cases you take dish N , with a score of .5. Therefore, you
are expecting to score .625, and you will take dish N − 2 if it scores more than
that.
•
In .375 of the cases, dish N − 3 will score higher than that.
•
Et cetera.
From this example we see some characteristics:
Stages The optimization problem involves a sequence of stages, each involving a choice.
Principle of optimality Once you arrive at a certain stage, the optimal solution for the rest
of the path is independent of how you got to that stage.
Stepwise solution The solution (here: solution strategy) is arrived at by starting at the final
and working backward. We will also examples that are solved forward; in general
it is a characteristic that dynamic programming problems are solved stage-bystage.
TEX – LATEX – CS 594
3.1. SOME EXAMPLES
91
Often, the fact that the problem is solved starting at the last stage is considered an essential feature. We will not do so: many dynamic programming problems can also be solved
forward.
Exercise 30. For this example, draw up the recurrence relation between the expected scores. Prove that the relation is monotonically increasing (for decreasing index), and bounded above by 1, in fact with
limit 1. Bonus: solve this relation explicitly.
3.1.2
A manufacturing problem
Suppose a factory has N months time to produce a quantity S of their product, which we
will for now assume to be bulk. Because of seasonal variations, in month k the cost of
producing an amount pk is wk p2k . The problem is to produce the requested amount in the
given time, at minimal cost, that is
X
wk p2k .
Pmin
pk =S
We break down the problem by looking at the cost for producing the remaining amount in
the remaining time. Define the minimum cost as
X
wk p2k
v(s|n) = P min
k>N −n
pk =s
and p(s|n) as the work that has to be done n months from the end, given that s work is left,
then, by splitting this into the cost this month plus the cost for the then remaining time, we
get
X
v(s|n) = min wn p2n +
wk p2k
pn ≤s
k>N −n+1
P
pk =s−pn
= min wn p2n + v(s − pn |n − 1)
pn ≤s
That is, we get a recurrence where the remaining work for n months is expressed in that
for n − 1 months.
Starting off is easy: p(s|1) = s, and v(s|1) = w1 s2 . By the above rule then
v(s|2) = min{w2 p22 + v(s − p2 |1)} = min c(s, p2 )
p2
p2
w2 p22 + w1 (s − p2 )2 .
where c(s, p2 ) =
We find the minimum by taking δc(s, p2 )/δp2 = 0,
which gives us p(s|2) = w1 s/(w1 + w2 ) and v(s|2) = w1 w2 s2 /(w1 + w2 ).
Solving one or two more steps like this, we notice the general form:
n
X
1/wn
s,
v(s|n) = s2
1/wi .
p(s|n) = Pn
i=1 1/wi
i=1
This solution can in fact be derived by a variational approach to the constraint minimization
problem
X
X
wk p2k + λ(
pk − S)
k
Victor Eijkhout
k
92
CHAPTER 3. BREAKING THINGS INTO PIECES
for which we set the derivatives to both pn and λ to zero.
This problem shows another characteristic of dynamic programming:
State The cost (yield) function that we define as a function of the stage, also has a state
parameter. It typically describes the amount of some constrained quantity that is
still left. In this case, that is the amount still to be produced.
We see that for this problem there is both an analytical solution, and one found by dynamic programming, using analytical techniques for local optimization. However, these
techniques would become increasingly cumbersome, if we imposed restrictions such as
that the factory does not have unlimited production capacity, or even impossible, if the
product can only be made in discrete units, meaning that the pn have to be integers.
The next problem is a good example of the sort of discrete choices that dynamic programming is well suited for.
3.1.3
The stagecoach problem
A business man in the Old West needs to travel from city 0 to city 8. For this, he has to go
through 3 other cities, but in each stage of the trip there are choices. This being the Wild
West, he decides to get travel insurance. However, not all routes are equally safe, so the
cost of insurance varies. The problem is to find the trip that minimizes the total cost of the
insurance.
3
1
5
1
1
3
4
4
0
6
5
2
2 4
8
2
4
4
2
7
3
2
5
We will look at various ways of solving this problem. First let us define the data.
table = [ [0, 5, 4,
[0, 0, 0,
[0, 0, 0,
[0, 0, 0,
[0, 0, 0,
[0, 0, 0,
[0, 0, 0,
[0, 0, 0,
]
final = len(table);
0,
1,
4,
0,
0,
0,
0,
0,
0,
3,
2,
0,
0,
0,
0,
0,
0,
4,
2,
0,
0,
0,
0,
0,
0,
0,
0,
5,
2,
4,
0,
0,
0,
0,
0,
1,
4,
3,
0,
0,
0],
0],
0],
0],
0],
0],
5],
2]
# first stage: 0
# second: 1 & #2
# third: 3, #4, #5
# fourth: 6 & #7
TEX – LATEX – CS 594
3.1. SOME EXAMPLES
3.1.3.1
93
The wrong way to solve this
The solution to the stage coach problem is easy to formulate recursively, given that you are
in some city:
•
If you are in the final city, the cost is zero;
•
Otherwise it is the minimum – over all cities reachable from here – of the cost of
the next leg plus the minimum cost from that city.
# the wrong way
def cost_from(n):
# if you’re at the end, it’s free
if n==final: return 0
# otherwise range over cities you can reach
# and keep minimum value
val = 0
for m in range(n+1,final+1):
# for all later cities
local_cost = table[n][m]
if local_cost==0: continue
# if there is a connection from here,
# compute the minimum cost
local_cost += cost_from(m)
if val==0 or local_cost<val:
val = local_cost
return val
print "recursive minimum cost is",cost_from(0)
If there are N cities, divided into S stages of L cities in each stage, and assuming that in
each stage all cities of the next stage are reachable, the cost of this algorithm is O(LS ).
3.1.3.2
Dynamic programming solution
The implementation strategy of the previous section is wasteful. Consider some city 1, and
cities 2 and 3 in the stage before that. The cost computations from 2 and 3 both compute
the cost from 1, which is clearly redundant. We call this characteristic
Overlapping subproblems: A straightforward (recursive) solution to the problem would
revisit a subproblem, meaning that different solution attempts have a common
subproblem.
Recognizing this leads us to a better solution strategy: once the minimum cost from one
city has been computed, we store it for future reference.
An equivalent interpretation of this strategy is that we compute in each stage a cost function fn (xn ), describing the cost of reaching the end starting at the nth step, giving that we
start there in city xn . This is in fact a dynamic programming solution, with xn the state
variable in the nth stage.
Formally, fk (s) is the minimum cost for traveling the from stage k given that your are in
city s in that stage. Then
fk−1 (s) = min{cst + fk (t)
t
Victor Eijkhout
94
CHAPTER 3. BREAKING THINGS INTO PIECES
where cst is the cost of traveling from city s to t.
Initially, the cost from every city till the final one (in particular from the final one itself) is
zero:
# initialization
cost = (final+1)*[0]
Now we loop backwards over the stages, computing in each stage the cost of all city we
can go through. These two loops – the stages, and the cities in each stage – can actually be
collapsed into one loop:
# compute cost backwards
for t in range(final-1,-1,-1):
# computing cost from t
For each city t we consider the ones i that are reachable from it. By induction, for these
later ones we already know the cost of going from them to the final one, so the cost from t
to the final one is the cost from t to i plus the cost from i:
for i in range(final+1):
local_cost = table[t][i]
if local_cost==0: continue
local_cost += cost[i]
If there was no cost yet associated with t, we set it now, otherwise we compare the cost
from t over i with the cost of an earlier evaluated route:
if cost[t]==0 or local_cost<cost[t]:
cost[t] = local_cost
In the end, the minimum cost is given in cost[0].
We see that the main difference between this solution and the recursive one given above, is
that the recursive function call has been replaced by a lookup in a table.
The running time of this algorithm is O(N · L) or O(L2 S), which is a considerable improvement over LS for the straightforward implementation. This solution carries an extra
cost of N memory locations; on the other hand, it does not build up a recursion stack.
3.1.3.3
Forward dynamic programming solution
This problem was solved by starting at the end point. We can also work our way to the
solution forwards, with a code that is essentially the same. Instead of computing the cost
of reaching the final city from an intermediate, we now compute the cost of reaching the
intermediate city from the initial one.
We loop over all cities and all their connections:
cost = (final+1)*[0]
for t in range(final):
for i in range(final+1):
local_cost = table[t][i]
if local_cost == 0: continue
TEX – LATEX – CS 594
3.1. SOME EXAMPLES
95
Now we can compute the cost to the connecting city as the transition cost, plus the known
minumum cost to get where we are:
cost_to_here = cost[t]
newcost = cost_to_here+local_cost
if cost[i]==0 or newcost<cost[i]:
cost[i] = newcost
The result is now in cost[final].
The minimization problem corresponding to this algorithm concerns fk s, the cost to get to
city s in stage k. Then
fk+1 (t) = min{cst + fk (s)
s
which is equivalent to the earlier problem.
Exercise 31. A ‘sparse matrix’ is a matrix where a number of matrix
elements are zero, sufficiently many that you do not want to store them.
To compute the matrix
P vector product y = Ax you then do not compute
the full sum yi = j aij xj , but only those terms for which aij 6= 0.
This sort of operation is easy enough to code, but is pretty inefficient in
execution.
Suppose that for small k the product with k consecutive matrix elements
(that is aij xj + aij+ xj+1 + · · · + aij+k−1 xj+k−1 can be executed more
efficiently than doing it as k separate operations. For instance, suppose
that with k = 3 the time per ai· x· reduced to .4 of the normal multiply
time, that is, doing three consecutive multiplications as a block takes
time 1.2, while doing them separately takes time 3.
Now, if a11 6= 0, a12 = 0, a13 6= 0, the multiplication a11 x1 + a13 x3
takes time 2, but if we store the intermediate zero at a12 , the size 3 block
multiplication takes time 1.2. This means that doing some superfluous
operations (multiplying by zero) we can actually speed up the matrixvector product.
Let a pattern of nonzeros and reduction factors be given. The pattern
stands for the locations of the nonzeros in a matrix row, for instance
row = [1,0,0,1,0,1,1,0,0,1,0,0,0,1,1,0,1,1]
redux = [1, .75, .4, .28]
Formulate a principle of optimality for this problem, and write a dynamic programming solution for finding the covering of the sparse row
that will lead to the shortest multiplication time. Tinker with the redux
times (but make sure the n-th is more than 1/n in base-1 indexing) and
see how the solution changes.
3.1.4
Traveling salesman
The above problems all had dynamic programming solutions with a cost slightly more than
linear in the input problem size. Dynamic programming does not always give that low a
complexity.
Victor Eijkhout
96
CHAPTER 3. BREAKING THINGS INTO PIECES
The traveling salesman problem looks a bit like the stagecoach problem above. However,
now the traveler does not only need to go from a starting to a final city, he also has to visit
every city on his travel.
This problem can be solved by dynamic programming, but the concept of stage is now
more complicated. We can no longer map the cities into a linear ordered set of stages since
they can be visited in any sequence.
The solution is to let stage n correspond to picking the nth city, and to define the current
state as the last visited city, plus the set of the remaining ones. Initially we loop over all
possible last cities, so the cost is the sum of the single leg trip to the end point, plus the
minimum cost through remaining cities. unvisited cities.
To be precise: a state is a pair (S, f ) of a set of cities left to be visited, and the current
city f ∈ S.
We can now construct a cost function that depends on the stage and the current state.
C({1}, 1) = 0
C({f }, f ) = a1f for f = 2, 3, . . .
C(S, f ) =
min [C(S − f, m)] + amf ]
m∈S−f
This is easily enough implemented:
def shortest_path(start,through,lev):
if len(through)==0:
return table[start][0]
l = 0
for dest in through:
left = through[:]; left.remove(dest)
ll = table[start][dest]+shortest_path(dest,left,lev+1)
if l==0 or ll<l:
l = ll
return l
to_visit = range(1,ntowns);
s = shortest_path(0,to_visit,0)
This solution has factorial complexity.
3.2
Discussion
In the example above we saw various properties that a problem can have that make it
amenable to dynamic programming.
Stages The optimization problem involves a sequence of stages, each involving a choice,
or a discrete or continuous parameter to be determined.
Stepwise solution The solution is arrived at by solving the subproblems in the stages one
by one. Often this is done starting at the final stage and working backward.
TEX – LATEX – CS 594
3.2. DISCUSSION
97
State The cost (yield) function that we define as a function of the stage, also has a state
parameter. It typically describes the amount of some constrained quantity that is
still left to be consumed or produced.
Overlapping subproblems This is the property that a straightforward (recursive) solution to the problem would revisit a subproblem, meaning that different solution
attempts have a common subproblem.
Principle of optimality This is the property that the restriction of a global solution to a
subset of the stages is also an optimal solution for that subproblem.
The principle of optimality is often phrased thus:
An optimal policy has the property that whatever the initial state and
initial decision are, the remaining decisions must be an optimal policy
with regard to the state resulting from the first decision.
It is easy to show that the principle of optimality holds for P
certain functions and conN
straints.PFor example, consider the problem of maximizing i gi (xi ) under the constraint i xi = X where xi ≥ 0. Call this maximal sum fN (X), then
fN (X)
=
max
PN
i xi =X
N
X
gi (xi )
i
(
=
=
max
xN <X
gN (xN ) + P
max
N −1
i
N
−1
X
xi =X−xN
)
gi (xi )
i
max {gN (xN ) + fN −1 (X − xN )}
xN <X
We see here that the form of the gi functions does not enter the argument, but the fact that
the total utility is a sum of gi s does. Utility functions that are not symmetric in the component gi functions clearly can not straightforwardly be solved with dynamic programming.
Victor Eijkhout
98
CHAPTER 3. BREAKING THINGS INTO PIECES
TEX paragraph breaking.
Breaking a paragraph into lines is the problem
of, given a string of words and other material with intervening spaces, breaking that string
into chunks (lines) of approximately equal length, and doing so in a visually attractive way.
Simple strategies (see the ‘first fit’ algorithm below) give a result that is easy to compute,
but that can be visually very unappealing. While the problem of finding globally optimal
line breaks sounds very hard – with n words there are 2n ways of breaking the paragraph;
also, this problem resembles the bin-packing problem which is NP-complete – it can actually be solved fairly efficiently.
TEX’s basic strategy is to calculate the badness of breaking the lines at certain points, and
to minimize the badness over the whole paragraph.
3.3
The elements of a paragraph
TEX’s paragraph breaking algorithm is based around the concepts of
•
Boxes: this comprises letters, formulas, TEX boxes, and other material of a fixed
with.
•
Glue: this is white space; a glue item has a natural width, stretchability, and
shrinkability.
•
Penalties: these are items that express the desirability or undesirability of breaking
a line at a particular point.
The same elements are also present in a vertical list; in both cases there are some other,
more rare items, that we will ignore here.
3.3.1
Some details
3.3.1.1
Boxes
The boxes in a paragraph are such things as words, rules, math formulas, and actual TEX
\boxes. A box can not be broken: it is completely described by its height, depth, width.
Actually, looking at words as boxes is too simplistic, since words can often be hyphenated.
This means that a word is a sequence of boxes alternating with penalties.
3.3.1.2
Penalties
A penalty item describes the desirability or undesirability of breaking a list at some point.
Penalties can be inserted by hand (most often as the \nobreak macro, which is equivalent
to \penalty10000), or in a macro, or are inserted by TEX itself. An example of the latter
case is the \hyphenpenalty which discourages breaking at a hyphen in a word.
Hyphenating a word can be simulated by having a penalty of zero at the hyphenation location. Since this usually introduces a hyphen character, TEX internally pretends that penalties
can have a width if the list is broken at that place.
The most common types of penalty are the infinite penalty that starts a non-breaking space,
and the penalty associated with breaking by hyphenating a word. The latter kind is called
TEX – LATEX – CS 594
3.3. THE ELEMENTS OF A PARAGRAPH
99
a ‘flagged penalty’, and TEX has an extra amount of demerits associated with them, for
instance to prevent two consecutive lines ending in a hyphen.
Penalties can have positive and negative values, to discourage or encourage breaking at a
specific place respectively. The values +∞ and −∞ are also allowed, corresponding to a
forbidden and forced break respectively.
3.3.1.3
Glue
A ‘glue’ is a dimension with possible stretch and/or shrink. There are glue denotations,
such as 2cm plus .5cm minus .1cm, or glue parameters, such as \leftskip or
\abovedisplayskip. The parameters are inserted automatically by the various TEX
mechanisms.
Glue can be measured in points pt, centimeters cm, millimeters mm, inches in. There is
also infinite glue: fil, fill, and filll. Presence of TEX’s infite glue (fill) causes
all other glue to be set at natural width, that is, with zero stretch and shrink.
If there is more than one glue item in a list, the natural widths and the stretch and shrink
amounts are added together. This means that a list with both 2cm plus 1cm and 2cm
plus -1cm has no stretch since the stretch amounts add up to zero. On the other hand,
with 2cm plus 1cm and 2cm minus 1cm it has both stretch and shrink.
The stretch and shrink components of glue are not treated symmetrically. While in a pinch
we can allow a line to be stretched further than the indicated maximum, we can not allow
spaces to be shrunk to zero, or even close to that.
3.3.1.4
Stretch and shrink
Each space can have stretch and shrink. When we consider a line, we add up all the stretch
and shrink and compute an ‘adjustment ratio’ as the ratio of the shortfall or excess space to
the available stretch or shrink repectively. This ratio r is negative for lines that need to be
shrunk.
A simple algorithm would be to impose a limit of |r| ≤ 1 (and then to minimize the number
of hyphenations under that restriction), but that might be too restrictive. Instead, TEX uses
a concept of ‘badness’. The badness of a line break is infinite if r < −1; otherwise it is
cubic in the absolute size of r.
3.3.1.5
Line break locations
Here are the main (but not the only) locations where TEX can decide to break a line.
•
•
•
•
At a penalty
At a glue, if it is preceeded by a non-discardable item, meaning, not a penalty or
other glue
At a hyphen in a word
At a place where TEX knows how to hyphenate the word. (There is actually a
mechanism, called ‘discretionaries’ that handles these last two cases.)
Victor Eijkhout
100
3.3.2
CHAPTER 3. BREAKING THINGS INTO PIECES
Examples
Here are a few examples of the things that the boxes/penalties/glue mechanism is capable
of.
3.3.3
Centered text
By using \leftskip and \rightskip we can get centered text.
\begin{minipage}{4cm}
\leftskip=0pt plus 1fil \rightskip=0pt plus 1fil
\parfillskip=0pt
This paragraph puts infinitely stretchable glue at
the left and right of each line.
The effect is that the lines will be centered.
\end{minipage}
Output:
This paragraph puts
infinitely stretchable glue at
the left and right of each
line. The effect is that the
lines will be centered.
The following centers only the last line. This is done by letting the \leftskip and
\rightskip cancel each other out, except on the last line.
\begin{minipage}{5cm}
\leftskip=0pt plus 1fil \rightskip=0pt plus -1fil
\parfillskip=0pt plus 2fil
This paragraph puts infinitely stretchable glue at
the left and right of each line, but the amounts cancel out.
The parfillskip on the last line changes that.
\end{minipage}
Output:
This paragraph puts infinitely
stretchable glue at the left and right
of each line, but the amounts cancel
out. The parfillskip on the last line
changes that.
3.3.3.1
Hanging punctuation
Hanging punctuation is a style of typesetting where punctuation that would wind up against
the right margin is actually set in the right margin. Optically, this makes the margin look
straighter.
\newbox\pbox \newbox\cbox
\setbox\pbox\hbox{.} \wd\pbox=0pt
\setbox\cbox\hbox{,} \wd\cbox=0pt
TEX – LATEX – CS 594
3.3. THE ELEMENTS OF A PARAGRAPH
101
\newdimen\csize \csize=\wd\cbox
\newdimen\psize \psize=\wd\pbox
\catcode‘,=13 \catcode‘.=13
\def,{\copy\cbox \penalty0 \hskip\csize\relax}
\def.{\copy\pbox \penalty0 \hskip\psize\relax}
3.3.3.2
Mathematical Reviews
In ‘Mathematical Reviews’ the name of the reviewer should be separated sufficiently from
the review, but fit on the same line if space allows.
We do this by having two separate infinite glues with a break in between, and with a total
natural width equal to the minimum separation. The trick is to make sure that the second
glue is not discarded after a break, which we do by putting an invisible box at the beginning.
\def\signed#1{\unskip
\penalty10000 \hskip 40pt plus 1fill
\penalty0
\hbox{}\penalty10000
\hskip 0pt plus 1fill
\hbox{#1}%
\par
}
Victor Eijkhout
102
CHAPTER 3. BREAKING THINGS INTO PIECES
3.4
TEX’s line breaking algorithm
3.4.1
Elements
3.4.1.1
Glue setting and badness
In order to make a list fit a prescribed dimension, there is a process called ‘glue setting’.
The natural size of the list and the desired size are compared. Let ρ be the ratio of the
amount stretched (shrunk) to the possible amount of stretch (shrink). The exact definition
is such that the ratio is positive for stretch and negative for shrink: let ` be the desired length
of the line, L the natural width, X the available stretch and Y the available shrink, then
0
`=L
(` − L)/X (stretch:) ` > L and X > 0
ρ=
(` − L)/Y (shrink:) ` < L and Y > 0
undefined otherwise
Then the ‘badness’ of the needed glue setting is
10 000
ρ < 1 or undefined
b=
min 10 000, 100|ρ|3
otherwise
Since 10 000 is considered infinite in glue arithmetic, this algorithm allows glue to be
stretched further than the indicated amount, but not shrunk beyond what is available.
A list that is stretched or shrunk is put in one of the following four categories:
tight (3) if it has shrunk with b ≥ 13
decent (2) if b ≤ 12
loose (1) if it has stretched with 100 > b ≥ 13
very loose (0) if it has stretched with b ≥ 100
Note that 100×(1/2)3 = 12.5, so the crossover values denote that half the stretch or shrink
is used.
Lines that differ by more than one in their classifications are called ‘visually incompatible’.
3.4.1.2
Demerits
Breaking a line at a certain points gives the penalty p associated with that point, and the
badness b of the resulting stretch/shrink. These are combined into a ‘demerits’ figure:
b2 + p2 0 ≤ p < 10 000
d=
b2 − p2 −10 000 < p < 0
The demerits for breaking a paragraph along a certain sequence of break points is then the
sum of the demerits of the lines, plus \adjdemerits for every two lines that are not
visually compatible (section 3.4.1.1), \doublehyphendemerits for pairs of lines that
end in a hyphen, and \finalhyphendemerits if the last full line ends in a hyphen.
TEX acts as if before the first line there is a line that is ‘decent’; the last line will typically
contain infinite glue, so all spaces are set at natural width.
For full generality, the last line of a paragraph is handled like any other. Filling out the
line to the margin is realized by added infinite glue plus a trailing penalty that forces a line
break at the end of the paragraph.
TEX – LATEX – CS 594
3.4. TEX’S LINE BREAKING ALGORITHM
3.4.2
103
Breaking strategies
We will now look at a couple of line breaking strategies. The first two will be strictly local;
the third – TEX’s algorithm – is able to optimize in a more global sense.
The problem with local algorithms is that they can not take a slightly worse solution in
one line to prevent much worse from happening later. This will for instance allow tight and
very loose lines to occur next to each other.
3.4.2.1
First fit
The traditional algorithm for line breaking only looks at the current line. Once a word is
starting to exceed the right margin, the following cases are investigated.
1.
2.
3.
4.
If the spaces in the line can be compressed without exceeding some maximum
shrinkage, break after this word.
Otherwise, if the spaces can be stretched short of some maximum, break before
this word.
Otherwise, try hyphenating this word.
If no hyphenation point can be found, accept a line with spaces stretched to whatever degree is needed to break before this word.
If you have set text with TEX, you will have noticed that TEX’s solution to the last point
is slightly different. It will let a word protrude into the right margin as a visual indicator
that no good breakpoint could be found. (TEX’s tolerance to bad breaks can be increased
by setting the \emergencystretch parameter.)
This method can be called ‘first fit’, because it will the first option (compress), without
comparing if later options (stretching) look better. This is remedied in TEX by, instead of
having an all-or-nothing it fits / it doesn’t fit distinction, there is a continuous scale of
evaluation.
3.4.2.2
Best fit
A slight improvement on the first fit algorithm results from deciding between the possibilities 1–3 based on badness calculations. This can be termed ‘best fit’, and while it may work
slightly better than fit, it is still a local decision process.
3.4.2.3
Total fit
TEX’s actual algorithm calculates the ‘demerits’ of a line break as a compound of badness,
the breakpoint penalty, plus any flagged penalties. It then adds together the demerits of the
whole paragraph, and minimizes this number. This makes it possible to use a slightly worse
line break early in the paragraph, to prevent a much worse one later.
Exercise 32. In dynamic programming, many solutions start from a
final stage and work backwards. Why is this approach inappropriate for
TEX’s line breaking algorithm? Why would it be even less appropriate
for a page breaking algorithm?
Victor Eijkhout
104
3.4.3
CHAPTER 3. BREAKING THINGS INTO PIECES
Model implementation
We will here only discuss implementations of solutions based on dynamic programming.
The line breaking algorithm goes linearly through the items in the horizontal list, and for
each considers whether there is a valid breakpoint after it, and with what cost. For the latter
point, it needs to know what the beginning of the line is, so there is an inner loop over
all preceeding words. This would make the running time of the algorithm quadratic in the
number of words, which is much better than the initial estimate of 2n .
However, the situation is better than that. The number of words that can fit on a line is
limited by what can fit when all spaces are sqeezed to zero. The easiest way to implement
this is by maintaining an ‘active list’ of possible words to begin the line with. A word can
be removed from the active list if the material from it to the current word does not fit: if it
does not fit now, it will certainly not fit when the next word is considered.
This is then the main program; we will mainly vary the function that computes the breakpoint cost.
active = [0]
nwords = len(paragraph)
for w in range(1,nwords):
# compute the cost of breaking after word w
for a in active:
line = paragraph[a:w+1]
ratio = compute_ratio(line)
if w==nwords-1 and ratio>0:
ratio = 0 # last line will be set perfect
print "..line=",line,"; ratio=",ratio
if ratio<-1:
active.remove(a)
print "active point",a,"removed"
else:
update_cost(a,w,ratio)
report_cost(w)
active.append(w)
print
The only thing different between various strategies is how the cost of a line break is computed by update_cost(a,w,ratio).
Exercise 33. Not every word should be added to the active list. For
instance, for any realistic line length, the second word in the paragraph
will not have a valid breakpoint after it, so we need not consider it.
Take the model implementation and add this modification. Measure the
improvement in running time, for instance by counting the number of
calls to some inner routine. Give a theoretical argument for how this
reduces the complexity of the algorithm.
TEX – LATEX – CS 594
3.4. TEX’S LINE BREAKING ALGORITHM
3.4.3.1
105
First fit implementation
Since at first we are looking only locally, for each breakpoint we only keep track of the
cost and the previous breakpoint that the cost was computed from. Here we set up the
data structure cost. Element cost[w] describes the cost of breaking after word w; the
’from’ component is the number of the first word of the line.
def init_costs():
global cost
cost = len(paragraph)*[0]
for i in range(len(paragraph)):
cost[i] = {’cost’:0, ’from’:0}
cost[0] = {’cost’:10000, ’from’:-1}
The essential function is the cost computation. In first fit we accept any stretch or shrink
that is |ρ| < 1.
def update_cost(a,w,ratio):
global cost
if a>0 and cost[a-1][’cost’]<10000:
if ratio<=1 and ratio>=-1:
to_here = abs(ratio)
else: to_here = 10000
if cost[w][’cost’]==0 or to_here<cost[w][’cost’]:
cost[w][’cost’] = to_here; cost[w][’from’] = a-1
(The first test serves to make sure that the previous point being considered is in fact a valid
breakpoint.)
Here is the remaining function that constructs the chain of breakpoints:
def final_report():
global cost,nwords,paragraph
print "Can break this paragraph at cost",\
cost[nwords-1][’cost’]
cur = len(paragraph)-1; broken = []
while cur!=-1:
prev = cost[cur][’from’]
line = paragraph[prev+1:cur+1]
broken.insert(0,line)
cur = prev;
set_paragraph(broken)
A small example text, faked in monospace:
You may never have thought of it, but fonts (better:
typefaces) usually have a mathematical definition somehow.
If
a
font
is
given
as
bitmap,
this
is often
a result originating from a more compact description.
Imagine the situation that you have bitmaps at 300dpi, and
you
buy
a 600dpi printer. It wouldn’t look pretty.
There
is
then
a
need
for a mathematical way of
describing arbitrary shapes. These shapes can also be
three-dimensional; in fact, a˜lot of the mathematics in
Victor Eijkhout
-0.111111111111
-0.666666666667
0.888888888889
0.0
-0.777777777778
0.25
0.555555555556
0.0
-0.285714285714
106
CHAPTER 3. BREAKING THINGS INTO PIECES
this chapter was developed by a car manufacturer for
modeling
car
body shapes. But let us for now only
look
in two dimensions, which means that the curves
are lines, rather than planes.
0.0
0.222222222222
0.125
We see ugly stretched break in line 3, especially after the compressed line 2. However, both
of them fit the test.
It is in fact simple to turn this into a dynamic programming solution that considers a global
minimum:
def update_cost(a,w,ratio):
global cost
if ratio<=1 and ratio>=-1:
to_here = abs(ratio)
else: to_here = 10000
if a>0:
from_there = cost[a-1][’cost’]
to_here = to_here+from_there
else: from_there = 0
if cost[w][’cost’]==0 or to_here<cost[w][’cost’]:
cost[w][’cost’] = to_here; cost[w][’from’] = a-1
3.4.3.2
Best fit
In the best fit strategy, we compute badness from the stretch/shrink ratio. This involves only
a slight change in the cost computation function:
def update_cost(a,w,ratio):
global cost
to_here = 100*abs(ratio)**2
if a>0:
from_there = cost[a-1][’cost’]
to_here = to_here+from_there
else: from_there = 0
if cost[w][’cost’]==0 or to_here<cost[w][’cost’]:
cost[w][’cost’] = to_here; cost[w][’from’] = a-1
The same text:
You may never have thought of it, but fonts (better:
typefaces) usually have a mathematical definition somehow.
If
a
font
is
given
as bitmap, this is often a
result
originating
from
a more compact description.
Imagine the situation that you have bitmaps at 300dpi, and
you
buy
a 600dpi printer. It wouldn’t look pretty.
There
is
then
a
need
for a mathematical way of
describing arbitrary shapes. These shapes can also be
three-dimensional; in fact, a˜lot of the mathematics in
this chapter was developed by a car manufacturer for
modeling
car
body shapes. But let us for now only
look
in two dimensions, which means that the curves
are lines, rather than planes.
-0.111111111111
-0.666666666667
0.5
0.5
-0.777777777778
0.25
0.555555555556
0.0
-0.285714285714
0.0
0.222222222222
0.125
TEX – LATEX – CS 594
3.4. TEX’S LINE BREAKING ALGORITHM
107
While there are no lines stretched with ρ >, the quadratic function has improved the break
in line 3.
3.4.3.3
Total fit
For the algorithm that TEX uses, we have to distinguish between lines that are tight, decent,
loose. This makes our datastructure more complicated:
def init_costs():
global cost
nul = [0,0,0]
cost = len(paragraph)*[ 0 ]
for i in range(len(paragraph)):
cost[i] = nul[:]
for j in range(3):
cost[i][j] = {’cost’:10000, ’from’:-2}
for j in range(3):
cost[0][j] = {’cost’:10000, ’from’:-1}
An element cost[i] is now an array of three possible breakpoints, one in each of the
classifications. An actual breakpoint is now in cost[word][type][’from’] and
cost[word][type][’cost’].
The cost computation becomes more complicated:
def minimum_cost_and_type(w):
global cost
c = 10000; t = 0
for type in range(3):
nc = cost[w][type][’cost’]
if nc<c:
c = nc; t = type
return [c,t]
def update_cost(a,w,ratio):
global cost
type = stretch_type(ratio)
to_here = 100*abs(ratio)**2
if a>0:
[from_there,from_type] = minimum_cost_and_type(a-1)
to_here += from_there
else: from_there = 0
if cost[w][type][’cost’]==0 or\
to_here<cost[w][type][’cost’]:
cost[w][type][’cost’] = to_here;
cost[w][type][’from’] = a-1
Exercise 34. The total fit code does not yet contain the equivalent of
TEX’s \adjdemerits. Add that.
Let us look at the same test text again:
Victor Eijkhout
108
CHAPTER 3. BREAKING THINGS INTO PIECES
You may never have thought of it, but fonts (better:
typefaces)
usually
have
a
mathematical
definition
somehow. If a font is given as bitmap, this is often a
result
originating
from
a more compact description.
Imagine
the
situation
that
you
have
bitmaps
at
300dpi, and you buy a 600dpi printer. It wouldn’t look
pretty. There is then a need for a mathematical way of
describing arbitrary shapes. These shapes can also be
three-dimensional; in fact, a˜lot of the mathematics in
this chapter was developed by a car manufacturer for
modeling
car
body shapes. But let us for now only
look
in two dimensions, which means that the curves
are lines, rather than planes.
-0.111111111111
1.2
-0.454545454545
0.5
1.0
-0.333333333333
-0.4
0.0
-0.285714285714
0.0
0.222222222222
0.125
In this output, line 2 is stretched further than before, to prevent lower badnesses later.
Exercise 35. Add the functionality for hanging indentation to this code.
Exercise 36. (bonus point exercise) TEX has the possibility of forcing
a paragraph to be a line longer or shorter than optimal. Implement that.
3.4.3.4
Utility parts
File header: we read a text and store it.
#! /usr/bin/env python
import sys
max_line_length = 60
paragraph = []
while 1:
try:
a = raw_input()
paragraph.extend(a.split())
except (EOFError):
break
In order to fake stretch and shrink with a monospace font, we let a ‘space’ be two spaces
by default.
def line_length(words):
l = 2*(len(words)-1)
for w in words:
l += len(w)
return l
#
# ratio = -1 : shrink each double space to one
# ratio = 1 : stretch each double space to three
#
def compute_ratio(line):
spaces = len(line)-1
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109
need = 1.*(max_line_length-line_length(line))
#print "ratio:",need,spaces
if spaces==0: return 10000
else: return need/spaces
Output formatting with this idea:
def set_paragraph(para):
for l in range(len(para)-1):
line = para[l]
set_line(line)
set_last_line(para[len(para)-1])
def set_line(line):
shortfall = max_line_length-line_length(line)
for w in range(len(line)-1):
sys.stdout.write(line[w])
if shortfall>0:
sys.stdout.write(’
’); shortfall = shortfall-1
elif shortfall<0:
sys.stdout.write(’ ’); shortfall = shortfall+1
else:
sys.stdout.write(’ ’)
sys.stdout.write(line[len(line)-1])
print " ",compute_ratio(line)
def set_last_line(line):
for w in range(len(line)-1):
sys.stdout.write(line[w])
sys.stdout.write(’ ’)
sys.stdout.write(line[len(line)-1])
print
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NP completeness.
3.5
Introduction
The ‘NP’ stands for ‘nondeterministic polynomial time’, which stands for the fact that
a solution can be checked (not: found) in polynomial time. This class of algorithms is
informally characterized by the fact there is polymomial time for checking their solution.
However, it is also true that there is no polynomial algorithm known for solving them.
The fact that there is no efficient algorithms known would not be bad if it could be proved
that no efficient algorithm exists.
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111
However, also there exists no non-polynomial lower bound on the solution time. Thus, the
question whether they can be solved in polynomial time is still open. Since methods in this
class can all be translated into each other, having a solution method for one implies that
methods exist for all of them. This also means that none of the problems in this class have
been solved so far.
3.6
Basics
3.6.1
Optimization versus decision problems
Many problems that are said to be NP-complete are optimization problems. For instance, in
the traveling salesman problem the shortest route through a set of cities is asked. However,
it is more convenient to look at decision problems, that is, problems that have a yes or no
answer for each input.
It is not hard to transform an optimization problem into a decision problem. Rather than
asking for an optimal solution, we determine a bound B, and ask whether there is a solution
that is within that bound.
Exercise 37. Finish this argument. Show that, if we can solve the optimization problem, we can solve the decision problem. Now, supposing
we can solve the decision problem, how does that give a solution of
the optimization problem? Assume that the outcome of the optimization
problem is an integer quantity. Do you have to make other assumptions;
discuss? What is the complexity of the one solution method given a certain complexity for the other?
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CHAPTER 3. BREAKING THINGS INTO PIECES
Language of a problem
For each optimization or decision problem we can defined ‘instances’, which are ways of
setting all the free variables of the problem. Since these variables are in sets of types that
depend on the problem, we can not be more precise here. A problem can then be phrased as
a question over this set of instances: which instances optimize the cost function, or which
give a yes answer. That last set we will denote YΠ .
Again depending on the problem, we can encode instances of a problem. For instance, in
the traveling salesman problem, an instance would be encoded as the ordered list of cities
to pass through.
With this, we can define the language of a problem:
L[Π, e] = {the instances in YΠ encoded under e}
3.6.3
Turing machines
A Turing machine, given some input, can halt with the yes state qY , the no state qN , or can
never halt. We say that a string is accepted if it leads the Turing machine to halt with qY .
The language LM of a Turing machine M is the set of strings that are accepted.
A deterministic Turing machine (DTM) M is said to solve a problem Π (under some encoding e), or equivalently to recognize L[Π, e], if
•
•
it halts for all strings over its input alphabet, and
its language LM is L[Π, e].
Note that ‘solving a problem’ here actually means ‘recognizing a solution of the problem’.
This DTM is a solution checker, not a solution generator.
As an example, consider the recast the traveling salesman problem ‘does a route, shorter
than B, exist?’. The set of purported solutions are then lists of cities, and the DTM gives
for each list a verdict ‘yes, this route is shorter than B’ or ‘no, this route is not shorter
than B’.
3.7
Complexity classes
3.7.1
Class P
This allows us to define class P :
P = {L : there is DTM that recognizes L in polynomial time}
and with this
Π∈P
≡ L[Π, e] ∈ P for some encoding e
≡ there is a polynomial time DTM that recognizes L[Π, e]
What this means is that for problems in P there is a polynomial time Turing machine that
recognizes strings in YΠ as valid, and that on strings not in YΠ it both halts, and gives a
negative verdict.
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3.7.2
113
Class NP
Solving the traveling salesman problem may be hard, but if we have a network and some
bound, and someone gives us an itinerary with the claim that it satisfies that bound, we can
check this with a DTM in polynomial time. We can now build a non-deterministic Turing
machine (NDTM) which ‘solves’ such problems: given a decision problem it ‘guesses’
some purported solution, and then checks (with a DTM) whether that solves the decision
problem. The guessed solution is called a ‘certificate’.
Clearly, only if the decision problem has an answer of ‘true’ can the NDTM guess a solution, so the YΠ of this problem is precisely the set of decision problem instances with a yes
answer.
For instance, for the traveling salesman problem, the instances in YΠ are a combination
of a cities network plus a feasible bound on the travel time. The non-deterministic Turing
machine would then guess an itinerary, and it can be checked in polynomial time that that
is indeed a valid route, and that is satisfies the bound.
We can write this whole story compactly: a problem Π is in NP if there is a polynomial
time function A(·, ·) such that
w ∈ YΠ ⇔ ∃C : A(w, C) = true
and C itself can be polynomially generated.
The final condition on the generation of the certificate is needed for a total polynomial
runtime, but in practice it is not much of a limitation. For instance, in the traveling salesman
problem, a list of cities can be guessed in linear time.
Exercise 38. Prove that NP is closed under union and intersection.
What difficulty is there in showing that it is closed under complement
taking?
3.7.3
Examples
As was indicated above, while finding a solution to a problem is often hard, checking that
something is a solution is often fairly simply and doable in polynomial time. A nontrivial example of a polynomial time problem is checking whether a number is prime. This
question was only settled in 2002. Before that there were polynomial time probabilistic
testers, which would test a number and return a verdict with high reliability, or with a high
probability of polynomial running time.
Exercise 39. Why is the following algorithm not a linear time solution
to the P RIME√problem?
for i = 0 . . . n:
if mod(n, i) ≡ 0
return true
Other algorithms have provably an exponential running time. Examples here are finding
the best move in chess, or checking statements in Pressburger arithmetic.
It is possible to find levels in between polynomial and exponential. The problem of factoring an integer (note that this is more powerful than primality testing) has a runtime of
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CHAPTER 3. BREAKING THINGS INTO PIECES
O(exp((n · 64/9)1/3 )(log n)2/3 ). Interestingly, on a quantum computer, a polymial algorithm is known; see http://en.wikipedia.org/wiki/Shors_algorithm.
In the next section we will go further into the middle ground, of algorithms for which no
polymomial time algorithm is known, but for which no exponential bound is known either.
3.8
NP-completeness
3.8.1
Transformations
Often it is possible to transform one problem into another. This has the advantage that, if
you prove that one problem is in a certain class, the other one is also in that class. Of course
this depends on the nature of the transformation.
We will here consider ‘polynomial
transformations’. Let L1 and L2 be the languages of two
P∗
P
∗
problems over alphabets 1 and 2 respectively, then f is a polymomial transformation
of problem 1 into problem 2 if
•
•
There is a DTM that computes f (x) in time Tf (x) ≤ pf (|x|) for some polynomial pf , andP
∗
For all x ∈ 1 , x ∈ L1 iff f (x1 ) ∈ L2 .
The transformation does not have to be a one-to-one mapping, so it is sometimes explicitly
terms a ‘many-to-one polynomial transformation’.
Lemma 1 Suppose f is a polynomial transformation from L1 to L2 , then
L2 ∈ P ⇒ L1 ∈ P
Proof: assume that M2 : L2 → {0, 1} is a DTM that recognizes L2 , then M2 ◦ f is a
DTM that recognizes L1 , and this composition still runs in polynomial time TM2 ◦f (x) ≤
pT2 (|pf (|x|)|).
If L1 transforms to L2 (in polynomial time), we notate that as L1 ≤ L2 . This notation also
suggests the idea that L1 is easier than L2 .
It is easy to see that
L1 ≤ L2 ∧ L2 ≤ L3 ⇒ L1 ≤ L3 ,
that is, the ‘transforms into’ relation is transitive.
3.8.2
NP-complete
A language L is said to be NP-complete if
•
•
L ∈ N P , and
for all L0 ∈ N P : Ł0 ≤ L
(Languages that satisfy the second clause but not the first are called ‘NP-hard’. One example is the halting problem, which is known not to be decidable. In other words, the DTM
that should recogize the language does not always halt with yes or no.)
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115
Informally, the class NP-complete is that of the problems where a solution can be verified
quickly (meaning, in polynomial time). On the other hand, P is the class of problems where
the solution can be computed quickly. The question whether these classes are disjoint is
open. In fact, you can win a million dollars by settling it one way or another.
Lemma 2 If L1 , L2 ∈ N P , L1 is NP-complete, and L1 ≤ L2 , then L2 is NP-complete.
Proof: the only thing we need to check is that every L0 ≤ L2 for all L2 ∈ N P . Since L1 is
NP-complete, L0 ≤ L1 . Now use the transitivity of the transform relation.
3.8.3
Proving NP-completeness
Above we saw that, given one NP-complete problem, others can easily be proved NPcomplete by constructing a mapping between the one problem and the other. This raises a
bootstrapping question.
Stephen Cook was the first to prove the NP-completeness of any problem (1971), in his
case the satisfiability problem. This is the problem of, given boolean variables x1 . . . xn
and a logical formula F (x1 , . . . , xn ), deciding whether there is a way of specifying the
variables such that the result is true.
Examples: the formula x1 ∨ =
6 x1 is always true; x1 ∧ =
6 x1 is always false, and x1 ∧ =
6 x2
is only true for the pair (x1 = T, x2 = F ). For the first and third examples, there are values
of x1 , x2 for which the formula is true, so these are satisfiable. The second example is not
satisfiable.
The Boolean satisfiability problem is in NP because a non-deterministic Turing machine
can guess an assignment of truth values to the variables, determine the value of the expression under that assignment, and accept if the assignment makes the entire expression
true.
Now, to prove that the satisfiability problem is NP-complete, it remains to show that any
language L ∈ N P can polynomially be transformed into it. This is done by assuming a
NDPT Turing machine for L, and transforming it into a logical formula, so that there is
a correspondence between successful computation in the NDTM, and a satisfied logical
formula.
Let the Turing machine be
M = hQ, s, Σ, F, δi
where
Q is the set of states, and s ∈ Q the initial state,
Σ the alphabet of tape symbols,
F ⊂ Q the set of accepting states, and
δ ⊂ Q × Σ × Q × Σ × {−1, +1} the set of transitions,
and that M accepts or rejects an instance of the problem in time p(n) where n is the size
of the instance and p(·) is a polynomial function.
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We describe for each instance I a Boolean expression which is satisfiable if and only if the
machine M accepts I.
The Boolean expression uses the variables set out in the following table, where q ∈ Q,
−p(n) ≤ i ≤ p(n), j ∈ Σ, and 0 ≤ k ≤ p(n):
Variables
Tijk
Hik
Qqk
Intended interpretation
True iff tape cell i contains symbol j
at step k of the computation
True iff the M ’s read/write head is
at tape cell i at step k of the computation.
True iff M is in state q at step k of
the computation.
How many
O(p(n)2 )
O(p(n)2 )
O(p(n))
Define the Boolean expression B to be the conjunction of the clauses in table ??, for all
−p(n) ≤ i ≤ p(n), j ∈ Σ, and 0 ≤ k ≤ p(n).
For all:
initial conditions
Tape cell i of the
input I contains
symbol j.
Add the clauses
Interpretation
How many
clauses?
Tij0
Initial contents of the tape.
O(p(n))
Qs0
H00
Initial state of M
Initial
position
read/write head.
O(1)
O(1)
physical constraints
symbols j 6= j 0
Tijk → ¬Tij 0 k
0
states q 6= q
Qqk → ¬Qq0 k
0
cells i 6= i
Hik → ¬Hi0 k
Turing machine basics
i, j, k
Tijk = Tij(k+1) ∨ Hik
f ∈F
transition table
(q, σ, q 0 , σ 0 , d) ∈
δ
The disjunction
clauses Qf,p(n)
of
the
The disjunction of the
clauses
(Hik ∧ Qqk ∧ Tiσk ) →
(H(i+d)(k+1) ∧ Qq0 (k+1) ∧
Tiσ0 (k+1) )
of
One symbol per tape cell.
Only one state at a time.
Only one head position at a
time.
O(p(n)2 )
O(p(n))
O(p(n))
Tape remains unchanged
unless written.
Must finish in an accepting
state.
O(p(n)2 )
Possible transitions at
computation step k when
head is at position i.
O(p(n)2 )
O(1)
Table 3.1: Translation table from a NDPT Turing machine to a logic formula
This table describes how to construct a logical formula in the variables Tijk , Hik , Qqk
(describing tape contents, head positions, and states, respectively) that corresponds to the
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117
Turing machine. If there is an accepting computation for M on input I, then B is satisfiable,
by assigning Tijk , Hik and Qik their intended interpretations. On the other hand, if B is
satisfiable, then there is an accepting computation for M on input I that follows the steps
indicated by the assignments to the variables.
How large is B? There are O(p(n)2 ) Boolean variables, each of which may be encoded in
space O(log p(n)). The number of clauses is O(p(n)2 ). So the size of B is O((log p(n))p(n)2 ).
This is polynomial in n, the size of the input, so the transformation is certainly a polynomialtime reduction, as required.
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Page breaking.
3.9
Introduction
TEX’s page breaking algorithm is simpler than the line breaking one. The reason for this
is probably that global optimization of breakpoints, the way it is done in the paragraph
algorithm, would take prohibitively much memory, at least, for the amount of memory that
computers around 1980 had. The algorithm used is related to the ‘best fit’ algorithm we
discussed in the line breaking chapter.
Theoretically, page breaking is a more complicated problem than line breaking. We will
explore the issues in section 3.11, but first we will briefly go into the algorithms that TEX
actually uses.
3.10
TEX’s page breaking algorithm
The problem of page breaking has two components. One is that of stretching or shrinking
available glue (mostly around display math or section headings) to find typographically
desirable breakpoints. The other is that of placing ‘floating’ material, such as tables and
figures. These are typically placed at the top or the bottom of a page, on or after the first
page where they are referenced. These ‘inserts’, as they are called in TEX, considerably
complicate the page breaking algorithms, as well as the theory.
3.10.1
Typographical constraints
There are various typographical guidelines for what a page should look like, and TEX has
mechanisms that can encourage, if not always enforce, this behaviour.
1.
2.
3.
4.
The first line of every page should be at the same distance from the top. This
changes if the page starts with a section heading which is a larger type size.
The last line should also be at the same distance, this time from the bottom. This
is easy to satisfy if all pages only contain text, but it becomes harder if there are
figures, headings, and display math on the page. In that case, a ‘ragged bottom’
can be specified.
A page may absolutely not be broken between a section heading and the subsequent paragraph or subsection heading.
It is desirable that
(a) the top of the page does not have the last line of a paragraph started on the
preceding page
(b) the bottom of the page does not have the first line of a paragraph that continues on the next page.
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3.10. TEX’S PAGE BREAKING ALGORITHM
3.10.2
119
Mechanisms
The basic goal of page breaking in TEX is to fill up a box of height \vsize. The is the
goal size of the material without header and footer lines. The box is constructed by adding
material to the vertical list until an optimal breakpoint is found. The material before that
breakpoint is then put in \box255, and the code in \output, the ‘output routine’ is
executed. The command to send a box to the output file is \shipout.
The typographical rules in the previous section can be realized fairly simply in TEX.
1
The vertical location of the first line on a page is controlled by \topskip. If the
baseline of the first line is closer to the top than this parameter, glue is inserted to
make up for the difference.
2
The vertical location of the last line of a page is controlled by \maxdepth. If
the last line of the page is deeper than this amount, the reference point of the box
is shifted down accordingly.
3
Preventing page breaks between vertical material of different kinds can be done
by the proper use of penalties and glue.
4a
A break after the first line of a paragraph is prevented by setting the \clubpenalty.
4b
A break before the last line of a paragraph is prevented by setting the \widowpenalty.
3.10.3
Breakpoints
TEX builds up a current page that is a vertical list of material. It regularly tests whether
there is enough material to fill a box of height \vsize while incurring a badness less
than 10, 000. The breakpoints are similar to those in the line breaking algorithm, mostly
occurring at a penalty, or at a glue that follows non-discardable material.
3.10.4
Some output routines
The very simplest output routine simply takes the vertical list and ships it to the output file:
\output={\shipout\box255}
Slighly more sophisticated, a header and footer are added:
\output={
\setbox255=\vbox{ <header>
\box255
<footer>
}
\shipout\box255
}
The following example makes the page one line longer if a widow (break before the last line
of a paragraph) occurs. First we save the original \vsize and we declare a recognizable
value for the \widowpenalty:
\newif\ifEnlargedPage \widowpenalty=147
\newdimen\oldvsize \oldvsize=\vsize
The output routine now works by testing for the widow penalty, and if it is found, increasing
the \vsize and returning the page material to the list by \unvbox255:
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CHAPTER 3. BREAKING THINGS INTO PIECES
\output={
\ifEnlargedPage <output the page>
\else \ifnum \outputpenalty=\widowpenalty
\global\EnlargedPagetrue
\global\advance\vsize\baselineskip
\unvbox255 \penalty\outputpenalty
\else \shipout\box255
\fi
\fi}
Here is the missing bit of code that outputs the enlarged page:
\ifEnlargedPage \shipout\box255
\global\LargePagefalse
\global\vsize=\oldvsize
3.10.5
Insertions
Floating material, such as tables and figures, are handled by a mechanism called ‘insertions’. Insertions fall in different classes, and insertion material is specified by
\insert<class number>{ <material> }
If the class number is n, then
•
•
•
When the output routine is active, \boxn contains the material of insertion class n.
\dimenn specifies the maximum amount of insertion material that is allowed
to be placed on one page. If there is more material, it is split with the remainder
carried over to the next page.
There are further fine points to this algorithm.
Insertions are thus added, for as far as possible, to the page being formed when the \insert
command is given. TEX has no way of moving insertions to an earlier page, although moving material to a later page – presumable where more space is available – is possible.
3.11
Theory of page breaking
At first glance, the page breaking problem is much like the line breaking problem, except
with larger basic blocks, and vertically instead of horizontally. In the line breaking problem,
a long list of words is broken into lines, adjusting the margins by stretching or shrinking
the interword space. In the page breaking problem, a vertical list of lines, display formulas,
and other vertical material, is broken into equal sized pages, using the various amounts of
vertical space for taking up the slack.
However, the page breaking problem becomes much harder if we include the possibility of
figures and other floating material. In that case, the computed badness (the measure we try
to minimize in the breaking process) will include reflect that we want a figure to be close
to pages that reference it, or satisfy other ordering rules involving references to it. Maybe
surprisingly, even rather simple cost functions make page breaking into an NP-complete
problem.
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3.11. THEORY OF PAGE BREAKING
121
To get an appreciation of the issues, consider this sequence of pages with figures:
References to the figures are here indicated with parenthesized numbers. We see that out
of 5 references, 3 are not to a figure on the same page. However, if we move one line from
page 1 to 2, and move figure 3 forward a page, we get:
where we see that only one reference is not to the same page. Unfortunately this is a backward reference.
In the remainder of this chapter we will investigate theoretical aspects of functions that try
to optimize placement of floating material in a global fashion. It should be noted that this is
considerably more sophisticated than what is implemented in TEX. The available algorithm
is closer to a ‘first fit’ strategy.
We will investigate two closely related page breaking algorithms. We show how a particular form of the page breaking problem (the ‘MQ’ problem) is equivalent to the 2satisfyability problem, which is known NP-complete. As is usual in the context of proving
NP-completeness, we transform our minimization problem ‘give the solution with minimum cost’ into a decision problem by asking ‘is there a solution with a cost ≤ B’ where
B is some predetermined bound.
The very similar ‘ML’ problem, which only differs in the form of the badness function,
does have a polynomial time solution.
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CHAPTER 3. BREAKING THINGS INTO PIECES
The MQ problem is NP-complete
We consider the MQ page breaking problem: Multiply referenced figures, and Quadratic
badness measure. Under the simplifying assumptions that each figure takes a whole page,
we then have a set T = {t1 , . . . , tN } of text blocks and a set F = {f1 , . . . , fN } of figures
and a function W : T × F such that W (ti , fj ) describes how many times text block i
references figure j. We assume a bound W (ti , fj ) ≤ q(N ) (where q is a polynomial)
dependent on the size of the problem.
The MQ problem is now the question whether there is an page ordering of text blocks and
figures, that is, a mapping P : (T ∪ F ) → {1, 2, . . . , 2N } such that
P (ti ) < P (tj )
P (fi ) < P (fj )
∀1≤i<j≤N
and so that
S=
X
W (ti , fj )(P (ti ) − P (fj ))2 ≤ B
i,j
In order to show that this problem is NP-complete, we show that it can be transformed into
an instance of the maximum 2-satisfiability problem. This means that solving the one problem is equivalent to solving the other, and since the transformation is done in polynomial
time, the two problems have the same complexity.
The maximum 2-satisfiability (MAX 2- SAT problem can be formulated as follows. Let there
be given n binary variables x1 , . . . , xn and m clauses {u1 ∨ v1 , . . . , um ∨ vm }, where
ui = xj or ui = ¬xj for some j. Given a bound K, is there a way of setting the xi variables
such that at least K of the clauses are satisfied? This problem is known to be NP-complete.
We will now make a pagination problem that ‘translates’ an instance of the 2-satisfyability
problem. That is, given a configuration of binary variables and clauses and a bound K
to satisfy, we will make a pagination problem with bound B such that the one bound is
satisfied if and only if the other is. Then, since MAX 2- SAT is NP-complete, it follows that
this particular pagination problem is also NP-complete.
3.11.1.1 Construction of the translation
We make the translation between the two types of problems by constructing the page assignment function P , and the weight function W . There are three aspects we need to take
care of, so we let B = B1 + B2 + B3 , and we will determine the Bi bounds separately.
First of all we set W (ti , fi ) = b1 sufficiently large that only configuration with |P (ti ) −
P (fi )| = 1 will satisfy the bound. (Recall that the badness is a sum of W (ti , fj )(P (ti ) −
P (fj ))2 terms.) To allow for the pages to be ordered this way, we let B1 = N b1 . The b1
quantity will be defined in terms of B2 and B3 as
b1 = d(B2 + B3 )/3e + 1
Supposing that at least one ti , fi pair is not adjacent, then it follows from this bound that
the badness will be
S ≥ (N − 1)b1 + 22 b1 = (N + 3)b1 > B
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3.11. THEORY OF PAGE BREAKING
123
where the N − 1 corresponds to the pairs that are adjacent, and the 22 to at least one pair
at distance 2.
Since text blocks and are next to each other, the only remaining question is which comes
first. This limits the number of cases we need to consider in the remainder of this proof.
Now let parameters n for the number of variables and m for the number of clauses be
as described above, then our pagination problem will have N text blocks and figures,
where N = 2n + 2m. The first 4n pages encode the values of the variables xi , with
each consecutive 4 pages corresponding to one variable:
4i − 3 4i − 2 4i − 1 4i
t2i−1
f2i−1 f2i
t2i
f2i−1 t2i−1
t2i
f2i
if xi is true
if xi is false
To ensure that these are the only configuration that will satisfy the bound, set W (t2i−1 , f2i ) =
W (f2i−1 , t2i ) = b2 large enough. Either of the above patterns then contributes 2 · 22 b2 =
8b2 , while the other possibilities (t f t f and f t f t) would contribute (12 + 32 )b2 = 10b2 .
P
Correspondingly, we allow a bound of B2 = 4b2 (i − j)2 where i, j range over the pairs
that satisfy W (ti , fj ) = b2 . Defining
b2 = 8(m − k) + 5
is sufficient to make violation of this condition outweigh the gain from more clauses being
satisfied.
Next, the 4m remaining pages will encode the clauses in a similar manner:
4n + 4j − 3 4n + 4j − 2
t2n+2j−1
f2n+2j−1
f2n+2j−1
t2n+2j−1
4n + 4j − 1
4n + 4j
if uj is true
if uj is false
t2n+2j
f2n+2j
if vj is true
f2n+2j
t2n+2j
if vj is false
Further conditions on W ensure that the uj variables indeed correspond to the proper xi .
For instance
W (t2n+2j−1 , f2i ) = W (t2i = f2n+2j−1 ) = b2
if uj = xi
2
This term contributes 2d b2 to the badness, where d is twice the difference between the
subscripts, in this case d = (2n + 2j − 2i). With a mismatch, a t and f page assignment
are reversed, so the contribution becomes (d − 1)2 + (d + 1)2 = 2(d2 + 1)b2 .
Proper truth values of the clauses are enforced as follows. Observe that the combination
where uj and vj are both false is the only one that gives a false result. This corresponds to
the pagination
f2n+2j−1
t2n+2j−1
f2n+2j
t2n+2j
In this configuration f2n+2j1 and t2n+2j are spread the furthest apart, so we penalize that
with
W (t2n+2j , f2n+2j1 ) = 5,
W (t2n+2j−1 , f2n+2j ) = 3.
This gives a contribution of 32 for the three true cases, and 48 for the false result. Correspondingly, to allow for K clauses to be satisfied we allow B3 = 48(m − K) + 32K.
Victor Eijkhout
124
CHAPTER 3. BREAKING THINGS INTO PIECES
Finally, by defining the polynomial q(i) = 64(i + 1)4 , we have q(N ) > b1 ≥ b2 > 5 > 3,
so W is bounded as required.
3.11.1.2 NP-completeness
In order to establish NP-completeness of the problem MQ, we need to establish that something is a true instance of M AX 2- SAT iff the translated instance is true for MQ.
Given a truth assignment of the xi s that makes K clauses true, we can now construct a
pagination P with a satisfied bound of B.
Conversely, let a pagination with bound B be given, then the corresponding truth assignment makes K clauses true. To see this, inspect each of the details of the translation between the problems, and observe that any violation will cause the bound B to be exceeded.
3.11.2
The ML problem has a polynomial time solution
The ‘ML’ problem (Multiply referenced figures, Linear badness function) is identical to
MQ, except for the form of the badness function. Having a linear badness function makes
it possible to solve this problem by dynamic programming in linear time.
As in MQ, we have text blocks ti and figures fj that take up whole pages. We generalize
the problem slightly to having different numbers of text and figure blocks:
T = {t1 , . . . , tN },
F = {f1 , . . . , fM }
The function W : T × F is such that W (ti , fj ) ≥ 0 describes how many times text block i
references figure j.
The ML problem is now the question whether, given the above, and given a bound B,
there is an page ordering of text blocks and figures, that is, a mapping P : (T ∪ F ) →
{1, 2, . . . , M + N } such that
P (ti ) < P (tj )
∀1≤i≤N,1≤j≤M
P (fi ) < P (fj )
and so that
X
S=
W (ti , fj ) |P (ti ) − P (fj )| ≤ B
i,j
3.11.2.1 Dynamic programming solution
The key to a dynamic programming solution of ML is to identify subproblems. The subproblem we consider is
Given i text blocks and j figures, what is the least badness of placing
these on the first i + j pages. Call this partial badness Bij .
The problem here is the ‘dangling’ references (tr , fs ) with r > i, s ≤ j or r ≤ i, s > j.
The measure Ri,j is defined as the number of dangling references after these blocks and
figures have been placed.
A dangling reference is either
TEX – LATEX – CS 594
3.11. THEORY OF PAGE BREAKING
125
A forward reference: A text block refering to a figure not yet placed. The number of
forward references
X from the i + j block is
Fij =
W (tr , fs )
1≤r≤i
j<s≤M
A backward reference: A figure that will be reference on a text block not yet placed.)
The number ofX
backward references from the i + j block is
Bij =
W (tr , fs )
i<r≤N
1≤s≤j
which makes Rij = Fij + Bij .
F
(i,j)
s
U
B
r
Figure 3.1: The Fij , Bij , and Uij areas in (r, s) space
For badness calculations involving dangling references, we count only the part to the
boundary of the i, j region. Formally:
(1)
(2)
Bij = Bij + Bij
where
(1)
Bij =
X
W (tr , fs ) |P (tr ) − P (fs )|
r≤i
s≤j
is the part of the badness due to references that are fully resolved within the pages already
placed; the part of the badness due to dangling references is
(1)
(2)
Figure 3.2: Resolved and dangling references of a block of pages
Victor Eijkhout
126
CHAPTER 3. BREAKING THINGS INTO PIECES
(2)
Bij =
X
W (tr , fs ) `(i, j; r, s) +
X
r>i
s≤j
W (tr , fs ) `(i, j; r, s)
r≤i
s>j
where
i + j − P (fs ) if r > i
i + j − P (tr ) if s > j
describes the part of the arc between tr and f2 that lies in the first i + j pages. These two
types of arcs are illustrated in figure 3.2.
`(i, j; r, s) =
Figure 3.3 illustrates how reference arcs change status when we go from i + j − 1 to i + j
pages, say by placing text block ti :
(2)
(1)
(3)
i+j-1
i+j
Figure 3.3: Change in status of resolved and dangling references upon extending a block of
pages
References that were unresolved with references originating in ti move their contribution from the B (2) term to the B (1) term. (Note that a reference to a page one
location outside the current block is already fully counted in the badness.)
(2)
DanglingP
references thatP
stay unresolved increase their contribution to the B (2)
term by ( r≤i−1,s>j + r>i−1,s≤j )W (tr , fs )
(3)
Arcs that used to fall completely outside the P
page block, but that are now dangling
in the new page block, add a contribution of r=i,s>j W (tr , fs ) to the B (2) term.
P
r>i,s≤j W (tr , fs ) In sum, Bij = Bi−1,j + Rij . The same story holds for extending
i + j − 1 pages by placing figure fj , so we have the recurrence
Bij = min(Bi−1,j , Bi,j−1 ) + Rij .
We still need to compute the Rij quantities, describing the number of dangling references
from placing i text blocks and j figures. Above we saw Rij = Fij + Bij . For efficient
calculation of these sums, it is convenient to make a table of
X
Uij =
W (tr , fs )
(1)
1≤r≤i
1≤s≤j
which takes time O(N M ), that is, quadratic in the total number of pages. Then
Rij = UiM + UN j − 2Uij ,
as is easily seen from figure 3.1.
3.11.3
Discussion
It is interesting to note that certain details of the NP-completeness proof of MQ rely on the
quadratic badness function, rather than on more ‘structural’ properties of the problem.
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3.11. THEORY OF PAGE BREAKING
127
Exercise 40. Find a place in the NP-completeness proof of MQ that
uses the quadratic badness function, and show that the underlying fact
does not hold for linear functions. Does it hold for other functions than
quadratic?
Similarly, ML only has a dynamic programming solution thanks to the linear badness function.
Exercise 41. Explain how the linearity of the badness function is essential for the dynamic programming solution of ML.
Exercise 42. The discussion of the ML problem only gave the cost
computation. How can the actual page assignment be derived from the
given construction? What is the time and space complexity of this part
of the algorithm?
Victor Eijkhout
128
CHAPTER 3. BREAKING THINGS INTO PIECES
Projects for this chapter.
Project 3.1. What is the paragraph breaking algorithm of OpenOffice? Replace by TEX’s
algorithm.
Project 3.2. Write a paragraph breaking algorithm that prevents ‘rivers’ in the text.
Project 3.3. TEX’s line breaking algorithm is not just for good looks. Many aesthetic decisions in typography actually influence readability of the document. Read ‘Digital
Typography’ by Rubinstein [19] and find an issue to investigate.
Project 3.4. Many page layout parameters (Rubinstein [19] and the references therein)
have an influence on legibility. Has typographic design deteriorated in that sense
now that authors publish their own works? Do a study, using various books in the
library.
Project 3.5. The following sentence
Only the fool would take trouble to verify that this sentence was composed of ten a’s, three b’s, four c’s, four d’s, forty-six e’s, sixteen f’s,
four g’s, thirteen h’s, fifteen i’s, two k’s, nine l’s, four m’s, twenty-five
n’s, twenty-four o’s. five p’s, sixteen r’s, forty-one s’s, thirty-seven t’s,
ten u’s, eight v’s, eight w’s, four x’s, eleven y’s, twenty-seven commas, twenty-three apostrophes, seven hyphens and, last but not least,
a single !
is called a pangram. (There are other pangrams. Google for the combination of
‘pangram’ and ‘Lee Sallows’ for this particular type.) Given a beginning of the
sentence (‘Only the fool. . . ’), solve the rest of the sentence by dynamic programming.
TEX – LATEX – CS 594
Chapter 4
Fonts
Knuth wrote a font program, Metafont, to go with TEX. The font descriptions involve some
interesting mathematics.
Handouts and further reading for this chapter
Bezier curves and raster graphics are standard topics in computer graphics. The book by
Foley and Van Dam (section 11.2 about Splines) has been placed on reserve, T385.C587 .
More theoretical information can be found de Boor’s Practical Guide to Splines [2], which
unfortunately limits itself to spline functions.
Digital typography is a very wide area, spanning from perception psychology and physiology to the electronics and mathematics of display equipment. The book by Rubinstein [19]
is a good introduction. This book has been placed on reserve, Z253.3.R8 .
The relation between Bezier curves and aesthetics is explicitly discussed in http://
www.webreference.com/dlab/9902/examples-ii.html.
129
130
CHAPTER 4. FONTS
Bezier curves.
4.1
Introduction to curve approximation
You may never have thought of it, but fonts (actually, typefaces) usually have a mathematical definition somehow. If a font is given as a bitmap, this is typically the result from a
more compact description. Imagine the situation that you have bitmaps at 300dpi, and you
buy a 600dpi printer. It wouldn’t look pretty.
There is then a need for a mathematical way of describing arbitrary shapes. These shapes
can also be three-dimensional; in fact, a lot of the mathematics in this chapter was developed by a car manufacturer for modeling car body shapes. But let us for now only look in
two dimensions, which means that the curves are lines, rather than planes.
A mathematical formalism for curves should have these properties:
•
•
•
•
•
The description should be clear and unambiguous.
It should be easy to tinker with the shapes. This is important for the design phase.
Constructing and evaluating the curves should be computationally cheap.
The curves should be well behaved: small changes in the definition should not
lead to unexpected bulges or spikes.
Curves should be easily composable: if two curves describe adjacent pieces of the
shape, they should connect smoothly.
We actually have two problems that we want to solve:
1.
2.
The exact curve is known, and we want to approximate it, for instance by something that is cheaper to compute, or
Only certain points are given, and we want to draw a smooth curve through them.
We will tackle the second problem first.
4.1.1
Interpolation
The interpolation problem is that of, given points (x1 , f1 ) . . . (xn , fn ), drawing a curve
through them, for instance for purposes of computing intermediate values. Suppose that
we have decided on a polynomial for the interpolating curve. With n points we need an
n − 1st degree polynomial p(x) = pn−1 xn−1 + · · · + p1 x + p0 , which takes n coefficients.
We can draw up the set of equations p(xi ) = fi and solve that. The system
pn−1 xn−1
+ · · · + p1 x1 + p0
1
pn−1 xn−1
+ · · · + p1 xn + p0
n
can be written as X p¯ = f¯, where
p1
.
j
X = xi , p¯ = .. ,
pn−1
=
...
=
f1
fn
f1 − p0
.
..
f¯ =
fn − p0
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4.1. INTRODUCTION TO CURVE APPROXIMATION
131
Figure 4.1: A family of functions that converges pointwise but not uniformly.
Solving this system is not overly expensive, but its numerical stability is questionable. A
better way of computing the same polynomial p is to define auxiliary polynomials p(k) :
p(k) (x) = ck (x − x1 ) · · · (x − xk−1 ) (x − xk+1 ) · · · (x − xn )
where ck is chosen so that p(k) (xk ) = 1. From the fact that p(i) (xj ) = δij , it follows that
X
Y x − xj
p(x) =
fi p(i) (x),
p(i) (x) =
(4.1)
xi − xj
i
j6=i
interpolates the points as intended. It is easy enough to prove that polynomials are uniquely
defined by these interpolation points, so we have now computed the same polynomial in a
more stable way. A polynomial that is based on exact interpolation of values in a number
of points is called a ‘Lagrange interpolation’ polynomial.
Another type of interpolation is ‘Hermite interpolation’, where the derivatives are dictated,
rather than function values. Analogous to the above construction, we can define polynomials
q (k) = ck (x − x1 )2 · · · (x − xk−1 )2 · (x − xk ) · (x − xk+1 )2 · · · (x − xn )2
0
where ck is chosen so that q (k) (xk ) = 1.
4.1.2
Approximation
The above notion of interpolation is sometimes applied to known curves. For instance, by
finding an interpolating polynomial we may aim to find a cheaper way of computing values
on the curve. This raises the question how well the interpolation curve approximates the
original curve.
In classical approximation theory there is usually a family of functions {fn }, typically
polynomials of increasing degree, such that kfn − f k → 0, typically on a closed interval I.
The Weierstrass approximation theorem tells us that every continuous function on a closed
bounded interval can be approximated by polynomials.
Note that this is uniform convergence:
∀ ∃N ∀x∈I,n≥N : |fn (x) − f (x)| ≤ .
Victor Eijkhout
132
CHAPTER 4. FONTS
Figure 4.2: The Runge effect of interpolating with a high degree polynomial
This is a stronger statement than pointwise convergence:
∀x∈I, ∃N ∀n≥N : |fn (x) − f (x)| ≤ .
It is easy to find families of functions fn that convergence in the pointwise sense, but not
uniformly; see figure 4.1.
The spline curves that we will study later in this chapter are a special case of Bernstein
polymials: the n-th Bernstein polynomials for a function f is
Bn (f )(t) =
n
X
n p
f
(1 − t)n−p tp .
p
n
p=0
If f is continuous on [0, 1], this sequence converges uniformly to f . It is worth remarking
that these polynomials do not require computation of derivatives.
While the ideas behind Lagrange and Hermite interpolation will find applicability later in
this story, the idea of interpolating with a single, high degree, polynomial may not be a
good one from a point of uniform convergence. The error can be unacceptably large, as
can be seen in figure 4.2, where the dashed line is an interpolation on equispaced points. In
this case there is in fact not even pointwise convergence. There are a few ways out of that,
such as better choice of interpolation points or of basis functions. In figure 4.2 the dotted
line uses Tchebyshev interpolation points which is seen to remedy the problem to a large
extent.
However, the approach we will use here is that of piecewise approximations with relatively
low degree polynomials. This simplifies certain aspects, but we need to take care to piece
together such curves smoothly. For instance, with Lagrange interpolation the direction of
the curve at the end points can not be specified.
TEX – LATEX – CS 594
4.1. INTRODUCTION TO CURVE APPROXIMATION
4.1.3
133
Computation with interpolation curves
While we will mostly investigate theoretical properties of interpolation curves, the practical
matter of how efficient it is to work with them, also deserves attention. In equation (4.1)
there are n terms involving n multiplications and additions each, making for an O(n2 ) cost.
This can be considerably reduced by rewriting the formula as
Y
Y
X yi
,
yi = fi / (xi − xj ),
p(x) =
(x − ti ) ·
x − ti
i
i
j6=i
which takes n additions, multiplications, and additions if the yi quantities are precomputed.
We will now see a formulation that dispenses with the divisions, and that will also be
simpler if derivatives need to be computed.
The k-th ‘divided difference’ of a function g in the points τ1 . . . τk+1 , notation [τ1 , . . . , τk+1 ]g,
is the leading coefficient of the k-th order1 polynomial pk+1 that agrees with g in the
points τ1 . . . τk+1 .
The simplest examples of divided differences are:
•
•
The zeroeth divided difference of a function is the leading coefficient of a zeroth
order polynomial p(x) = c that agrees with that function in one point: g(τ1 ) = g1 .
Clearly [τ1 ]g = g1 .
The first divided difference is the leading coefficient of a linear function that
agrees in two points:
g(τ2 ) − g(τ1 )
[τ2 ]g − [τ1 ]g
[τ1 , τ2 ]g =
=
τ2 − τ1
τ2 − τ1
This equation may suggest to the reader a relation for higher divided differences.
We shall see in lemma 4 that this indeed holds.
We now prove some facts about divided differences.
Lemma 3 Let pk+1 ∈
τ1 . . . τk , then
Q
<k+1
agree with g in τ1 . . . τk+1 , and pk ∈
pk+1 (x) − pk (x) = [τ1 , . . . , τk+1 ]g
k
Y
(x − τi ).
Q
<k
with g in
(4.2)
i=1
Proof. Since pk is of a lower order, we immediately have
pk+1 − pk = [τ1 , . . . , τk+1 ]gxk + cxk−1 + · · · .
Observing that pk+1 − pk is zero in ti for i ≤ k, it follows that
pk+1 − pk = C
k
Y
(x − τi ).
i=1
From this we get that C = [τ1 , . . . , τk+1 ]g.
•
1.
It is convenient to talk about polymomials of aQcertain order rather than degree: a polynomial of order k has
a degree no more than k. We denote this set with <k+1 . One advantage of this set is that it is closed under
summing.
Victor Eijkhout
134
CHAPTER 4. FONTS
If we repeat this lemma we find that
pk+1 (x) =
k+1
X
[τ1 , . . . , τm ]g
m=1
m−1
Y
(x − τi ),
(4.3)
i=1
which we can evaluate as
pk+1 (x)
Qk
Qk−1
= [τ1 , . . . , τk+1 ]g
(x − τi ) + [τ1 , . . . , τk ]g
(x − τi )
= [τ1 , . . . , τk+1 ]g(x − τk ) ck + [τ1 , . . . , τk ]g(x − τk−1 ) ck−1 + · · ·
where ck = [τ1 , . . . , τk ]g/[τ1 , . . . , τk+1 ]g. This is a very efficient evaluation by Horner’s
rule.
The remaining question is how to construct the divided differences. We approach this recursively.
Lemma 4 Divided differences can be constructed by, eh, dividing differences
[τ1 , . . . , τn+1 ]g = ([τ1 , . . . , τn ]g − [τ2 , . . . , τn+1 ]g) /(τ1 − τn+1 ).
(4.4)
Proof. Let three polynomials be given:
Q
(1)
•
pn ∈ <n agrees with g on τ1 . . . τn ;
Q
(2)
•
pn ∈ <n agrees with g on τ2 . . . τn+1 ;
Q
•
pn−1 ∈ <n−1 agrees with g on τ2 . . . τn .
Then by lemma 3
(1)
pn − pn−1
(2)
pn − pn−1
Qn
= [τ1 , . . . , τn ]g j=2 (x − τj )
Qn
= [τ2 , . . . , τn+1 ]g j=2 (x − τj )
Now let pn+1 be the polynomial that agrees with g on τ1 . . . τn+1 , then
Qn
pn+1 − p(1) = [τ1 , . . . , τn+1 ]g j=1 (x − τj )
Qn+1
pn+1 − p(2) = [τ1 , . . . , τn+1 ]g j=2 (x − τj )
(1)
(2)
Subtracting both pairs of equations, we find two expressions for pn − pn :
n
n+1
n
Y
Y Y
(x − τj )
([τ1 , . . . , τn ]g − [τ2 , . . . , τn+1 ]g)
(x − τj ) = [τ1 , . . . , τn+1 ]g
−
j=2
j=2
j=1
Filling in τ2 . . . τn in this equation, we find zero for both sides. Using x = τ1 we find
([τ1 , . . . , τn ]g − [τ2 , . . . , τn+1 ]g)
n
Y
j=2
from which the lemma follows.
(τ1 − τj ) = [τ1 , . . . , τn+1 ]g
n+1
Y
(τ1 − τj )
j=2
•
From this lemma, we see easily that [τ1 , . . . , τn ]g can be computed in approximately n2 /2
additions and divisions.
TEX – LATEX – CS 594
4.2. PARAMETRIC CURVES
4.2
135
Parametric curves
So far we have been looking at approximation by a function of a single value. That is,
we have y as a function of x. This rules out many curves, such as circles. We could try
expressing x as a function of y, or more generally rotating them, but that is a big hassle,
and it would still rule out some curves.
Another approach is to let the curve be defined implicitly by f (x, y, z) = 0. This suffers
from several problems. We could get too many solutions, for instance as in x2 +y 2 −1 = 0,
requiring constraints. Tracking a path is possible in theory, but is not trivial in practice.
Finally, many shapes are defined piecewise, and joining shapes in a smooth way is also
tricky.
Rather than considering a curve in the plane as a function, it is more productive to describe
the shape of the curve, as some imaginary point moves over the curve. That is, we have a
description of the points on the curve as
x(t)
P = P (t) =
.
y(t)
(Feel free to think of t as time.)
A simple example of how this works is to consider two points P1 and P2 , and the curve
P = tP2 + (1 − t)P1 . Then for t = 0, P (0) = P1 , for t = 1, P (1) = P2 , and for
intermediate values of t we get some point between P1 and P2 .
That this solves our problems with multiple solutions that were present in both function
and implicit approaches is clear if we look at the example P (t) = (cos 2πt, sin 2πt), which
traces out a circle as t goes from 0 to 1.
While a description in terms of piecewise linear functions would be very simple, it is not
smooth. Using the various ideas sketched above, we will now concentrate on curves that
are piecewise parametric cubic splines. Cubics have the following property that they are the
lowest degree that allows specification of location and direction in the end points. Higher
degree functions would allow for instance the specification of higher derivatives, but unless
great care is taken, they would introduce unwanted ‘wiggles’ in the curves.
Using piecewise cubic parametric curves then is a good mean between ease of use and
power of approximation.
4.2.1
Parametrized basis functions
To begin with we will concentrate on a single curve Q(t) where t = 0 . . . 1. We often write
this as Q(t) = C · T where the coefficient matrix
1
c11 c12 c13 c14
t
C = c21 c22 c23 c24 ,
T = 2
t
c31 c32 c33 c34
t3
Victor Eijkhout
136
CHAPTER 4. FONTS
The direction of the curve is then given by
0
dQ(t)
dT
1
=C·
=C ·
2t
dt
dt
2
3t
We see that the C matrix can be derived if we know a total of four locations or directions
of the curve. For instance, if P1 = Q(0), R1 = Q0 (0), P2 = Q(1), and R2 = Q0 (1) are
given, then
1 0 1 0
0 1 1 1
C ·
= [P1 , R1 , P2 , R2 ],
0 0 1 2
0 0 1 3
from which C follows.
Now, often we have a set of basis polynomials given, and we want to take combinations of
them to satisfy the constraints. That can be done by splitting C = GM , where M describes
the basis polynomials, and G is the ‘geometry matrix’. We get the equation
m11 . . . m14
g11 g12 g13 g14
.
..
Q(t) = G · M · T = g21 g22 g23 g24 · ..
·T
(4.5)
.
g31 g32 g33 g34
m41 . . . m44
If we introduce new basis polynomials πi (t) = Mi∗ · T , then we see that Qx = G11 π1 +
G12 π2 + G13 π3 + G14 π4 , Qy = G21 π1 + · · ·, et cetera.
4.2.2
Hermite basis functions
In equation (4.5) the matrix M describes the basis functions, so it is fixed for a certain
class of curves: we will have one set of basis functions for Lagrange type curves, one for
Hermite curves, et cetera. However, we have not yet seen a way to compute the matrix M .
The geometry matrix G is used to derive a specific curve in the class described by M : each
choice of G corresponds to one curve. The columns of G will be points or direction vectors
that somehow describe the intended curve.
Let us now consider Hermite curves. Here we want G to be the matrix of geometric constraints, [P1 , R1 , P2 , R2 ] in the above example. Recall that these constraints, using the locations of the end points and the derivatives of the curve there, give us indeed an example
of Hermite interpolation.
We write out the equations. From Q = G · M · T we get
Q(t) = GH · MH · T (t),
Q0 (t) = GH · MH · T 0 (t).
Applying both these formulas to t = 0 and t = 1, we get
QH ≡ [Q(0), Q0 (0), Q(1), Q0 (1)] = GH · MH · TH
where
1 0 1 0
0 1 1 1
TH = [T (0), T 0 (0), T (1), T 0 (1)] =
0 0 1 2
0 0 1 3
TEX – LATEX – CS 594
4.2. PARAMETRIC CURVES
137
1.2
P1(x)
P2(x)
P3(x)
1
0.8
0.6
0.4
0.2
0
-0.2
0
0.2
0.4
0.6
0.8
1
Figure 4.3: Hermite polynomials
But this QH is the matrix that we had stipulated as the matrix of geometry constraints, in
other words: GH = QH . It now follows that
1 0 −3 2
0 1 −2 1
−1
MH = T H
=
.
0 0 3 −2
0 0 −1 1
Writing this out, we find the cubic Hermite polynomials
P1 (t) = 2t3 − 3t2 + 1,
P2 (t) = t3 − 2t2 + t,
P3 (t) = −2t3 + 3t2 ,
P1 (t) = t3 − t2
illustrated in figure 4.3, and we get Q(t) = G · BH where BH = M · T is the matrix of
‘Hermite blending functions’.
With this formula, we can take a set of geometric constraints, in this case the two endpoints
and the directions there, and derive the formula for the Hermite curve that satisfies these
constraints. As an example, figure 4.5 is the curve .3P1 − 2P2 + P3 − 2P 4, that is, the
curve through (0, .3) and (1, 1), with slope −2 in both x = 0, 1.
We make the transition to parametric curves by expressing both components as Hermite
curves. For instance, figure 4.7 shows the curve
Qx (t) = .1 ∗ P1 (t) + P2 (t) + .9 ∗ P3 (t),
#
# 4
#
set
set
set
cubic Hermite polynomials
terminal pdf
xrange [0:1]
yrange [-.2:1.2]
Qy (t) = .2 ∗ P1 (t) + .3 ∗ P3 (t) − P4 (t)
P1(x) = 2*x**3-3*x**2+1
P2(x) = x**3-2*x**2+x
P3(x) = -2*x**3+3*x**2
P4(x) = x**3-x**2
plot P1(x), P2(x), P3(x), P4(x) title ""
Figure 4.4: The gnuplot source for figure 4.3
Victor Eijkhout
138
CHAPTER 4. FONTS
1.2
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
Figure 4.5: An example of Hermite interpolation
that is
.1
1
.9
0
Q=
P1 +
P2 +
P3 +
P4 .
.2
0
.3
−1
There is one aspect of these Hermite
curves worth
remarking on. In figure 4.9 we have
replaced the direction vector 10 in (0, 0) by x0 , where x = 1, 1.5, 2, 2.5, which all
have the same direction, but a different magnitude. Clearly there is a visual effect.
4.2.3
Splines
We will now take a close look at Bernshtein polynomials of degree 3:
z(t) = (1 − t)3 z1 + 3(1 − t)2 tz2 + 3(1 − t)t2 z3 + t3 z4 ,
(4.6)
also known as Bezier curves or Bezier cubics after Pierre Bezier, an engineer at Renault2 .
2.
Pierre B´ezier was born September 1, 1910 and died November 25, 1999 at the age of 89. In 1985 he
was recognized by ACM SIGGRAPH with a ‘Steven A. Coons’ award for his lifetime contribution to computer
graphics and interactive techniques.
#
# Hermite interpolation
#
set terminal pdf
set multiplot
set xrange [0:1]
set yrange [0:1.3]
P1(x) = 2*x**3-3*x**2+1
P2(x) = x**3-2*x**2+x
P3(x) = -2*x**3+3*x**2
P4(x) = x**3-x**2
p1y = .3
p1slope = -2
p2y = 1
p2slope = -2
plot p1y*P1(x) + p1slope*P2(x) \
+ P3(x) + p2slope*P4(x) title ""
set parametric
set style function lines
plot [t=0:.1] t, p1y+t*p1slope \
title "" with lines 2
plot [t=0:.1] 1-t,p2y-t*p2slope \
title "" with lines 2
Figure 4.6: The source for figure 4.5
TEX – LATEX – CS 594
4.2. PARAMETRIC CURVES
139
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
Figure 4.7: A Hermite interpolation curve
There are a couple of ways of looking at these polynomials. Consider the function z(t) to
be the sum of four basis functions, (1 − t)3 , (1 − t)2 t, (1 − t)t2 , and t3 , each multiplied by
a factor deriving from a control point. From the formulas, and from a picture (figure 4.11)
we see that the first term p1 (t) = (1 − t)3 is the only one with p(0) 6= 0. Likewise, p4 is the
only one with p(1) 6= 0. That means z(0) = z1 and z(1) = z4 . Furthermore, the second
term is (after the first term) the only remaining one with p0 (0) 6= 0, so by choosing z2 we
can change z 0 (0) without changing z(0) or z(1). Likewise z3 for z 0 (1).
Bezier curves can be derived from cubic Hermite splines by replacing the direction vectors R1 , R2 by control points P10 , P20 , so that R1 = 3(P10 − P 1) and R2 = 3(P2 − P20 ). For
the Bezier geometry vector we then have
GB = [P1 , P10 , P20 , P2 ]
and the relation with the Hermite geometry vector
GH = [P1 , R1 , P2 , R2 ] = [P1 , P10 , P20 , P2 ]MBH = GB · MBH
#
# Parametric Hermite curve
#
set terminal pdf
set parametric
set multiplot
set xrange [0:1]
set yrange [0:.7]
P1(t) = 2*t**3-3*t**2+1
P2(t) = t**3-2*t**2+t
P3(t) = -2*t**3+3*t**2
P4(t) = t**3-t**2
p1x = .1 ; p1y = .2
p1dx = 1 ; p1dy = 0
p2x = .9 ; p2y = .3
p2dx = 0 ; p2dy = -1
plot [t=0:1] \
p1x*P1(t)+p1dx*P2(t)+p2x*P3(t)+p2dx*P4(t), \
p1y*P1(t)+p1dy*P2(t)+p2y*P3(t)+p2dy*P4(t) \
title ""
plot [t=0:.2] p1x+t*p1dx,p1y+t*p1dy \
title "" with lines 2
plot [t=0:.2] p2x-t*p2dx,p2y-t*p2dy \
title "" with lines 2
Figure 4.8: The source of figure 4.7
Victor Eijkhout
140
CHAPTER 4. FONTS
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
Figure 4.9: The same curve, tinkering with the direction vector
where
1 −3
0
3
=
0 0
0 0
MBH
0 0
0 0
0 −3
1 3
(4.7)
# direction 2:
set terminal pdf
p1dx = 1.5 ; p1dy = 0
set parametric
plot [t=0:1] \
set multiplot
p1x*P1(t)+p1dx*P2(t)+p2x*P3(t)+p2dx*P4(t), \
set dummy t
p1y*P1(t)+p1dy*P2(t)+p2y*P3(t)+p2dy*P4(t) \
set xrange [0:1]
set yrange [0:.7]
title ""
# direction 3:
P1(t) = 2*t**3-3*t**2+1
p1dx = 2 ; p1dy = 0
P2(t) = t**3-2*t**2+t
plot [t=0:1] \
P3(t) = -2*t**3+3*t**2
p1x*P1(t)+p1dx*P2(t)+p2x*P3(t)+p2dx*P4(t), \
P4(t) = t**3-t**2
p1y*P1(t)+p1dy*P2(t)+p2y*P3(t)+p2dy*P4(t) \
p1x = .1 ; p1y = .2
title ""
p2x = .9 ; p2y = .3
p2dx = 0 ; p2dy = -1
# direction 4:
p1dx = 2.5 ; p1dy = 0
# direction 1:
plot [t=0:1] \
p1dx = 1 ; p1dy = 0
plot [t=0:1] \
p1x*P1(t)+p1dx*P2(t)+p2x*P3(t)+p2dx*P4(t), \
p1y*P1(t)+p1dy*P2(t)+p2y*P3(t)+p2dy*P4(t)
\
p1x*P1(t)+p1dx*P2(t)+p2x*P3(t)+p2dx*P4(t),
\
title
p1y*P1(t)+p1dy*P2(t)+p2y*P3(t)+p2dy*P4(t)
\ ""
title ""
Figure 4.10: The source for figure 4.9
TEX – LATEX – CS 594
4.2. PARAMETRIC CURVES
141
1
x**3
3*x**2*(1-x)
3*x*(1-x)**2
(1-x)**3
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
Figure 4.11: Bernshtein polynomials
Defining
1 −3 3 −1
0 3 −6 3
=
0 0
3 −3
0 0
0
1
MB = MBH · MH
(4.8)
we now get for Bezier curves
Q(t) = GH · MH · T (t) = GB · MBH · MH · T (t) = GB · MB · T (t)
We can also write that as Qx (t) = g11 B1 (t) + g12 B2 (t) + · · · where
B1 (t) = 1 − 3t + 3t2 − t3
B2 (t) = 3t − 6t2 + 3t3
B3 (t) = 3t2 − 3t3
B4 (t) =
= (1 − t)3
= 3t(1 − t)2
= 3t2 (1 − t)
= t3
which are the Bernstein polynomials we started this section with.
The sum of these polynomials (this is equivalent to setting the zi coefficients to one in
equation (4.6)) is z(t) = (t + (1 − t)))3 ≡ 1. Also, the polynomials are positive on [0, 1],
so the components Qx , Qy , Qz are weighted averages of the polynomials. This means that
the curve, consisting of weighted sums of the control points, is contained in the convex hull
of the control points.
Exercise 43. One could also define quadratic Bezier curves. These
have only a single control point, and the curve is such that in both the
endpoints it is aimed at the control point.
Victor Eijkhout
142
CHAPTER 4. FONTS
2
Quadratic Bezier curve
1.5
1
0.5
0
0
0.2
0.4
0.6
0.8
Derive the basis functions and geometry matrix for this case. Make a
gnuplot figure of a single quadratic Bezier curve, and of two curves
joined smoothly.
Hint: you can follow the construction of the cubic splines in the lecture
notes. The only problem is defining the control point. First draw up the
Hermite geometry matrix based on end points q0 and q1 , and the derivative q00 in the first end point. Derive from them the derivative q10 in the
other end point. The control point then lies on the intersection of two
lines. Solving this looks like a single equation in two unknowns, but it
can be solved: write it as a matrix-vector equation that is satisfied no
matter the choice of the geometry matrix.
4.2.4
Calculation of Bezier curves
Suppose we have a Bezier curve defined by four control points, and we want to draw points
on the curve, meaning that we have to evaluate the function Q(t) for a number of values
of t. The relation Q(t) = G · M · T (t) allows us to do this calculation in
2 multiplications to form the terms t2 and t3 in T ;
16 multiplications and 12 additions forming M · T ;
12 multiplications and 9 additions forming G · (M · T ).
˜ = G · M , which brings the cost down to two
An obvious improvement is to store M
multiplications for T and
˜ · T.
•
12 multiplications and 9 additions for forming M
•
•
•
A similar count is arrived at by looking at divided differences. Additionally, this way of
computing is more stable.
TEX – LATEX – CS 594
1
4.3. PRACTICAL USE
143
From the formula Q(t) = G · M · T (t) we get for each component
X
X
qi (t) =
Gij (M T )j =
Gij Mjk tk−1 .
j
j,k
Looking at only one component for simplicity, we find, for instance
X
X
x(t) =
ck tk−1
ck =
G1j Mjk .
j
k
We recall equation (4.8):
1 −3 3 −1
0 3 −6 3
MB =
0 0
3 −3
0 0
0
1
and writing gj ≡ G1j we find
c1 = g1 ,
c2 = 3(g2 − g1 ),
c3 = 3(g3 − 2g2 + g1 ),
c4 = g4 − 3g3 + 3g2 − g1 .
In this we recognize divided differences of g:
[2, 1]g
= g2 − g1 ,
[3, 2, 1]g = [3, 2]g − [2, 1]g = (g3 − g2 ) − (g2 − g1 )
= g3 − 2g2 + g1
[4, 3, 2, 1] = [4, 3, 2]g − [3, 2, 1]g = (g4 − 2g3 + g2 ) − (g3 − 2g2 + g1 )
= g4 − 3g3 + 3g2 − g1
using lemma 4.
4.3
Practical use
4.3.1
Piecewise curves
As we indicated earlier, complicated shapes can be approximated by piecewise cubics. Figure 4.12 shows two Hermite curves joined together. The curve is continuous and the directions at the join are proportional. This is called ‘geometric continuity’, and is denoted G1 .
So-called B-splines (‘basis splines’) are constructed by piecing together Bezier curves,
not only continuously, but differentiably. For this, if P1 . . . P4 and P4 . . . P7 are the control
points of the two curves, then we require
P4 = (P3 + P5 )/2.
This is shown in figure 4.14.
4.3.2
Drawing splines
Even if we can evaluate a Bezier curve efficiently (section 4.2.4) in a given point, we
can improve on this considerably in the context of line drawing. Consider the problem of
drawing a cubic curve on the interval [0, 1] by drawing consecutive points Q(nδ) for n =
0, 1, . . ..
We will discuss one line-drawing technique in the chapter on raster graphics. A technique
that is attractive for splines, and which is used in M ETAFONT, is recursive subdivision.
Victor Eijkhout
144
CHAPTER 4. FONTS
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
Figure 4.12: Two Hermite curves, joined together smoothly in (.5, .3)
Here, a curve is subdivided until a segment is a straight line within the pixel resolution, so
that a line drawing algorithm can be used. The test for a straight line can be implemented
efficiently through using the spline control points.
set terminal pdf
set parametric
set multiplot
set dummy t
set xrange [0:1]
set yrange [0:.8]
P1(t) = 2*t**3-3*t**2+1
P2(t) = t**3-2*t**2+t
P3(t) = -2*t**3+3*t**2
P4(t) = t**3-t**2
p1x = .1 ; p1y = .2
p2x = 1 ; p2y = 0
p3x = .5 ; p3y = .3
p4x = 0 ; p4y = -1
plot [t=0:1] \
p1x*P1(t)+p2x*P2(t)+p3x*P3(t)+p4x*P4(t), \
p1y*P1(t)+p2y*P2(t)+p3y*P3(t)+p4y*P4(t) \
title ""
p5x = .9 ; p5y = .6
p6x = 0 ; p6y = -1
plot [t=0:1] \
p3x*P1(t)+.5*p4x*P2(t)+p5x*P3(t)+p6x*P4(t), \
p3y*P1(t)+.5*p4y*P2(t)+p5y*P3(t)+p6y*P4(t) \
title "" with lines 2
plot [t=0:1] \
p3x*P1(t)+2*p4x*P2(t)+p5x*P3(t)+p6x*P4(t), \
p3y*P1(t)+2*p4y*P2(t)+p5y*P3(t)+p6y*P4(t) \
title "" with lines 2
Figure 4.13: The source for figure 4.12
TEX – LATEX – CS 594
4.4. INTRODUCTION
145
3
2.5
2
1.5
1
0.5
0
0
1
2
3
4
5
6
Figure 4.14: Two Bezier curves, joined together smoothly
Curve plotting with gnuplot.
The gnuplot utility can be used for
plotting sets of points. However, here we will only discuss drawing curves.
4.4
Introduction
The two modes for running gnuplot are interactive and from file. In interactive mode,
you call gnuplot from the command line, type commands, and watch output appear (see
next paragraph); in the second case you call gnuplot <your file>.
The output of gnuplot can be a picture on your screen, or drawing instructions in a file.
set terminal pdf
set xrange [0:6]
set yrange [0:3]
set parametric
B1(x) = x**3
B2(x) = 3*x**2*(1-x)
B3(x) = 3*x*(1-x)**2
B4(x) = (1-x)**3
P1x = .5 ; P1y = .2
P2x = 1.2 ; P2y = .4
P3x = 2.2 ; P3y = 1.3
P4x =
3 ; P4y = 1.2
P5x = 2*P4x-P3x
P5y = 2*P4y-P3y
P6x = 4.5 ; P6y = .2
P7x = 5
; P7y = 2.5
set multiplot
plot [t=0:1] \
P1x*B1(t)+P2x*B2(t)+P3x*B3(t)+P4x*B4(t), \
P1y*B1(t)+P2y*B2(t)+P3y*B3(t)+P4y*B4(t) \
title ""
plot [t=0:1] \
P4x*B1(t)+P5x*B2(t)+P6x*B3(t)+P7x*B4(t), \
P4y*B1(t)+P5y*B2(t)+P6y*B3(t)+P7y*B4(t) \
title ""
plot [t=-1:1] \
P4x+t*(P5x-P4x),P4y+t*(P5y-P4y) \
title "" with lines 2
plot [t=0:1] \
P1x+t*(P2x-P1x),P1y+t*(P2y-P1y) \
title "" with lines 3
plot [t=0:1] \
P7x+t*(P6x-P7x),P7y+t*(P6y-P7y) \
title "" with lines 3
Figure 4.15: The source for figure 4.14
Victor Eijkhout
146
CHAPTER 4. FONTS
Where the output goes depends on the setting of the terminal. By default, gnuplot will
try to draw a picture. This is equivalent to declaring
set terminal x11
or aqua, windows, or any choice of graphics hardware.
For output to file, declare
set terminal pdf
or fig, latex, pbm, et cetera.
4.5
Plotting
The basic plot command is plot. By specifying
plot x**2
you get a plot of f (x) = x2 ; gnuplot will decide on the range for x. With
set xrange [0:1]
plot 1-x title "down", x**2 title "up"
you get two graphs in one plot, with the x range limited to [0, 1], and the appropriate legends
for the graphs. The variable x is the default for plotting functions.
Plotting one function against another – or equivalently, plotting a parametric curve – goes
like this:
set parametric
plot [t=0:1.57] cos(t),sin(t)
which gives a quarter circle.
To get more than one graph in a plot, use the command set multiplot.
4.5.1
Styles
You can change the default drawing style with
set style function dots
(lines, dots, points, et cetera), or change on a single plot with
plot f(x) with points
TEX – LATEX – CS 594
4.6. VECTOR GRAPHICS AND RASTER GRAPHICS
147
Figure 4.16: Line with slope ≤ 1 and one pixel per column on
Raster graphics.
4.6
Vector graphics and raster graphics
We may use fancy techniques such as Bezier curves for describing shapes, but at some
point the graphic needs to be rendered on an actual device with pixels or ink dots. Thus we
need algorithms for deciding which pixels to turn on or off, and, in case the device has a
larger bitdepth, with what intensity.
The technical terms here are ‘vector graphics’ for a description of the lines and curves, and
‘raster graphics’ for the pixel-by-pixel description. A description of a raster is also called
‘bitmap’. Common bitmap-based formats are JPEG, GIF, TIFF, PNG, PICT, and BMP.
Vector graphics make for a much more compact file, but they rely on the presence of a final
rendering stage. For instance, Macromedia’s Flash format (now an open Internet standard)
is a vector graphics format, that relies on a browser plugin. However, presence of a Flash
renderer is pretty much standard in browsers these days. Adobe Postscript is also a vector
format. The first popular printer to incorporate it, the Apple Laserwriter, had a Motorola
68000 processor, exactly as powerful as the Macintosh computer it connected to.
Two vector standards are being proposed to the W3C: the Precision Graphics Markup
Language and the Vector Markup Language. PGML is backed by Adobe Systems, IBM,
Netscape, and Sun Microsystems. VML is supported by Microsoft, Hewlett-Packard, Autodesk, Macromedia, and Visio. Both standards are based on Extensible Markup Language
(XML).
4.7
Basic raster graphics
4.7.1
Line drawing
We limit the discussion to lines with slope less than 1. For those, on a grid of pixels, one
pixel per column will be switched on, and the question to be addressed is which pixel to
turn on in each column.
Probably the simplest way to draw a line is by using an ‘incremental’ drawing algorithm.
Let a line y = mx + B be given, and we write the slope as m = δy/δx. In the case of pixel
graphics, we set δx ≡ 1, so δy = m and we can recursively state
yi+1 = yi + δy.
Victor Eijkhout
148
CHAPTER 4. FONTS
Figure 4.17: The midpoint algorithm for line drawing
The simplest implementation of this is
let x0 , y0 and m be given, then
for i = 0 . . . n − 1
WritePixel(xi , Round(yi ))
xi+1 = xi + 1
yi+1 = yi + m
Since (xi , yi ) is never related to the actual formula for the line, there will be an accumulation of round-off error. However, this will be negligible. More seriously, the rounding
operation is relatively expensive. In the next algorithm we will eliminate it. If possible, we
want to operate completely in integer quantities.
The ‘midpoint algorithm’ proceeds fully by integer addition. First we write the equation
for the line in two different ways as
dy
y=
x + B,
F (x, y) = ax + by + c = 0.
dx
Clearly, a = dy, b = −dx, c = B, and we can derive dx, dy from the end pixels of the line.
The function F is zero on the line, positive in the half plane under it, and negative above it.
Now we consider how to progress given that we have switched on a pixel at (xp , yp ). With
the initial assumption that the line has a slope between 0 and 1, the next pixel will be either
(xP + 1, yP + 1), or (xp + 1, yp ), depending on which is closer to the line.
Instead of measuring the distance of the candidate next pixels to the line, we decide whether
their midpoint M is above or under the line. For this, we use the function F (·, ·) introduced
above, and evaluate the ‘decision value’ of the midpoint:
d = F (xp + 1, yp + 1/2).
The two cases to consider then are
d < 0: M lies over the line, so we take yp+1 = yp ;
d ≥ 0: M lies under the line, so we take yp+1 = yp + 1.
Similarly we update the mid point: if d ≥ 0, the midpoint moves up. Note that the new
midpoint is at xp+1 + 1.
TEX – LATEX – CS 594
4.7. BASIC RASTER GRAPHICS
149
Now, we do not actually use the midpoint, only the value of d. The algorithm is then
complete once we find a way to update d cheaply. For this we look at its next value
d0 = F (xp+1 + 1, yp+1 + 1/2).
Corresponding to the above two cases:
d0 = a(xp+1 + 1) + b(yp+1 + 1/2) + c =
d < 0 : = a(xp + 2) + b(yp + 1/2)
= d + a = d + dy
d ≥ 0 : = a(xp + 2) + b(yp + 3/2) + c
= d + a + b = d + dy − dx
In other words, we update d with dy or dy − dx depending on whether it’s negative or
non-negative.
To start off the algorithm, dx and dy are computed from the endpoints, and the initial value
of d follows from
d0 = F (x0 + 1, y0 + 1/2) = F (x0 , y0 ) + a + b/2 = 0 + dy − dx/2.
To get rid of the division by 2, which would cause real rather than integer values to be
used throughout the algorithm, we can redefine F˜ (x, y) = 2F (x, y); correspondingly we
update d with 2dy and 2(dy − dx) in the two cases.
Exercise 44. Can you modify the DDA line drawing algorithm so that
it works (as well as possible) when the line is given between points that
are not on pixels?
These algorithms are sometimes referred to as ‘Digital Differential Analyzers’, since they
trace out a curve by proceeding with small differences. The line algorithm was first derived
by Bressenham.
4.7.2
Circle drawing
Circles can be drawn with a similar algorithm. We observe that, because of 8-fold symmetry, we can limit the algorithm to the part of a circle from x = 0 to x = y. The midpoint
argument is now slightly more complicated. The function for the circle is
F (x, y) = x2 + y 2 − R2 ,
and the decision value in the midpoint M is
d = F (xp + 1, yp + 1/2) = x2 + 2x + y 2 + y + 5/4.
The two cases to consider are similar to before:
d < 0: M lies in the circle, so we take yp+1 = yp ;
d ≥ 0: M lies outside the circle, so we take yp+1 = yp + 1.
To update the decision value we get the two cases
d0 = F (xp+1 + 1, yp+1 + 1/2) =
d < 0 : = x2 + 4x + y 2 + y + 4 1/4
= d + 2x + 3
d ≥ 0 : = x2 + 4x + y 2 + 3y + 6 1/4 = d + 2(x + y) + 5
Exercise 45. Why is there no need to consider bigger increments of yp
in the circle drawing algorithm? After all, a circle has curvature so the
slope increases.
The circle algorithm can be further improved by observing that the quantities 2x and 2y
can themselves easily by constructed by updating. This removes the need for any multiplication.
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Figure 4.18: The midpoint algorithm for circle drawing
4.7.3
Cubics
Suppose we have a cubic function f (t) = at3 + bt2 + ct + d. Instead of evaluating this
polynomial directly, using Horner’s rule, we compute the value f (t + δ) by updating:
f (t + δ) = f (t) + ∆f (t).
We find
∆f (t) = f (t + δ) − f (t)
= a(3t2 δ + 3tδ 2 + δ 3 ) + b(2tδ + δ 2 ) + cδ
= 3aδt2 + (3aδ 2 + 2bδ)t + aδ 3 + bδ 2 + cδ
This still leaves us with a quadratic function to evaluate, so we define
∆2 f (t) = ∆f (t + δ) − ∆f (t)
= 3aδ(2tδ + δ 2 ) + (3aδ 2 + 3bδ)δ
= 6aδ 2 t + 6aδ 3 + 2bδ 2
Finally, we derive ∆3 f (t) = ∆2 f (t + δ) − ∆2 f (t) = 6aδ 2 . Taken all together, we can
now compute fn+1 ≡ f ((n + 1)δ) by initializing
∆3 f0 = 6aδ 2 , ∆2 f0 = 6aδ 3 + 2bδ 2 , ∆f0 = aδ 3 + bδ 2 + cδ
and computing by update
fn+1 = fn + ∆fn , ∆fn+1 = ∆fn + ∆2 fn , ∆2 fn+1 = ∆2 fn + ∆3 f0
The advantage of this algorithm is its low operation count, and the fact that it works fully
by integer operations.
4.8
Rasterizing type
Typefaces can be described by curves, but several aspects to them make it necessary to do
more than just rendering these curves, when rasterizing them. Part of the problem is that
characters in a font are relatively small, and satisfy all sorts of constraints that both may be
hard to satisfy (especially at low resolution), and are immediately noticed when rendered
incorrectly.
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Figure 4.19: Problems in rasterizing type, and resulting illegible output
Such problems result from using too simple algorithms for converting the character outlines
to rasters. For instance, an obvious algorithm is
Figure 4.20: A bad and a good way of rendering a Times Roman ‘e’ at low resolution
A pixel is turned on if its center is within the curve.
Now consider the case where a curve with large radius exceeds location y = n + 1/2 for
only one x. This results in the ‘pimple’ on top of the ‘e’ in figure 4.20. On the other hand,
if such a curve stays just under such a halfpoint, we get a long plateau, as in the left side
curve of the ‘e’.
4.8.1
Scaled fonts
These rasterizing problems are largely due to the facts that
•
Characters are scalable, so the relations between top/bottom or left/right are not
always mapped the same way to a pixel grid;
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CHAPTER 4. FONTS
•
Even if characters are used at the same size, they need to be displayed on various
kinds of rasters (printer, screen);
Characters can be placed in any horizontal or vertical location, so relations to
pixel boundaries are also flexible.
•
The conversion process goes more or less as follows3 :
Figure 4.21: Scaled and rasterized character outline
•
Character outlines are based on coordinates on some grid, often expressed as fixed
point, or other scheme with a finite mesh size.
The character is scaled to the raster on which it will be rendered.
The result is rounded to the raster, in figure 4.21 this puts the left and right sides on
pixel boundaries; note that other vertical parts of the character are not necessarily
pixel-aligned.
The scaled and rounded outline is then rasterized by turning a set of pixels on.
•
•
•
We see that in both final steps, rounding to the raster, and switching on pixels, we can have
unwanted effects.
4.8.2
Pixelation
Above we said that pixels are switched on if their center falls within the curve. There are
two problems with this:
•
•
Sometimes not enough pixels are turned on to render the shape accurately, and
Deciding whether a pixel is within the shape is not trivial to begin with. For instance, letters such as ‘o’ or ‘p’ have an inner region that should not be filled. In
another example, sometimes it is convenient to describe a shape as non-disjoint
union of other shapes; see figure 4.22
The second problem can be solved a number of ways. We could for instance look at a scan
line, and switch to on/off mode every time we cross a curve boundary. This approach does
not work for intersecting curves.
3.
Much of this discussion is based on the definition of TrueType fonts.
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153
Figure 4.22: A shape consisting of overlapping contours
Figure 4.23: The effects of different winding number rules
Better solutions are based on looking at the so-called ‘winding number’. This number
counts, for a given point in the plane, how often a curve winds around the point. If this
is zero, the point is outside the curve, otherwise it is inside it.
That implementing winding number rules is not trivial can be seen from two screen shots
of Acrobat Reader version 4; figure 4.24.
4.8.3
Font hinting / instructing
To prevent some of the problems indicated above, characters in a font file consist of more
than just the outline description. Additionally, each character can have a short program in a
language defined by the font file format. Such a program can enforce that certain distances
in the font as exact multiples of pixel distances.
For instance, the letter ‘O’ in figure 4.25 has the following constraints
1.
A certain amount of white space buffering the character; distance 3 is the ‘transport’;
2.
The width of the band
5,6
Visual under and overshoot.
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Figure 4.24: Acrobat 4 rendering of a complicated figure at different magnifications
7
The height of the band
Distances 5 and 6 are over and undershoot: a letter with curved top/bottom like ‘O’ would
seem too small if it stayed between the baseline and the cap height. To compensate for that,
the letter is made slightly higher and deeper. However, for small sizes and low resolutions,
this compensation needs to be switched off, since it would look too drastic.
4.8.4
Dropouts
In certain cases, pixels can not be switched on that are necessary for continuity of the figure
drawn. Figure 4.26 shows a case where a connecting bit is too thin to hit the centers of any
pixels. To cover such cases, the renderer usually has an algorithm that detects when a scan
line enters and leaves a contour without setting any pixels. It will then set, for instance, the
left pixel.
4.9
Anti-aliasing
In the algorithms so far, the decision was between switching a pixel on or off. There are a
few problems with this. For instance, for certain slopes the rendered curve can have a very
‘jagged’ look. Also, a curve at slope 1 will
√ have the same number of pixels on as a curve at
slope 0, but on a path that is longer by 2. Thus, it may look lighter or thinner.
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Figure 4.25: Constraints on the letter ‘O’
If the display supports a range of values (‘grayscale’), we can try to use this and find a
better visual rendering. First we will look at an example, then briefly consider the general
theory of this phenomenon.
4.9.1
Raster graphics with larger bitdepths
In the algorithms above, pixels close to a line or other curve were switched on. If the display
supports it, we can compute a measurement of proximity of the pixel to the line, and set a
brightness based on that. There are various ways to implement this. For instance, one could
consider the line to have an area, and to compute the intersection area of the line and the
pixel box. Here we will use a ‘filter function’. The support of this function will be larger
than the pixel box.
We will modify the midpoint method for line drawing so that in each column three pixels
will be nonzero, with an intensity based on the Euclidean distance to the line. Let v be the
(signed) vertical
p distance from the midpoint to the line, and D the euclidean distance, then
D = vdx/ dx2 + dy 2 . The denominator we can compute once and for all at the start of
the algorithm, but a simple way of computing vdx is needed.
Consider the case where d < 0, in which case we choose yp+1 = yp . Now we have
0 = F (xp + 1, yp + v) = F (xp + 1, yp ) + 2bv ⇒ 2vdx = F (xp + 1, yp ),
and
d = F (M ) = F (xp + 1, yp + 1/2) = F (xp + 1, yp ) + b
so 2vdx = d + dx. Likewise, if d ≥ 0, 2vdx = d − dx. Since we know how to update d
cheaply, we can now iteratively compute D.
√
√
For the top and bottom point, D = 2(1−v)dx/ . . . and D = 2(1 +v)/ . . ., respectively.
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Figure 4.26: A case of ‘dropout’: missing pixels give a disconnected curve
4.9.2
The general idea
Smoothing out a picture with grayscales is usually called ‘anti-aliasing’. To see why, consider how the parts of a graphics system fit together. After we derive a curve or other
two-dimensional object (this could be a projected surface) to display, the exact values are
sampled according to the pixel resolution. By computing a Fourier transform of these samples, we get an impression of the visual ‘frequencies’ in the picture.
If, instead of simply switching on pixels based on the sampled curve, we compute pixel
values so that sampling them reproduces the frequency spectrum, we get a picture that
looks closer to the intended one.
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Projects for this chapter.
Project 4.1. Bezier curves can also be used for graphics problems such as enlarging a
bitmat or projecting text on a waving flag. Consult http://www.tinaja.
com/cubic01.asp and report on the techniques used.
Project 4.2. (very mathematical) Explain elliptical integrals and how they are used to compute the length of a Bezier curve. Explain approximations. Same resource as the
previous project.
Project 4.3. (very mathematical) Study the theory of NURBS (Non-Uniform Rational BSplines); what are their capabilities and limitations? Start at http://devworld.
apple.com/dev/techsupport/develop/issue25/schneider.html
for an introduction.
Project 4.4. Investigate perception issues in font design or display technology. Start by
browsing through Rubinstein’s book.
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Chapter 5
TEX’s macro language – unfinished chapter
The programming language of TEX is rather idiosyncratic. One notable feature is the difference between expanded and executed commands. The expansion mechanism is very
powerful: it is in fact possible to implement lambda calculus in it.
Handouts and further reading for this chapter
The inspiration for this chapter was the article about lists by Alan Jeffrey [8].
159
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CHAPTER 5. TEX’S MACRO LANGUAGE – UNFINISHED CHAPTER
Lambda calculus in TEX.
5.1
Logic with TEX
5.1.1
Truth values, operators
We start by defining a couple of simple tools.
\def\Ignore#1{}
\def\Identity#1{#1}
\def\First#1#2{#1}
\def\Second#1#2{#2}
For example:
Taking first argument:
input : \First {first}{second}
output : first
Taking second argument:
input : \Second {first}{second}
output : second
We define truth values:
\let\True=\First
\let\False=\Second
and logical operators:
\def\And#1#2{#1{#2}\False}
\def\Or#1#2{#1\True{#2}}
\def\Twiddle#1#2#3{#1{#3}{#2}}
\let\Not=\Twiddle
Explanation: And x y is y if x is true, false is x is false. Since True and False are defined as taking the first and second component, that gives the definition of And as above.
Likewise Or.
To test logical expressions, we attach TF to them before evaluting; that was \True TF
will print T, and \False TF will print F.
Let us test the truth values and operators:
True takes first of TF:
Not true is false:
input : \True
input : \Not \True
output : T
output : F
False takes second of TF:
And truth table TrueTrue:
input : \False
input : \And \True \True
output : F
output : T
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5.1. LOGIC WITH TEX
And truth table TrueFalse:
input : \And \True \False
output : F
And truth table FalseTrue:
input : \And \False \True
output : F
And truth table FalseFalse:
input : \And \False \False
output : F
Or truth table TrueTrue:
input : \Or \True \True
output : T
5.1.2
161
Or truth table TrueFalse:
input : \Or \True \False
output : T
Or truth table FalseTrue:
input : \Or \False \True
output : T
Or truth table FalseFalse:
input : \Or \False \False
output : F
Conditionals
Some more setup. We introduce conditionals
\def\gobblefalse\else\gobbletrue\fi#1#2{\fi#1}
\def\gobbletrue\fi#1#2{\fi#2}
\def\TeXIf#1#2{#1#2 \gobblefalse\else\gobbletrue\fi}
\def\IfIsPositive{\TeXIf{\ifnum0<}}
with the syntax
\TeXIf <test> <arg>
We test this:
Numerical test:
input : \IfIsPositive {3}
output : T
Numerical test:
input : \IfIsPositive {-2}
output : F
5.1.3
Lists
A list is defined as a construct with a head, which is an element, and a tail, which is another
list. We will denote the empty list by Nil.
\let\Nil=\False
We implement a list as an operator with two arguments:
•
If the list is not empty, the first argument is applied to the head, and the tail is
evaluated;
•
If the list is empty, the second argument is evaluated.
In other words
a2 if L = ()
L a1 a2 =
a1 (x) Y if L = (x, Y )
In the explanation so far, we only know the empty list Nil. Other lists are formed by taking
an element as head, and another list as tail. This operator is called Cons, and its result is a
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list. Since a list is a two argument operator, we have to make Cons itself a four argument
operator:
% \Cons <head> <tail> <arg1> <arg2>
\def\Cons#1#2#3#4{#3{#1}{#2}}
Since Cons#1#2 is a list, applied to #3#4 it should expand to the second clause of the
list definition, meaning it applies the first argument (#3) to the head (#1), and evaluates
the tail (#2).
The following definitions are typical for list operations: since a list is an operator, applying
an operation to a list means applying the list to some other objects.
\def\Error{{ERROR}}
\def\Head#1{#1\First\Error}
\def\Tail#1{#1\Second\Error}
Let us take some heads and tails of lists. As a convenient shorthand, a singleton is a list
with an empty tail:
\def\Singleton#1{\Cons{#1}\Nil}
Head of a singleton:
input : \Head {\Singleton \True }
output : T
Head of a tail of a 2-elt list:
input : \Head {\Tail {\Cons \True {\Singleton \False
}}}
output : F
We can also do arithmetic tests on list elements:
Test list content:
input : \IfIsPositive {\Head {\Singleton {3}}}
output : T
Test list content:
input : \IfIsPositive {\Head {\Tail {\Cons
{3}{\Singleton {-4}}}}}
output : F
Exercise 46.
Write a function \IsNil and test with
\test{Detect NIL}{\IsNil\Nil}
\test{Detect non-NIL}{\IsNil{\Singleton\Nil}}
5.1.3.1
A list visualization tool
If we are going to be working with lists, it will be a good idea to have a way to visualize
them. The following macros print a ‘1’ for each list element.
\def\Transcribe#1{#1\TranscribeHT\gobbletwo}
\def\TranscribeHT#1#2{1\Transcribe{#2}}
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5.1.3.2
163
List operations
Here are some functions for manipulating lists. We want a mechanism that takes a function f , an initial argument e, and a list X, so that
Apply f e X ⇒ f x1 (f x2 (. . . (f xn e) . . .))
% #1=function #2=initial arg #3=list
\def\ListApply#1#2#3{#3{\ListApplyp{#1}{#2}}{#2}}
\def\ListApplyp#1#2#3#4{#1{#3}{\ListApply{#1}{#2}{#4}}}
This can for instance be used to append two lists:
\def\Cat#1#2{\ListApply\Cons{#2}{#1}}
For example:
Cat two lists:
input : \Transcribe {\Cat {\Singleton \Nil }{\Cons
\Nil {\Singleton \Nil }}}
output : 111
From now on the \Transcribe macro will be implicitly assumed; it is no longer displayed in the examples.
5.1.4
Numbers
We can define integers in terms of lists: zero is the empty list, and to add one to a number
is to Cons it with an empty list as head element. In other words,
n + 1 ≡ (0, n).
This defines the ‘successor’ function on the integers.
\let\Zero\Nil
\def\AddOne#1{\Cons\Nil{#1}}
Examples:
Transcribe zero:
input : \Zero
output :
Transcribe one:
input : \AddOne \Zero
output : 1
Transcribe three:
input : \AddOne {\AddOne {\AddOne \Zero }}
output : 111
Writing this many \AddOnes get tiring after a while, so here is a useful macro:
\newtoks\dtoks\newcount\nn
\def\ndef#1#2{\nn=#2 \dtoks={\Zero}\nndef#1}
\def\nndef#1{
\ifnum\nn=0 \edef\tmp{\def\noexpand#1{\the\dtoks}}\tmp
\else \edef\tmp{\dtoks={\noexpand\AddOne{\the\dtoks}}}\tmp
\advance\nn by -1 \nndef#1
\fi}
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which allows us to write
\ndef\One1 \ndef\Two2 \ndef\Three3 \ndef\Four4 \ndef\Five5
\ndef\Seven7\ndef\Six6
et cetera.
It is somewhat surprising that, even though the only thing we can do is compose lists, the
predecessor function is just as computable as the successor:
\def\SubOne#1{#1\Second\Error}
Predecessor of two:
input : \SubOne {\AddOne {\AddOne \Zero }}
output : 1
(If we had used \Ignore instead of \Second a subtle TEXnicality would come into play:
the list tail would be inserted as {#2}, rather than #2, and you would see an Unexpected }
error message.)
Some simple arithmetic: we test if a number is odd or even.
\def\IsEven#1{#1\IsOddp\True}
\def\IsOddp#1#2{\IsOdd{#2}}
\def\IsOdd#1{#1\IsEvenp\False}
\def\IsEvenp#1#2{\IsEven{#2}}
Zero even?:
input : \IsEven \Zero
output : T
Zero odd?:
input : \IsOdd \Zero
output : F
Test even:
input : \IsEven {\AddOne
{\AddOne {\AddOne \Zero
}}}
output : F
Test odd:
input : \IsOdd {\AddOne
{\AddOne {\AddOne \Zero
}}}
output : T
Test even:
input : \IsEven {\AddOne
{\AddOne {\AddOne {\AddOne
{\Zero }}}}}
output : T
Test odd:
input : \IsOdd {\AddOne
{\AddOne {\AddOne {\AddOne
{\Zero }}}}}
output : F
Exercise 47. Write a test \IsOne that tests if a number is one.
Zero:
input : \IsOne \Zero
output : F
One:
input : \IsOne \One
output : T
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165
Two:
input : \IsOne \Two
output : F
5.1.4.1
Arithmetic: add, multiply
Above, we introduced list concatenation with \Cat. This is enough to do addition. To save
typing we will make macros \Three and such that stand for the usual string of \AddOne
compositions:
\let\Add=\Cat
Adding numbers:
input : \Add {\Three }{\Five }
output : 11111111
Instead of adding two numbers we can add a whole bunch
\def\AddTogether{\ListApply\Add\Zero}
For example:
Adding a list of numbers:
input : \AddTogether {\Cons \Two {\Singleton \Three
}}
output : 11111
Adding a list of numbers:
input : \AddTogether {\Cons \Two {\Cons \Three
{\Singleton \Three }}}
output : 11111111
This is one way to do multiplication: to evaluate 3 × 5 we make a list of 3 copies of the
number 5.
\def\Copies#1#2{#1{\ConsCopy{#2}}\Nil}
\def\ConsCopy#1#2#3{\Cons{#1}{\Copies{#3}{#1}}}
\def\Mult#1#2{\AddTogether{\Copies{#1}{#2}}}
Explanation:
•
If #1 of \Copies is empty, then Nil.
•
Else, \ConsCopy of #2 and the head and tail of #1.
•
The tail is one less than the original number, so \ConsCopy makes that many
copies, and conses the list to it.
For example:
Multiplication:
input : \Mult {\Three }{\Five }
output : 111111111111111
However, it is more elegant to define multiplication recursively.
\def\MultiplyBy#1#2{%
\IsOne{#1}{#2}{\Add{#2}{\MultiplyBy{\SubOne{#1}}{#2}}}}
Multiply by one:
input : \MultiplyBy \One \Five
output : 11111
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Multiply bigger:
input : \MultiplyBy \Three \Five
output : 111111111111111
5.1.4.2
More arithmetic: subtract, divide
The recursive definition
of subtraction is
m
if n = 0
m−n=
(m − 1) − (n − 1) otherwise
Exercise 48. Implement a function \Sub that can subtract two numbers. Example:
Subtraction:
input : \Sub \Three \Five
output : 11
5.1.4.3
Continuing the recursion
The same mechanism we used for defining multiplication from addition can be used to
define taking powers:
\def\ToThePower#1#2{%
\IsOne{#1}{#2}{%
\MultiplyBy{#2}{\ToThePower{\SubOne{#1}}{#2}}}}
Power taking:
input : \ToThePower {\Two }{\Three }
output : 111111111
5.1.4.4
Testing
Some arithmetic tests. Greater than: if
X = (x, X 0 ), Y = (y, Y 0 )
then Y > X is false if Y ≡ 0:
\def\GreaterThan#1#2{#2{\GreaterEqualp{#1}}\False}
Otherwise, compare X with Y 0 = Y − 1: Y > X ⇔ Y 0 ≥ X; this is true if X ≡ 0:
\def\GreaterEqualp#1#2#3{\GreaterEqual{#1}{#3}}
\def\GreaterEqual#1#2{#1{\LessThanp{#2}}\True}
Otherwise, compare X 0 = X − 1 with Y 0 = Y − 1:
\def\LessThanp#1#2#3{\GreaterThan{#3}{#1}}
Greater (true result):
input : \GreaterThan \Two
\Five
output : T
Greater (false result):
input : \GreaterThan \Three
\Two
output : F
Greater (equal case):
input : \GreaterThan \Two
\Two
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5.1. LOGIC WITH TEX
output : F
Greater than zero:
input : \GreaterThan \Two
\Zero
output : F
167
Greater than zero:
input : \GreaterThan \Zero
\Two
output : T
Instead of just printing ‘true’ or ‘false’, we can use the test to select a number or action:
Use true result:
input : \GreaterThan \Two \Five \Three \One
output : 111
Use false result:
input : \GreaterThan \Three \Two \Three \One
output : 1
Let’s check if the predicate can be used with arithmetic.
3 < (5 − 1):
input : \GreaterThan \Three {\Sub \One \Five }
output : T
3 < (5 − 4):
input : \GreaterThan \Three {\Sub \Four \Five }
output : F
Equality:
\def\Equal#1#2{#2{\Equalp{#1}}{\IsZero{#1}}}
\def\Equalp#1#2#3{#1{\Equalx{#3}}{\IsOne{#2}}}
\def\Equalx#1#2#3{\Equal{#1}{#3}}
Equality, true:
input : \Equal
output : T
Equality, true:
input : \Equal
output : T
Equality, false:
input : \Equal
output : F
Equality, false:
input : \Equal
output : F
\Five \Five
\Four \Four
\Five \Four
(1 + 3) ≡ 5: false:
input : \Equal {\Add \One
\Three }\Five
output : F
(2 + 3) ≡ (7 − 2): true:
input : \Equal {\Add \Two
\Three }{\Sub \Two \Seven
}
output : T
\Four \Five
Fun application:
\def\Mod#1#2{%
\Equal{#1}{#2}\Zero
{\GreaterThan{#1}{#2}%
{\Mod{#1}{\Sub{#1}{#2}}}%
{#2}%
}}
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Mod(27, 4) = 3:
input : \Mod \Four \TwentySeven
output : 111
Mod(6, 3) = 0:
input : \Mod \Three \Six
output :
With the modulo operation we can compute greatest common divisors:
\def\GCD#1#2{%
\Equal{#1}{#2}%
{#1}%
{\GreaterThan{#1}{#2}%
{\IsOne{#1}\One
{\GCD{\Sub{#1}{#2}}{#1}}}%
{\IsOne{#2}\One
{\GCD{\Sub{#2}{#1}}{#2}}}}}
GCD(27,4)=1:
input : \GCD \TwentySeven \Four
output : 1
GCD(27,3)=3:
input : \GCD \TwentySeven \Three
output : 111
% #2>#1
% then take GCD(#2-#1,#1)
% else GCD(#1-#2,#2)
and we can search for multiples:
\def\DividesBy#1#2{\IsZero{\Mod{#1}{#2}}}
\def\NotDividesBy#1#2{\GreaterThan\Zero{\Mod{#1}{#2}}}
\def\FirstDividesByStarting#1#2{%
\DividesBy{#1}{#2}{#2}{\FirstDividesByFrom{#1}{#2}}}
\def\FirstDividesByFrom#1#2{\FirstDividesByStarting{#1}{\AddOne{#2}}}
5|25:
input : \DividesBy \Five \TwentyFive
output : T
5 6 |27:
input : \DividesBy \Five \TwentySeven
output : F
5 6 |27:
input : \NotDividesBy \Five \TwentySeven
output : T
10 = min{i : i ≥ 7 ∧ 5|i}:
input : \FirstDividesByFrom \Five \Seven
output : 1111111111
5.1.5
Infinite lists
So far, we have dealt with lists that are finite, built up from an empty list. However, we can
use infinite lists too.
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169
\def\Stream#1{\Cons{#1}{\Stream{#1}}}
Infinite objects:
input : \Head {\Tail {\Stream 3}}
output : 3
Infinite objects:
input : \Head {\Tail {\Tail {\Tail {\Tail {\Tail
{\Stream 3}}}}}}
output : 3
Even though the list is infinite, we can easily handle it in finite time, because it is never
constructed further than we ask for it. This is called ‘lazy evaluation’.
We can get more interesting infinite lists by applying successive powers of an operator to
the list elements. Here is the definition of the integers by applying the AddOne operator a
number of times to zero:
% \StreamOp <operator> <initial value>
\def\StreamOp#1#2{\Cons{#2}{\StreamOp{#1}{#1{#2}}}}
\def\Integers{\StreamOp\AddOne\Zero}
\def\PositiveIntegers{\Tail\Integers}
\def\TwoOrMore{\Tail\PositiveIntegers}
Again, the Integers object is only formed as far as we need it:
Integers:
input : \Head {\Tail {\Integers }}
output : 1
Integers:
input : \Head {\Tail {\Tail {\Tail {\Tail {\Tail
{\Integers }}}}}}
output : 11111
Let us see if we can do interesting things with lists. We want to make a list out of everything
that satisfies some condition.
\def\ConsIf#1#2#3{#1{#2}{\Cons{#2}{#3}}{#3}}
\def\Doubles{\ListApply{\ConsIf{\DividesBy\Two}}\Nil\PositiveIntegers}
\def\AllSatisfy#1{\ListApply{\ConsIf{#1}}\Nil\PositiveIntegers}
\def\FirstSatisfy#1{\Head{\AllSatisfy{#1}}}
third multiple of two:
input : \Head {\Tail {\Tail \Doubles }}
output : 111111
old enough to drink:
input : \FirstSatisfy {\GreaterThan \TwentyOne }
output : 1111111111111111111111
We add the list in which we test as a parameter:
\def\AllSatisfyIn#1#2{\ListApply{\ConsIf{#1}}\Nil{#2}}
\def\FirstSatisfyIn#1#2{\Head{\AllSatisfyIn{#1}{#2}}}
:
input : \FirstSatisfyIn {\NotDividesBy
{\FirstSatisfyIn {\NotDividesBy \Two }\TwoOrMore
Victor Eijkhout
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CHAPTER 5. TEX’S MACRO LANGUAGE – UNFINISHED CHAPTER
}} {\AllSatisfyIn {\NotDividesBy \Two }\TwoOrMore
}
output : 11111
And now we can repeat this:
\def\FilteredList#1{\AllSatisfyIn{\NotDividesBy{\Head{#1}}}{\Tail{#1}}}
\def\NthPrime#1{\Head{\PrimesFromNth{#1}}}
\def\PrimesFromNth#1{\IsOne{#1}\TwoOrMore
{\FilteredList{\PrimesFromNth{\SubOne{#1}}}}}
Third prime; spelled out:
input : \Head {\FilteredList {\FilteredList
\TwoOrMore }}
output : 11111
Fifth prime:
input : \NthPrime \Four
output : 1111111
However, this code is horrendously inefficient. To get the 7th prime you can go make a cup
of coffee, one or two more and you can go pick the beans yourself.
%\def\FilteredList#1{\AllSatisfyIn{\NotDividesBy{\Head{#1}}}{\Tail{#1}}}
\def\xFilteredList#1#2{\AllSatisfyIn{\NotDividesBy{#1}}{#2}}
\def\FilteredList#1{\xFilteredList{\Head{#1}}{\Tail{#1}}}
Fifth prime:
input : \NthPrime \Five
output : 11111111111
TEX – LATEX – CS 594
Chapter 6
Character encoding
This chapter is about how to interpret the characters in an input file – no there ain’t such a
thing as a plain text file – and how the printed characters are encoded in a font.
Handouts and further reading for this chapter
There is very little printed material on this topic. A good introduction is http://www.
joelonsoftware.com/articles/Unicode.html; after that, http://www.cs.
tut.fi/˜jkorpela/chars.html is a good tutorial for general issues, and http:
//en.wikipedia.org/wiki/Unicode for Unicode.
For the technical details on Unicode consult http://www.unicode.org/. An introduction to ISO 8859: http://www.wordiq.com/definition/ISO_8859.
171
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CHAPTER 6. CHARACTER ENCODING
Input file encoding.
6.1
History and context
6.1.1
One-byte character sets; Ascii
Somewhere in the depths of prehistory, people got to agree on a standard for character codes
under 127, A SCII. Unlike another encoding scheme, E BCDIC, it has a few nice properties.
•
•
•
•
All letters are consecutive, making a test ‘is this a letter’ easy to perform.
Uppercase and lowercase letters are at a distance of 32.
The first 31 codes, everything below the space character, as well as position 127,
are ‘unprintable’, and can be used for such purposes as terminal cursor control.
Unprintable codes are accessible through the control modifier (for this reason
they are also called ‘control codes’), which zeros bits 2 and 3: hit Ctrl-[ to get
Esc1 .
The ISO 646 standard codified 7-bit A SCII, but it left certain character positions (or ‘code
points’) open for national variation. For instance, British usage put a pound sign (£) in the
position of the dollar. The A SCII character set was originally accepted as ANSI X3.4 in
1968.
6.1.2
Code pages
This left the codes with the high bit set (‘extended A SCII’) undefined, and different manufacturers of computer equipment came up with their own way of filling them in. These
standards were called ‘code pages’, and IBM gave a standard numbering to them. For
instance, code page 437 is the MS-DOS code page with accented characters for most European languages, 862 is DOS in Israel, 737 is DOS for Greeks.
1.
The way key presses generate characters is typically controlled in software. This mapping from keyboard
scan codes to 7 or 8-bit characters is called a ‘keyboard’, and can be changed dynamically in most operating
systems.
TEX – LATEX – CS 594
6.1. HISTORY AND CONTEXT
173
Here is cp473:
MacRoman:
and Microsoft cp1252:
More code pages are displayed on http://aspell.net/charsets/codepages.
html. These diagrams can be generated from Unicode mapping tables, which look like
=20
=21
=22
U+0020
U+0021
U+0022
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SPACE
EXCLAMATION MARK
QUOTATION MARK
174
CHAPTER 6. CHARACTER ENCODING
...
=A3
U+00A3 POUND SIGN
=A4
U+20AC EURO SIGN
=A5
U+00A5 YEN SIGN
...
The international variants were standardized as ISO 646-DE (German), 646-DK (Danish),
et cetera. Originally, the dollar sign could still be replaced by the currency symbol, but after
a 1991 revision the dollar is now the only possibility.
6.1.3
ISO 8859
The different code pages were ultimately standardized as ISO 8859, with such popular code
pages as 8859-1 (‘Latin 1’) for western European,
8859-2 for eastern, and 8859-5 for Cyrillic.
These ISO standards explicitly left the first 32 extended positions undefined. Microsoft
code page 1252 uses ISO 8859-1.
More useful information about A SCII: http://jimprice.com/jim-asc.htm. History of A SCII out of telegraph codes: http://www.wps.com/projects/codes/
index.html. A history, paying attention to multilingual use: http://tronweb.super-nova.
co.jp/characcodehist.html History as written by the father of A SCII: Bob Bemer
http://www.bobbemer.com/HISTORY.HTM.
A good inventory of ISO 8859, Latin-1: http://www.cs.tut.fi/˜jkorpela/latin1/
index.html, with a discussion by logical grouping: http://www.cs.tut.fi/˜jkorpela/
latin1/4.html.
6.1.4
DBCS
Since certain Asian alphabets do not fit in 256 positions, a system called the ‘Double Byte
Character Set’ was invented where some characters were stored in one, others in two bytes.
TEX – LATEX – CS 594
6.2. UNICODE
175
This is very messy, since you can not simply write s++ or s-- to traverse a string. Instead
you have to use functions from some library that understands these encodings. This system
is now only of historical interest.
6.2
Unicode
The systems above functioned quite well as long as you stuck to one language or writing
system. Poor dictionary makers. More or less simultaneously two efforts started that aimed
to incorporate all the world’s character sets in one standard: Unicode standard (originally
2-byte), and ISO 10646 (oringally 4-byte). Unicode then was extended, so that it has all
numbers up to 10FFFFF, which is slightly over a million.
6.2.1
ISO 10646 and Unicode
Two international standards organizations, the Unicode Consortium and ISO/IEC JTC1/SC2,
started designing a universal standard that was to be a superset of all existing character sets.
These standards are now synchronized. Unicode has elements that are not in 10646, but
they are compatible where it concerns straight character encoding.
ISO 10646 defines UCS, the ‘Universal Character Set’. This is in essence a table of official
names and code numbers for characters. Unicode adds to this rules for hyphenation, bidirectional writing, and more.
The full Unicode list of code points can be found, broken down by blocks, online at http:
//www.fileformat.info/info/unicode/index.htm, or downloadable at http:
//www.unicode.org/charts/.
6.2.2
BMP and earlier standards
Characters in Unicode are mostly denoted hexadecimally as U+wxyz, for instance U+0041
is ‘Latin Capital Letter A’. The range U+0000–U+007F (0–127) is identical to US-ASCII
(ISO 646 IRV), and U+0000–U+00FF (0–255) is identical to Latin 1 (ISO 8859-1).
The original 2-byte subset is now called ‘BMP’ for Basic Multilingual Plane.
From http://www.hyperdictionary.com/:
BMP (Basic Multilingual Plane) The first plane defined in Unicode/ISO 10646, designed
to include all scripts in active modern use. The BMP currently includes the Latin,
Greek, Cyrillic, Devangari, hiragana, katakana, and Cherokee scripts, among others, and a large body of mathematical, APL-related, and other miscellaneous characters. Most of the Han ideographs in current use are present in the BMP, but due
to the large number of ideographs, many were placed in the Supplementary Ideographic Plane.
SIP (Supplementary Ideographic Plane) The third plane (plane 2) defined in Unicode/ISO
10646, designed to hold all the ideographs descended from Chinese writing (mainly
found in Vietnamese, Korean, Japanese and Chinese) that aren’t found in the Basic Multilingual Plane. The BMP was supposed to hold all ideographs in modern
Victor Eijkhout
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CHAPTER 6. CHARACTER ENCODING
use; unfortunately, many Chinese dialects (like Cantonese and Hong Kong Chinese) were overlooked; to write these, characters from the SIP are necessary. This
is one reason even non-academic software must support characters outside the
BMP.
6.2.3
Unicode encodings
Unicode is basically a numbered list of characters. When they are used in a file, their
numbers can be encoded in a number of ways. To name the obvious example: if only the
first 128 positions are used, the long Unicode code point can be truncated to just one byte.
Here are a few encodings:
UCS-2 Obsolete: this was the original ‘native’ two-byte encoding before Unicode was
extended.
UTF-32 Little used: this is a four-byte encoding. (UTF stands for ‘UCS Transformation
Format’.)
UTF-16 This is the BMP.
UTF-8 A one-byte scheme; details below.
UTF-7 Another one-byte scheme, but now the high bit is always off. Certain byte values
act as ‘escape’, so that higher values can be encoded. Like UTF-1 and SCSU, this
encoding is only of historical interest.
There is an important practical reason for UTF-8. Encodings such as UCS-2 are wasteful
of space, if only traditional A SCII is needed. Furthermore, they would break software that
is expecting to walk through a file with s++ and such. Also, they would introduce many
zero bytes in a file, which would play havoc with Unix software that uses null-termination
for strings. Then there would be the problem of whether two bytes are stored in low-endian
or high-endian order. For this reason it was suggested to store FE FF or FF FE at the
beginning of each file as the ‘Unicode Byte Order Mark’. Of course this plays havoc with
files such as shell scripts which expect to find #! at the beginning of the file.
6.2.4
UTF-8
UTF-8 is an encoding where the positions up to 127 are encoded ‘as such’; higher numbers
are encoded in groups of 2 to 6 bytes. (UTF-8 is standardized as RFC 3629.) In a multibyte group, the first byte is in the range 0xC0–0xFD (192–252). The next up to 5 bytes are
in the range 0x80–0xBF (128–191, bit pattern starting with 10). Note that 8 = 1000 and
B = 1011, so the highest two bits are always 10, leaving six bits for encoding).
U-00000000
U-00000080
U-00000800
U-00010000
U-00200000
U-04000000
-
U-0000007F
U-000007FF
U-0000FFFF
U-001FFFFF
U-03FFFFFF
U-7FFFFFFF
7 bits
11 = 5 + 6
16 = 4 + 2 × 6
21 = 3 + 3 × 6
26 = 2 + 4 × 6
31 = 1 + 5 × 6
0xxxxxxx
110xxxxx
1110xxxx
11110xxx
111110xx
1111110x
10xxxxxx
10xxxxxx 10xxxxxx
10xxxxxx (3 times)
10xxxxxx (4 times)
10xxxxxx (5 times)
All bites in a multi-byte sequence have their high bit set.
Exercise 49. Show that a UTF-8 parser will not miss more than two
characters if a byte becomes damaged (any number of bits arbitrarily
changed).
TEX – LATEX – CS 594
6.2. UNICODE
177
IETF documents such as RFC 2277 require support for this encoding in internet software.
Here is a good introduction to UTF-8 use in Unix: http://www.cl.cam.ac.uk/
˜mgk25/unicode.html. The history of it: http://www.cl.cam.ac.uk/˜mgk25/
ucs/utf-8-history.txt.
6.2.5
Unicode tidbits
6.2.5.1
Line breaking
See http://www.cs.tut.fi/˜jkorpela/unicode/linebr.html and http:
//www.unicode.org/reports/tr14/
6.2.5.2
Bi-directional writing
Most scripts are left-to-right, but Arabic and Hebrew run right-to-left. Characters in a file
are stored in ‘logical order’, and usually it is clear in which direction to render them, even
if they are used mixed. Letters have a ‘strong’ directionality: unless overridden, they will
be displayed in their natural direction. The first letter of a paragraph with strong direction
determines the main direction of that paragraph.
However, when differently directional texts are embedded, some explicit help is needed.
The problem arises with letters that have only weak directionality. The following is a sketch
of a problematic case.
Memory: he said ”I NEED WATER!”, and expired.
Display: he said ”RETAW DEEN I!”, and expired.
If the exclamation mark is to be part of the Arabic quotation, then the user can select the
text ‘I NEED WATER!’ and explicitly mark it as embedded Arabic (<RLE> is Right-Left
Embedding; <PDF> Pop Directional Format), which produces the following result:
Memory: he said ”<RLE>I NEED WATER!<PDF>”, and expired.
Display: he said ”!RETAW DEEN I”, and expired.
A simpler method of doing this is to place a Right Directional Mark <RLM> after the
exclamation mark. Since the exclamation mark is now not on a directional boundary, this
produces the correct result.
Memory: he said ”I NEED WATER!<RLM>”, and expired.
Display: he said ”!RETAW DEEN I”, and expired.
For the full definition, see http://www.unicode.org/reports/tr9/.
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6.2.6
CHAPTER 6. CHARACTER ENCODING
Unicode and oriental languages
‘Han unification’ is the Unicode strategy of saving space in the oriental languages (traditional Chinese, simplified Chinese, Japanese, Korean: ‘CJK’) by recognizing common
characters. This idea is not uncontroversial; see http://en.wikipedia.org/wiki/
Han_unification.
6.3
More about character sets and encodings
6.3.1
Character sets
Informally, the term ‘character set’ (also ‘character code’ or ‘code’) used to mean something like ‘a table of bytes, each with a character shape’. With only the English alphabet to
deal with that is a good enough definition. These days, much more general cases are handled, mapping one octet into several characters, or several octets into one character. The
definition has changed accordingly:
A ‘charset is a method of converting a sequence of octets into a sequence
of characters. This conversion may also optionally produce additional
control information such as directionality indicators.
(From RFC 2978) A conversion the other way may not exist, since different octet combinations may map to the same character. Another complicating factor is the possibility of
switching between character sets; for instance, ISO 2022-JP is the standard A SCII character
set, but the escape sequence ESC $ @ switches to JIS X 0208-1978.
6.3.2
From character to encoding in four easy steps
To disentangle the concepts behind encoding, we need to introduce a couple of levels:
ACR Abstract Character Repertoire: the set of characters to be encoded; for example,
some alphabet or symbol set. This is an unordered set of characters, which can be
fixed (the contents of ISO 8859-1), or open (the contents of Unicode).
CCS Coded Character Set: a mapping from an abstract character repertoire to a set of nonnegative integers. This is what is meant by ‘encoding’, ‘character set definition’,
or ‘code page’; the integer assigned to a character is its ‘code point’.
There used to be a drive towards unambiguous abstract character names across
repertoires and encodings, but Unicode ended this, as it provides (or aims to provide) more or less a complete list of every character on earth.
CEF Character Encoding Form: a mapping from a set of nonnegative integers that are
elements of a CCS to a set of sequences of particular code units. A ‘code unit’ is
an integer of a specific binary width, for instance 8 or 16 bits. A CEF then maps
the code points of a coded character set into sequences of code point, and these
sequences can be of different lengths inside one code page. For instance
•
A SCII uses a single 7-bit unit
•
UCS-2 uses a single 16-bit unit
•
DBCS uses two 8-bit units
•
UTF-8 uses one to four 8-bit units.
TEX – LATEX – CS 594
6.3. MORE ABOUT CHARACTER SETS AND ENCODINGS
179
•
UTF-16 uses a mix of one and two 16-bit code units.
CES Character Encoding Scheme: a reversible transformation from a set of sequences of
code units (from one or more CEFs to a serialized sequence of bytes. In cases
such as A SCII and UTF-8 this mapping is trivial. With UCS-2 there is a single
‘byte order mark’, after which the code units are trivially mapped to bytes. On the
other hand, ISO 2022, which uses escape sequences to switch between different
encodings, is a complicated CES.
Additionally, there are the concepts of
CM Character Map: a mapping from sequences of members of an abstract character repertoire to serialized sequences of bytes bridging all four levels in a single operation.
These maps are what gets assigned MIBenum values by IANA; see section 6.3.3.
TES Transfer Encoding Syntax: a reversible transform of encoded data. This data may
or may not contain textual data. Examples of a TES are base64, uuencode, and
quoted-printable, which all transform a byte stream to avoid certain values.
6.3.3
A bootstrapping problem
In order to know how to interpret a file, you need to know what character set it uses. This
problem also occurs in MIME mail encoding (section 6.3.5), which can use many character sets. Names and numbers for character sets are standardized by IANA: the Internet
Assigned Names Authority (http://www.iana.org/). However, in what character
set do you write this name down?
Fortunately, everyone agrees on (7-bit) A SCII, so that is what is used. A name can be up to
40 characters from us-ascii.
As an example, here is the iana definition of A SCII:
name
ANSI_X3.4-1968
reference RFC1345,KXS2
MIBenum 3
source ECMA registry
aliases iso-ir-6, ANSI_X3.4-1986, ISO_646.irv:1991, ASCII, ISO646-US,
US-ASCII (preferred MIME name), us, IBM367, cp367, csASCII
The MIBenum (Management Information Base) is a number assigned by IANA2 . The full
list of character sets is at http://www.iana.org/assignments/character-sets,
and RFC 3808 is a memo that describes the IANA Charset MIB.
6.3.4
Character codes in HTML
HTML can access unusual characters in several ways:
•
With a decimal numerical code:   is a space token. (HTML 4 supports
hexadecimal codes.)
•
With a vaguely symbolic name: © is the copyright symbol. See http:
//www.cs.tut.fi/˜jkorpela/HTML3.2/latin1.html for a list of
symbolic names in Latin-1.
2.
Apparently these numbers derive from the Printer MIB, RFC 1759.
Victor Eijkhout
180
•
CHAPTER 6. CHARACTER ENCODING
The more interesting way is to use an encoding such as UTF-8 (section 6.2.3) for
the file. For this it would be nice if the server could state that the file is
Content-type: text/html;charset=utf-8
but it is also all right if the file starts with
<META HTTP-EQUIV="Content-Type" CONTENT="text/html;charset=utf-8">
It is requirement that user agents can at least parse the charset parameter, which means
they have to understand us-ascii.
Open this link in your browser, and additionally view the source: http://www.unicode.
org/unicode/iuc10/x-utf8.html. How well does your software deal with it?
See also section 6.7.1.
6.3.5
Characters in email
6.3.6
FTP
FTP is a very old ARPA protocol. It knows ‘binary’ and ‘text’ mode, but the text mode is
not well defined. Some ftp programs adjust line ends; others, such as Fetch on the Mac,
actually do code page translation.
6.3.7
Character encodings in editors and programming languages
Software must be rewritten to use character encodings. Windows NT/2000/XP, including
Visual Basic, uses UCS-2 as native string type. Strings are declared of type wchar_t
instead of char, and the programmer uses wcslen instead of strlen, et cetera. A literal
string is created as L"Hello world".
6.4
Character issues in TEX / LATEX
6.4.1
Diacritics
Original TEX is not very good at dealing with diacritics. They are implemented as things to
put on top of characters, even when, as with the cedilla, they are under the letter. Furthermore, TEX can not hypenate a word with accents, since the accent introduces a space in the
word (technically: an explicit kern). Both problems were remedied to a large extent with
the ‘Cork font encoding’, which contains most accented letters as single characters. This
means that accents are correctly placed by design, and also that the word can be hyphenated, since the kern has disappeared.
TEX – LATEX – CS 594
6.4. CHARACTER ISSUES IN TEX / LATEX
181
These fonts with accented characters became possible when TEX version 3 came out around
1990. This introduced full 8-bit compatibility, both in the input side and in the font addressing.
6.4.2
LATEX input file access to fonts
If an input file for LATEX is allowed to contain all 8-bit octets, we get all the problems of
compatibility that plagued regular text files. This is solved by the package inputenc:
\usepackage[code]{inputenc}
where codes is applemac, ansinew, or various other code pages.
This package makes all unprintable A SCII characters, plus the codes over 127, into active
characters. The definitions are then dynamically set depending on the code page that is
loaded.
6.4.3
LATEX output encoding
The inputenc package does not solve the whole problem of producing a certain font
character from certain keyboard input. It only mapped a byte value to the TEX command
for producing a character. To map such commands to actual code point in a font file, the
TEX and LATEX formats contain lines such as
\chardef\i="10
declaring that the dotless-i is at position 16. However, this position is a convention, and
other people – type manufacturers – may put it somewhere else.
This is handled by the ‘font encoding’ mechanism. The various people working on the
LATEX font schemes have devised a number of standard font encodings. For instance, the
OT1 encoding corresponds to the original 128-character set. The T1 encoding is a 256character extension thereof, which includes most accented characters for Latin alphabet
languages.
A font encoding is selected with
\usepackage[T1]{fontenc}
A font encoding definition contains lines such as
\DeclareTextSymbol{\AE}{OT1}{29}
\DeclareTextSymbol{\OE}{OT1}{30}
\DeclareTextSymbol{\O}{OT1}{31}
\DeclareTextSymbol{\ae}{OT1}{26}
\DeclareTextSymbol{\i}{OT1}{16}
6.4.4
Virtual fonts
Exercise 50. What does an ALT key do?
Exercise 51. What is E BCDIC? What is the basic idea?
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CHAPTER 6. CHARACTER ENCODING
Exercise 52. Find the Unicode definition. Can you find an example of
a character that has two functions, but is not defined as two characters?
Find two characters that are defined seperately for compatibility, but that
are defined equivalent.
Exercise 53. ISO 8859 has the ‘non-breaking space’ at position A0.
How does TEX handle the nbsp? How do TEX, HTML, Latin-1, MS
Word, et cetera handle multiple spaces? Discuss the pros and cons.
TEX – LATEX – CS 594
6.5. BASIC TERMINOLOGY
183
Figure 6.1: Different shapes of ‘lowercase roman a’
Font encoding.
6.5
Basic terminology
Terminology of fonts and typefaces is quite confused these days. Traditionally, a typeface
was a design, realized in number of fonts, that could be sorted in families, such as roman,
and italic. A font would then be a style (medium weight bold italic) in a particular size.
Somewhat surprisingly, once you start throwing computers at this problem, even talking
about characters becomes very subtle.
In Unicode, there are abstract characters and characters. They don’t differ by much: an
abstract character is a concept such as ‘Latin lowercase a with accent grave’, and a character
is that concept plus a position in the Unicode table. The actually visible representation of
a character is called a ‘glyph’. According to ISO 9541, a glyph is ‘A recognizable abstract
graphic symbol which is independent of any specific design’.
6.5.1
The difference between glyphs and characters
Often, the mapping between character and glyph is clear: we all know what we mean by
‘Uppercase Roman A’. However, there may be different glyph shapes that correspond to
the same character.
An abstract character is defined as
abstract character: a unit of information used for the organization, control, or representation of textual data.
This definition has some interesting consequences. Sometimes one glyph can correspond
to more than one character, and the other way around.
For example, in Danish, the ligature ‘æ’ is an actual character. On the other hand, the
ligature ‘fl’, which appears in English texts, is merely a typographical device to make the
combination ‘fl’ look better, so one glyph corresponds to two characters.
The opposite case is rarer. In Tamil, a certain character is split, because it is positioned
around other characters. It can then even happen that one of the split parts forms a ligature
with adjacent characters.
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CHAPTER 6. CHARACTER ENCODING
Figure 6.2: The f-i ligature
Figure 6.3: A split character in Tamil
A tricker question is how to handle accented letters: is ‘´e’ one character or a combination
of two? In math, is the relation in a 6= b one symbol, or an overstrike of one over another?
Another problem with ligatures is that a single glyph needs to be displayed, but two glyphs
need to be stored to make searching for the string possible.
6.5.2
The identity of a character
Another problem in defining a character is whether two glyphs that look the same, or sometimes ever are the same, should be the same character. For example, uppercase Latin a,
uppercase Greek α, and uppercase Cyrillic a, are all rendered ‘A’. Still, in Unicode they are
three distinct characters.
Similarly, in A SCII, there are no separate glyphs for minus, hyphen, and dash. In Unicode,
these are three characters. Is the character ‘superscript 2 ’ a separate glyph, or a typographical variant of the character ‘digit 2’? The latter should be the logical solution, but for
compatibility reasons with other standards it is defined as a separate glyph. There are quite
a few of these ‘compatibility characters.
Yet another example is the Greek letter Ω, which can be that letter, or the sign for electrical resistance in physics texts. Unicode defines them as two characters, but with identical
glyphs.
A capital ‘A’ in Times Roman and in Helvetica are the same character, but what about
italic?
All these matters are hard to settle objectively: everything is a matter of definition, or a
judgement call. The official Unicode white paper on characters versus glyphs is http:
//www.unicode.org/reports/tr17/.
Here are some of the guidelines of the Unicode project:
•
The Unicode Standard encodes characters, not glyphs.
TEX – LATEX – CS 594
6.6. ÆSTHETICS
185
Figure 6.4: Different interpretations of an accented character glyph
•
•
•
Characters have well-defined semantics.
The Unicode Standard encodes plain text.
And:
The Unicode Standard avoids duplicate encoding of characters by unifying them within scripts across languages; characters that are equivalent in form are given a single code. Common letters, punctuation
marks, symbols, and diacritics are given one code each, regardless of
language, [. . . ]
6.5.3
Diacritics
Unicode, a bit like TEX, has two ways of dealing with diacritics. It has precomposed accented characters, but it can also compose accented characters by listing accents (yes, plural: transliterated Vietnamese regularly has two accents over a letter one relating to vowel
quality, and one to tone) after the base symbol. This mechanism can also deal with languages such as Hangul (Korean) which have composite characters.
6.6
Æsthetics
6.6.1
Scaling versus Design sizes
Lots of attention is devoted to font scaling, with the implicit assumption that that is the
way to get a font to display at different sizes. This is only true to an extent: a small version
of a typeface was traditionally of a different design than the same typeface at larger sizes.
With metal type, independent designs for the different sizes were of course the only way
one could proceed at all, but with photo typesetters and computers the need went away,
and with it the realization that independent designs are visually actually a Good Thing.
Figure 6.5 shows the difference between a typeface set at its ‘design size’, and a scaled up
Figure 6.5: A typeface and a smaller version scaled up
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Figure 6.6: Adobe’s optical masters for a typeface
smaller version of it.
Adobe incorporated this idea in their Multiple Masters typefaces, which could interpolate
between different designs. This technology seems to have been abandoned, but Adobe’s
Originals now have so-called ‘Optical masters: four different designs of the same typeface,
to be set at different sizes. Adobe labels their purposes as ‘display’, ‘subhead’, ‘text’, and
‘caption’ in decreasing design size; see figure 6.6.
Apple developed their own version of multiple design sizes in TrueType GX, released in
1994. The ideas in TrueType GX are incorporated in Apple Advanced Typography (AAT)
in OS X, but there few AAT typefaces, and certainly very few non-Apple ones.
6.7
Font technologies
6.7.1
Unicode in fonts
It is unrealistic to expect any single font to support even a decent fraction of the Unicode
character repertoire. However, TrueType and OpenType do support Unicode.
The few fonts that support (almost) the whole of Unicode are called ‘pan-Unicode’. There
are only a few of those. However, software these days is pretty sophisticated in gathering together symbols from disparate fonts. Some browsers do this, prompting the user for
‘install on demand’ of fonts if necessary.
6.7.2
Type 1 and TrueType
Type 1 (‘Postscript fonts’) was the outline font format developed by Adobe that was adopted
by Apple in the mid 1980s. Since it was proprietary (Adobe had release the specifications
for Type 3 fonts, but not Type 1), Apple and Microsoft later developed TrueType.
With Type 1 fonts, information is stored in two files, one for shape data and one for hinting
and such. With TrueType, all information is in the one font file.
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6.7. FONT TECHNOLOGIES
6.7.2.1
187
Type1
Adobe Type 1 fonts are stored in two common formats, .pfa (PostScript Font ASCII) and
.pfb (PostScript Font Binary). These contain descriptions of the character shapes, with each
character being generated by a small program that calls on other small programs to compute
common parts of the characters in the font. In both cases, the character descriptions are
encrypted.
Before such a font can be used, it must be rendered into dots in a bitmap, either by the
PostScript interpreter, or by a specialized rendering engine, such as Adobe Type Manager,
which is used to generate low-resolution screen fonts on Apple Macintosh and on Microsoft
Windows systems.
The Type 1 outline files do not contain sufficient information for typesetting with the font,
because they have only limited metric data, and nothing about kerning (position adjustments of particular adjacent characters) or ligatures (replacement of adjacent characters by
a single character glyph, those for fi, ffi, fl, and ffl being most common in English typography).
This missing information is supplied in additional files, called .afm (Adobe Font Metric)
files. These are ASCII files with a well-defined easy-to-parse structure. Some font vendors,
such as Adobe, allow them to be freely distributed; others, such as Bitstream, consider them
to be restricted by a font license which must be purchased.
6.7.2.2
TrueType ⇔ Type1 conversion
Beware! There is no such thing as a one-to-one reversible conversion. There are several
problems:
The outlines are stored in different ways in both formats. In truetype, second-order Bezier
curves are used, and in type 1, third-order Bezier curves are employed. One second order
Bezier can be transformed into a third-order Bezier, but a third-order Bezier cannot be
transformed into one, two or seventeen second-order Beziers–approximations are in order
for that conversion. So, type 1 to truetype is problematic, right from the start. For truetype
to type 1, there is a snake in the grass, in the form of integer grid rounding (see below).
Both formats require all control points to be integers (whole numbers), falling in a grid.
Truetype uses a 2048x2048 grid, type 1 typically a 1000x1000 grid. For the truetype to type
1 direction, one could divide all grid values by two, but then what? Should 183.5 become
183 or 184? The type 1 to truetype direction is easier, at least from this point of view, as
we could multiply each grid coordinate by two, so no rounding loss would be involved.
However, in the truetype to type 1 direction, the rounding causes additional problems for
the new control points needed for the perfect third-order Bezier outlines mentioned above.
Placing ink on paper: the formats have different rules for placing ink on paper in case of
outlines that are nested or intersecting. These differences are not caught by many conversion programs. In most cases, the user should not worry about this—only rarely do we have
overlapping outlines (I was forced once to have them, for other reasons).
Complexity of the outlines: truetype permits more complex outlines, with more control
points. For example, I am sure you have all seen fonts made from scans of pictures of faces
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of people. Typically, these outlines are beyond the type 1 limit, so this restriction makes
the truetype to type 1 conversion impossible for ultra complex fonts.
Encoding: truetype can work with a huge number of glyphs. There are truetype fonts for
Chinese and Japanese, for example. In type 1, the number of active glyphs is limited to
256. Again, for most Latin fonts, this is a non-issue.
The remarks about grid rounding also apply to all metrics, the bounding boxes, the character widths, the character spacing, the kerning, and so forth.
Finally, there is the hinting. This is handled very differently in both formats, with truetype
being more sophisticated this time. So, in truetype to type 1 conversions of professionally
(hand-hinted) fonts, a loss will occur. Luckily, 99% of the truetype fonts do not make use
of the fancy hinting possibilities of truetype, and so, one is often safe.
All this to tell people to steer away like the plague from format conversions. And a plea
to the font software community to develop one final format. My recommendation: get rid
of truetype, tinker with the type 1 format (well, tinker a lot). More about that ideal format
elsewhere.
6.7.2.3
Downsampling bitmaps
In principle, given adequate resolution, the screen preview quality of documents set in
bitmap fonts, and set in outline fonts, should be comparable, since the outline fonts have to
be rasterized dynamically anyway for use on a printer or a display screen.
Sadly, this is not the case with versions of Adobe Acrobat Reader, acroread, and Exchange,
acroexch (version 5.x or earlier); they do a poor job of downsampling high-resolution
bitmap fonts to low-resolution screen fonts. This is particularly inexcusable, inasmuch as
the co-founder, and CEO, of Adobe Systems, is the author of one of the earliest publications on the use of gray levels for font display: [ John E. Warnock, The display of characters
using gray level sample arrays, Computer Graphics, 14 (3), 302–307, July, 1980. ]
6.7.3
FreeType
FreeType is an Open Source implementation of TrueType. Unfortunately this runs into
patent problems, since Apple has patented some of the hinting mechanism. Recently FreeType
has acquired an automatic hinting engine.
6.7.4
OpenType
OpenType is a standard developed by Adobe and Microsoft. It combines
bitmap, outline, and metric information in a single cross-platform file. It has Unicode support, and can use ‘Optical Masters’ (section 6.6.1) multiple designs. It knows about the
distinction between code points and glyphs, so applications can render a character differently based on context.
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6.8. FONT HANDLING IN TEX AND LATEX
6.8
189
Font handling in TEX and LATEX
TEX has fairly sophisticated font handling, in the sense that it knows a lot about the characters in a font. However, its handling of typefaces and relations between fonts is primitive.
LATEX has a good mechanism for that.
6.8.1
TEX font handling
Font outlines can be stored in any number of ways; TEX is only concerned with the ‘font
metrics’, which are stored in a ‘tfm file’. These files contain
•
•
•
Global information about the font: the \fontdimen parameters, which describe
the spacing of the font, but also the x-height, and the slant-per-point, which describes the angle of italic and slanted fonts.
Dimensions and italic corrections of the characters.
Ligature and kerning programs.
We will look at these in slightly more detail.
6.8.1.1
Font dimensions
The tfm file specifies the natural amount of space, with stretch and shrink for a font, but
also a few properties related to the size and shape of letters. For instance, it contains the
x-height, which is the height of characters without ascenders and descenders. This is, for
instance, used for accents: TEX assumes that accents are at the right height for characters
as high as an ‘x’: for any others the accent is raised or lowered.
The ‘slant per point’ parameters is also for use in accents: it determines the horizontal offset
of a character.
6.8.1.2
Character dimensions
The height, width, and depth of a character is used to determine the size of the enclosing
boxes of words. A non-trivial character dimension is the ‘italic correction’. A tall italic
character will protrude from its bounding box (which apparently does not always bound).
The italic correction can be added to a subsequent space.
‘TEX has’ versus ‘TEX has’
6.8.1.3
Ligatures and kerning
The tfm file contains information that certain sequences of characters can be replaced by
another character. The intended use of this is to replace sequences such as fi or fl by ‘fi’
or ‘fl’.
Kerning is the horizontal spacing that can bring characters closer in certain combinations.
Compare
‘Von’ versus ‘Von’
Kerning programs are in the tfm file, not accessible to the user.
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6.8.2
CHAPTER 6. CHARACTER ENCODING
Font selection in LATEX
Font selection in LATEX (and TEX) was rather crude in the early versions. Commands such
as \bf and \it switched to boldface and italic respectively, but could not be combined to
give bold italic. The New Font Selection Scheme improved that situation considerably.
With NFSS, it becomes possible to make orthogonal combinations of the font family (roman, sans serif), series (medium, bold), and shape (upright, italic, small caps). A quick
switch back to the main document font is \textnormal or \normalfont.
6.8.2.1
Font families
It is not necessary for a typeface to have both serifed and serifless (sans serif) shapes.
Often, therefore, these shapes are taken from different, but visually compatible typefaces,
for instance combining Times New Roman with Helvetica. This is the combination that
results from
\usepackage{times}
Loading the package lucidabr instead, gives Lucida Bright and Lucida Sans.
The available font families are
roman using the command \textrm and the declaration \rmfamily.
sans serif using the command \textsf and the declaration \sffamily.
typewriter type using the command \texttt and the declaration \ttfamily. Typewriter type is usually a monospaced font – all characters of the same width – and
is useful for writing about LATEX or for giving code samples.
6.8.2.2
Font series: width and weight
The difference between normal and medium width, or normal and bold weight, can be
indicated with font series commands:
medium width/weight using the command \textmd and the declaration \mdseries.
bold using the command \textbf and the declaration \bfseries.
6.8.2.3
Font shape
The final parameter with which to classify fonts is their shape.
upright This is the default shape, explicitly available through \textup or \upshape.
italic and slanted These are often the same; they are available through \textit, \textsl,
and \itshape, \slshape.
small caps Here text is set through large and small capital letters; this shape is available
through \textsc and \scshape.
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6.9. THE FONTENC PACKAGE
191
Input and output encoding in LATEX.
6.9
The fontenc package
Traditionally, in TEX accented characters were handled with control characters, such as
in \’e. However, many keyboards – and this should be understood in a software sense –
are able to generate accented characters, and other non-latin characters, directly. Typically,
this uses octets with the high bit set.
As we have seen, the interpretation of these octets is not clear. In the absense of some
Unicode encoding, the best we can say is that it depends on the code page that was used.
This dependency could be solved by having the TEX software know, on installation, what
code page the system is using. While this may be feasible for one system, if the input files
are moved to a different system, they are no longer interpreted correctly. For this purpose
the inputenc package was developed.
An input encoding can be stated at the load of the package:
\usepackage[cp1252]{inputenc}
or input encodings can be set and switched later:
\inputencoding{latin1}
With this, a (part of a) file can be written on one machine, using some code page, and still
be formatted correctly on another machine, which natively has a different code page.
These code pages are all conventions for the interpretation of singly octets. The inputenc
package also has limited support for UTF-8, which is a variable length (up to four octets)
encoding of Unicode.
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Projects for this chapter.
Project 6.1. What is the problem with ‘Han unification’? (section 6.2.6) Discuss history,
philology, politics, and whatever may be appropriate.
Project 6.2. How do characters get into a file in the first place? Discuss keyboard scan
codes and such. How do modifier keys work? How can an OS switch between
different keyboard layouts? What do APIs for different input methods look like?
Project 6.3. Dig into history (find the archives of alt.folklore.computers!) and
write a history of character encoding, focusing on the pre-ascii years. Describe
design decisions made in various prehistoric computer architectures. Discuss.
TEX – LATEX – CS 594
Chapter 7
Software engineering
In the course of writing TEX and Metafont, Knuth developed some interesting ideas about
software engineering. We will put those in the context of other writings on the subject. One
of the by-products of TEX is the Web system for ‘literate programming’. We will take a
look at that, as well as at the general idea of markup.
Handouts and further reading for this chapter
Knuth wrote a history of the TEX project in [10], reprinted in ‘Literate Programming’,
which is on reserve in the library, QA76.6.K644 1992 .
One of the classics of software engineering is Frederick Brooks’ ‘The Mythical ManMonth’ [6].
For software engineering research, consult the following journals:
•
Software practice and experience
•
Journal of systems and software
•
ACM Transactions on Software Engineering and Methodology
•
IEEE Transactions on Reliability
Some of these the library has available online.
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CHAPTER 7. SOFTWARE ENGINEERING
Literate programming.
7.1
The Web system
7.2
Knuth’s philosophy of program development
7.2.1
The Trip and Trap tests
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7.3. EXTREMELY BRIEF HISTORY OF TEX
Software engineering.
7.3
195
(Quotes by Knuth in this chapter taken from [10].)
Extremely brief history of TEX
Knuth wrote a first report on TEX in early 1977, and left it to graduate students Frank Liang
and Michael Plass to implement it over the summer. Starting from their prototype, he then
spent the rest of 1977 and early 1978 implementing TEX and producing fonts with the first
version of M ETAFONT.
TEX was used by Knuth himself in mid 1978 to typeset volume 2 of The Art of Computer
Programming; it was in general use by August of 1978. By early 1979, Knuth had written
a system called Doc that was the precursor of W EB, and that produced both documentation
and a portable Pascal version of the source; the original program was written in Sail.
In 1980 Knuth decided to rewrite TEX and M ETAFONT. He started on this in 1981, and
finished, including producing the five volumes of Computer and Typesetting, in 1985.
7.4
TEX’s development
7.4.1
Knuth’s ideas
Inspecting the work of his students, Knuth found that they had had to make many design
decisions, despite his earlier conviction to have produced ‘a reasonably complete specification of a language for typesetting’.
The designer of a new kind of system must participate fully in the implementation.
Debugging happened in about 18 days in March 1978. Knuth contrasts that with 41 days
for writing the program, making debugging about 30% of the total time, as opposed to
70% in his earlier experience. The whole code at that time was under 5000 statements. The
rewritten TEX82 runs to about 14 000 statements, in 1400 modules of W EB.
He considered this his first non-trivial program written using the structured programming
methodology of Dijkstra, Hoare, Dahl, and others. Because of the confidence this gave him
in the correctness of the program, he did not test TEX until both the whole program and the
fonts were in place. ‘I did not have to prepare dummy versions of non-existent modules
while testing modules that were already written’.
By mid 1979, Knuth was using TEX, and was improving TEX ‘at a regular rate of about one
enhancement for every six pages typed’.
Thus, the initial testing of a program should be done by the designer/implementor.
Triggered by a challenge of John McCarthy, Knuth wrote a manual, which forced him to
think about TEX as a whole, and which led to further improvements in the system.
The designer should also write the first user manual.
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CHAPTER 7. SOFTWARE ENGINEERING
‘If I had not participated fully in all these activities, literally hundreds of improvements
would never have been made, because I would never have thought of them or perceived
why they were important.’
Knuth remarks that testing a compiler by using it on a large, real, input typically leaves
many statements and cases unexecuted. He therefore proposes the ‘torture test’ approach.
This consists of writing input that is as far-fetched as possible, so that it will explore many
subtle interactions between parts of the compiler. He claims to have spent 200 hours writing
and maintaining the ‘trip test’; there is a similar ‘trap test’ for M ETAFONT.
7.4.2
Context
Software engineering is not an exact science. Therefore, some of Knuth’s ideas can be fit
in accepted practices, others are counter. In this section we will mention some schools of
thought in software engineering, and see how Knuth’s development of TEX fits in it.
7.4.2.1
Team work
The upshot of the development of TEX seems to be that software development is a one-man
affair. However, in industry, programming teams exist. Is the distinction between TEX and
commercial products then that between academic and real-world?
Knuth’s ideas are actually not that unusual. Programming productivity is not simply expressible as the product of people and time, witness the book ‘The Mythical man-Month’.
However, some software projects are too big, even for one really clever designer / programmer / tester / manual writer.
Dividing programming work is tricky, because of the interdependence of the parts. The
further you divide, the harder coordination becomes, and the greater the danger of conflicts.
Harlan Mills proposed that software should be written by groups, where each group works
like a surgical team: one chief surgeon who does all the real work, with a team to assist in
the more mundane tasks. Specifically:
•
•
•
•
•
•
•
•
The Chief Programmer designs the software, codes it, tests, it, and writes the
documentation.
The Co-Pilot is the Chief Programmer’s alter ego. He knows all the code but
writes none of it. He thinks about the design and discusses it with the Chief Programmer, and is therefore insurance against disaster.
The Administrator takes care of the mundane aspects of a programming project.
This can be a part-time position, shared between teams.
The Editor oversees production of documentation.
Two Secretaries, one each for the Administrator and Editor.
The Program Clerk is responsible for keeping records of all code and the test runs
with their inputs. This post is also necessary because all coding and testing will
be matter of public record.
The Toolsmith maintains the utilities used by the other team members.
The Tester writes the test cases.
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7.4. TEX’S DEVELOPMENT
197
•
The Language Lawyer investigates different constructs that can realize the Chief
Programmer’s algorithms.
With such teams of 10 people, coordination problems are divided by 10. For the overall
design there will be a system architect, or a small number of such people.
Recently, a development methodology name ‘Extreme Programming’ has become popular.
One aspect of this is pair programming: two programmers share one screen, one keyboard.
The advantage of this is that all code is immediately reviewed and discussed.
7.4.2.2
Top-down and bottom-up
Knuth clearly favours the top-down approach that was proposed by Nicklaus Wirth in ‘Program Development by Stepwise Refinement’ [20], and by Harlan Mills, who pioneered it
at IBM. The advantage of top-down programming is that the design of the system is set
from the beginning. The disadvantage is that it is hard to change the design, and testing
that shows inadequacies can only start relatively late in the development process.
Bottom-up programming starts from implementing the basic blocks of a code. The advantage is that they can immediately be tested; the disadvantage is that the design is in danger
of becoming more ad hoc.
An interesting form of bottom-up programming is ‘test-driven development’. Here, first a
test is written for a unit (a ‘unit test’), then the code. At all times, all tests need to be passed.
Rewriting code, usually to simplify it, with preservation of functionality as defined by the
tests, is known as ‘refactoring’.
7.4.2.3
Program correctness
The Trip test is an example of ‘regression testing’: after every change to the code, a batch
of tests is run to make sure that previous bugs do not reappear. This idea dates back to
Brooks; it is an essential part of Extreme Programming.
However, the Trip test only does regression testing of the whole code. TDD uses both
Unit tests and Integration tests. A unit is a specific piece of code that can easily be tested
since it has a clear interface. In testing a unit, the code structure can be used to design
the tests. Integration testing is usually done as Black Box testing: only the functionality
of an assemblage of units is known and tested, rather than the internal structure. One way
of doing integration testing is by ‘equivalence partitioning’: the input space is divided into
classes such that within each classes the input are equivalent in their behaviour. Generating
these classes, however, is heuristic, and it is possible to overlook cases.
On the opposite side of the testing spectrum is program proving. However, as Knuth wrote
in a memo to Peter van Emde Boas: ‘Beware of bugs in the above code; I have only proved
it correct, not tried it.’
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Markup.
7.5
History
The idea of markup has been invented several times. The ideas can be traced by to William
Tunnicliffe, chairman of the Graphic Communications Association (GCA) Composition
Committee, who presented a talk on the separation of information content of documents
from their format in 1967. He called this ‘generic coding’. Simultaneously, a New York
book designer named Stanley Rice was publishing articles about ”Standardized Editorial
Structures”, parameterized style macros based on the structural elements of publications.
7.5.1
Development of markup systems
Probably the first software incorporating these ideas comes out of IBM. Charles Goldfarb
recounts (http://www.sgmlsource.com/history/roots.htm) how in 1969 he
invented GML with Ed Mosher and Ray Lorie. They were tackling the problem of having a
document storage system, an editor, and a document printing system talk to each other, and
found that each was using different ‘procedural markup’ for its own purposes. Gradually
the idea grew to use markup for a logical description of the document. The project was
called ‘Integrated Text Processing’, and the first prototype ‘Integrated Textual Information
Management Experiment’: InTIME.
GML was officially published in 1973, and by 1980 an extension, SGML, was under development. This was published in 1986 as ISO 8879. Actually, SGML is a standard for
defining markup languages, rather than a language itself. Markup languages are defined in
SGML through ‘Document Type Definitions’ (DTDs).
The most famous application of SGML is HTML. However, HTML quickly began violating the separation of function and presentation that was the core idea of markup languages.
A renewed attempt was made to introduce a system for markup languages that truly defined content, not form, and this led to the ‘eXtensible Markup Language’. XHTML is a
realization of HTML as an XML ‘schema’. While SGML made some attempts at readability (and saving keystrokes for poor overworked typists), XML is aimed primarily at being
generated and understood by software, not by humans.
Another well-known application of SGML is ‘DocBook’. However, this has also been defined as an XML DTD1 , and this seems to be the current definition. DocBook is a good
illustration of the separation of content and presentation: there are XSL style sheets that
render DocBook files as Pdf, Rtf, HTML, or man pages.
1.
XML has both DTDs, which are SGML-like, and Schemas, which themselves XML. Schemas are the more
powerful mechanism, but also newer, so there are established DTDs that may not be redefined as Schemas. Furthermore, Schemas are more complicated to transmit and parse.
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7.5. HISTORY
7.5.2
199
Typesetting with markup
In the early 1970s, nroff/troff was written at Bell Labs, at first in PDP assembler and targetting a specific photo typesetter for producing Unix documentation. Later it was recoded
in C, with device independent output. Various tasks such as tables and equations were hard
in nroff/troff, so preprocessors existed: eqn for formulas, tbl for tables, and refer for bibliographies.
Brian Reid’s thesis of 1980 descibed a markup system called Scribe. Scribe source files can
be compiled to several target languages. For instance, recent versions Scribe can compile
to LATEX, HTML, or man pages.
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Projects for this chapter.
Project 7.1. Do a literature study of code/documentation development. Here are some
places to start:
POD Plain Old Documentation; used for Perl. http://www.perl.com/pub/
a/tchrist/litprog.html
JavaDoc http://java.sun.com/j2se/javadoc/
Doxygen http://www.stack.nl/˜dimitri/doxygen/
Fitnesse http://fitnesse.org/
Leo http://webpages.charter.net/edreamleo/front.html
What schools of thought are there about developing medium size codes such as
TEX? How does Knuth’s philosophy relate to the others?
Project 7.2. Compare the TeX ”way” to MS Word, PageMaker, FrameMaker, Lout, Griff,
previewLaTeX.
Project 7.3. TEX has been criticized for its arcane programming language. Would a more
traditional programming language work for the purpose of producing text output? Compare TEX to other systems, in particular lout, http://www.pytex.
org/, ant http://www-mgi.informatik.rwth-aachen.de/˜blume/
Download.html and write an essay on the possible approaches. Design a system of your own.
Project 7.4. TEX and HTML were designed primarily with output in mind. Later systems
(XML, DocBook) were designed so that output would be possible, but to formalize the structure of a document better. However, XML is impossible to write
by hand. What would be a way out? Give your thoughts about a better markup
system, conversion between one tool and another, et cetera.
Project 7.5. Several improvements on TEX and LATEX have been developed or are under
development. Investigate NTS, LaTeX3, Context, Lollipop TEX describe their
methodologies, and evaluate relative merits.
Project 7.6. Knuth has pretty liberal ideas about publishing software; somewhat against
the spirit of the times. Report on software patents, the difference between patents
and copyright, the state of affairs in the world. Read http://swpat.ffii.
org/gasnu/knuth/index.en.html
Project 7.7. Knuth devised the ‘torture test’ approach to program correctness. Report on
various schools of thought on this topic. Where does Knuth’s approach stand?
TEX – LATEX – CS 594
Bibliography
[1] Richard E. Bellman and Stuart E. Dreyfus. Applied Dynamic Programming. Princeton University Press, 1962.
[2] Carl de Boor. A Practical Guide to Splines, Revised Edition. Springer-Verlag, New
York, Berlin, Heidelberg, 2001.
[3] V. Eijkhout. An indentation scheme. TUGboat, 11:613–616.
[4] V. Eijkhout. A paragraph skip scheme. TUGboat, 11:616–619.
[5] V. Eijkhout and A. Lenstra. The document style designer as a separate entity. TUGboat, 12:31–34, 1991.
[6] jr Frederick P. Brooks. The Mythical Man-Month, essays on software engineering.
Addison-Wesley, 1995. Aniversary edition; originally published in 1975.
[7] Michael R. Garey and David S. Johnson. Computers and Intractibility, a guide to the
theory of NP-completeness. W.H. Freeman and company, San Francisco, 1979.
[8] A. Jeffrey. Lists in TEX’s mouth. TUGboat, 11:237–245, 1990.
[9] D.E. Knuth. Digital Typography.
[10] D.E. Knuth. The errors of TEX. Software Practice and Experience, 19:607–681.
[11] D.E. Knuth. TEX: the Program. Addison-Wesley, 1986.
[12] D.E. Knuth. The TEX book. Addison-Wesley, reprinted with corrections 1989.
[13] D.E. Knuth and M.F. Plass. Breaking paragraphs into lines. Software practice and
experience, 11:1119–1184, 1981.
[14] Donald E. Knuth. The Art of Computer Programming, Volume 3, Sorting and Searching. Addison Wesley Longman, 1998. Second edition.
[15] Helmut Kopka and Patrick W. Daly. A Guide to LATEX. Addison-Wesley, first published 1992.
[16] L. Lamport. LATEX, a Document Preparation System. Addison-Wesley, 1986.
[17] Frank Mittelbach, Michel Goossens, Johannes Braams, David Carlisle, and Chris
Rowley. The LATEX Companion, 2nd edition. Addison-Wesley, 2004.
[18] Michael F. Plass. Optimal Pagination Techniques for Automatic Typesetting Systems.
PhD thesis, 1981. also Xerox technical report ISL-81-1.
[19] Richard Rubinstein. Digital Typography. Addison-Wesley, 1988.
[20] Niklaus Wirth. Program development by stepwise refinement. Comm. ACM, 14:221–
227, 1971.
201
Index
&, 12
\˜, 15
˜, 15
accents, 187
active character, 6
\addcontentsline, 20
\addtocontents, 20
alphabet, 33
ambiguity, 50
analysis
lexical, 32
syntactical, 32
anti-aliasing, 154
Approximation, of curves, 129
ascii, 170
7-bit, 170
8-bit, 170
control codes, 170
extended, 170
unprintable, 170
associate storage, 77
automaton, 34
deterministic, 36
B-splines, 141
\backmatter, 8
baddness
of line breaks, 97
\bfseries, 14, 188
\Big, 18
\big, 18
bitmap, 145
BMP, 173
bold, 14
bottom-up parsing, 45
byte order mark, 177
202
charset, 176
code page, 170, 179
code point, 170, 176
code unit, 176
compatibility characters (Unicode),
182
control space, 6
control symbol, 6
CTAN, 25
cursive, 14
dangling else, 51
DBCS, 172
DDA, 147
demerit
of line breaking, 101
derivation, 41
design size, 183
divided difference, 131
DocBook, 196
DTD, 196
DVI file format, 8
ebcdic, 170
\emph, 14, 16
environment
abstract, 9
center, 9
flushleft, 9
flushright, 9
quotation, 9
quote, 9
tabular, 11
verbatim, 9, 10
verbatim*, 10
Equivalence partitioning, 195
Extreme programming, 195
INDEX
\fboxrule, 16
filter function, 153
Finite State Automaton, 34
Finite State Automaton, 32
Follow, 49
Font
Postscript, 184
TrueType, 184
Type 1, 184
Type 3, 184
font, 181
font encoding, 179
font metrics, 187
\footnote, 13
\footnotemark, 13
\footnotetext, 13
FreeType, 186
\frontmatter, 8
FSA, 32
FSA, see Finite State Automaton,
34
geometric continuity, 141
geometry matrix, 134
glyph, 181
GML, 196
Goldfarb, Charles, 196
grammar, 33
normal form, 42
handle, 46
Hash function, 78
Hermite interpolation, 129
\hline, 12
horizontal mode, 16
HTML, 196
IANA, 177
\include, 9
\includeonly, 9
incremental drawing, 145
\index, 20
\input, 9
inputenc package, 179
inserts, 116
Interpolation, 128
Victor Eijkhout
203
Hermite, 129
Lagrange, 129
ISO
10646, Unicode, 173
646, 7-bit ascii, 170
8859, Latin alphabets, 172
9541, 181
ISO 2022, 176
italic, 14
italic correction, 187
Item (in LR parser), 48
\itshape, 14, 188
label
after \item, 10
\label, 19
Lagrange interpolation, 129
language
context-free, 33
context-sensitive, 33
recursive, 33
regular, 33
language, formal, 33
lazy evaluation, 167
\leftmark, 22
leftmost derivation, 45
lexical analysis, 32
Linear Bounded automaton, 34
linear rehashing, 84
\linebreak, 15
list
description, 10
enumerate, 10
itemize, 10
\listoffigures, 17
\listoftables, 17
look ahead, 44
in FSAs, 40
LR parser, 47
macro, 4
\mainmatter, 8
\markboth, 22
\markright, 22
Markup
procedural, 196
204
INDEX
roman, 14
\mbox, 13, 15, 16
rule, 33
\mdseries, 14, 188
type of, 33
Multiple Master font technology,
184
\rule, 16
\newenvironment, 23
\newline, 15
\newpage, 15
\newtheorem, 23
NFA, see FSA, non-deterministic,
36
NFSS, 188
\nolinebreak, 15
non-deterministic FSA, 36
nonlinear rehashing, 84
\nonumber, 18
\nopagebreak, 15
\normalfont, 188
NP-hard, 112
scan code, 170
schema, 196
\scshape, 14, 188
\section*, 8
separate chaining, 80
\sffamily, 14, 188
SGML, 196
shift-reduce parsing, 45
shift/reduce conflict, 51
slanted, 14
\slshape, 14, 188
structured programming, 193
surgical team, 194
syntactical analysis, 32
open hashing, 80
OpenType, 186
operator grammar, 46
Optical masters, 184
OT1, 179
output routine, 117
T1, 179
\tableofcontents, 20
Test-driven development, 195
\texrm, 14
text
bold, 14
italic, 14
roman, 14
slanted, 14
\textbf, 14, 188
\textit, 14, 188
\textmd, 188
\textnormal, 188
\textrm, 188
\textsc, 188
\textsf, 188
\textsl, 14, 188
\texttt, 188
\textup, 188
tfm file, 187
\thanks, 13
\thispagestyle, 21
token, 32
tokenizer, 32
Torture test, 194
Trip test, 194
\pagebreak, 15
\pagestyle, 21
\par, 6, 16
parser, 32
parsing, 41
preamble, 7
\printindex, 20
production, 33
Pushdown automaton, 34
raster graphics, 145
Reduce/reduce conflict, 52
Refactoring, 195
Regression testing, 195
regular expressions, 33
regular expression, 35
\rightmark, 22
rightmost derivation, 45
\rmfamily, 14, 188
TEX – LATEX – CS 594
INDEX
\ttfamily, 14, 188
Turing machine, 34
typeface, 181
UCS, Universal Character Set,
173
underlining, 16
Unit test, 195
\upshape, 14, 188
vector graphics, 145
\verb, 10
\verb*, 10
\verbatiminput, 10
vertical mode, 16
XHTML, 196
XML, 196
yyerror, 69
Victor Eijkhout
205
COMPUTER SCIENCE COURSE 594, FALL 2004
VICTOR EIJKHOUT
DEPARTMENT OF COMPUTER SCIENCE,
UNIVERSITY OF TENNESSEE, KNOXVILLE TN 37996
DRAFT LECTURE NOTES
About this book.
These are the lecture notes of a course I taught in the fall of 2004. This was the first time
I taught the course, and the first time this course was taught, period. These lecture notes,
therefore, are probably full of inaccuracies, mild fibs, and gross errors. Ok, make that ‘are
definitely full of &c’, because I know of several errors that time has prevented me from
addressing.
However, I would be interested in hearing any comments and suggestions you have, or
errors you would like to point out.
Victor Eijkhout
[email protected]
Knoxville, TN, december 2004.
Enjoy!
2.7
2.8
Contents
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
2
2.1
2.2
2.3
2.4
2.5
2.6
2
About this book 1
TEX and LATEX 5
LATEX 6
Document markup 6
The absolute basics of
LATEX 8
The TEX conceptual
model of
typesetting 10
Text elements 11
Tables and figures 19
Math 19
References 21
Some TEXnical
issues 23
Customizing LATEX 23
Extensions to LATEX 27
TEX programming 30
TEX visuals 31
Projects for this
chapter 32
Parsing 33
Parsing theory 34
Levels of parsing 34
Very short
introduction 34
Lexical analysis 37
Finite state automata and
regular languages 37
Lexical analysis with
FSAs 41
Syntax parsing 43
Context-free
languages 43
Parsing context-free
languages 45
2.9
2.10
2.11
2.12
2.13
2.14
2.15
2.16
2.17
2.18
2.19
2.20
2.21
2.22
2.23
2.24
2.25
2.26
2.27
3
3.1
3.2
3.3
3.4
3.5
3.6
Lex 56
Introduction 56
Structure of a lex
file 56
Definitions section 57
Rules section 57
Regular expressions 60
Remarks 60
Examples 61
Yacc 65
Introduction 65
Structure of a yacc
file 65
Motivating example 65
Definitions section 67
Lex Yacc
interaction 67
Rules section 69
Operators; precedence
and associativity 69
Further remarks 70
Examples 73
Hashing 79
Introduction 79
Hash functions 79
Collisions 82
Other applications of
hashing 86
Discussion 87
Projects for this
chapter 88
Breaking things into
pieces 89
Dynamic
Programming 90
Some examples 90
Discussion 96
TEX paragraph
breaking 98
The elements of a
paragraph 98
TEX’s line breaking
algorithm 102
NP completeness 110
Introduction 110
Basics 111
CONTENTS
3.7
3.8
3.9
3.10
3.11
4
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5
5.1
6
Complexity
classes 112
NP-completeness 114
Page breaking 118
Introduction 118
TEX’s page breaking
algorithm 118
Theory of page
breaking 120
Projects for this
chapter 128
Fonts 129
Bezier curves 130
Introduction to curve
approximation 130
Parametric curves 135
Practical use 143
Curve plotting with
gnuplot 145
Introduction 145
Plotting 146
Raster graphics 147
Vector graphics and
raster graphics 147
Basic raster
graphics 147
Rasterizing type 150
Anti-aliasing 154
Projects for this
chapter 157
TEX’s macro language
– unfinished
chapter 159
Lambda calculus in
TEX 160
Logic with TEX 160
Character
encoding 171
Victor Eijkhout
3
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
7
7.1
7.2
7.3
7.4
7.5
Input file encoding 172
History and
context 172
Unicode 175
More about character
sets and encodings 178
Character issues in TEX /
LATEX 180
Font encoding 183
Basic terminology 183
Æsthetics 185
Font technologies 186
Font handling in TEX
and LATEX 189
Input and output
encoding in LATEX 191
The fontenc
package 191
Projects for this
chapter 192
Software
engineering 193
Literate
programming 194
The Web system 194
Knuth’s philosophy of
program
development 194
Software
engineering 195
Extremely brief history
of TEX 195
TEX’s
development 195
Markup 198
History 198
Projects for this
chapter 200
4
CONTENTS
TEX – LATEX – CS 594
Chapter 1
TEX and LATEX
In this chapter we will learn
•
The use of LATEX for document preparation,
•
LATEX style file programming,
•
TEX programming.
Handouts and further reading for this chapter
For LATEX use the ‘Not so short introduction to LATEX’ by Oetiker et al. For further reading
and future reference, it is highly recommended that you get ‘Guide to LATEX’ by Kopka and
Daly [15]. The original reference is the book by Lamport [16]. While it is a fine book, it has
not kept up with developments around LATEX, such as contributed graphics and other packages. A book that does discuss extensions to LATEX in great detail is the ‘LATEX Companion’
by Mittelbach et al. [17].
For the TEX system itself, consult ‘TEX by Topic’. The original reference is the book by
Knuth [12], and the ultimate reference is the published source [11].
5
6
CHAPTER 1. TEX AND LATEX
LATEX.
1.1
Document markup
If you are used to ‘wysiwyg’ (what you see is what you get) text processors, LATEX may
seem like a strange beast, primitive, and probably out-dated. While it is true that there is a
long history behind TEX and LATEX, and the ideas are indeed based on much more primitive
technology than what we have these days, these ideas have regained surprising validity in
recent times.
1.1.1
A little bit of history
Document markup dates back to the earliest days of computer typesetting. In those days,
terminals were strictly character-based: they could only render mono-spaced built-in fonts.
Graphics terminals were very expensive. (Some terminals could switch to a graphical character set, to get at least a semblance of graphics.) As a result, compositors had to key in
text on a terminal – or using punched cards in even earlier days – and only saw the result
when it would come out of the printer.
Any control of the layout, therefore, also had to be through character sequences. To set
text in bold face, you may have had to surround it with <B> .. the text .. </B>.
Doesn’t that look like something you still encounter every day?
Such ‘control sequences’ had a second use: they could serve a template function, expanding
to often used bits of text. For instance, you could imagine $ADAM$ expanding to ‘From
our correspondent in Amsterdam:’.
LATEX works exactly the same. There are command control sequences; for instance, you get
bold type by specifying \bf, et cetera. There are also control sequences that expand to bits
of text: you have to type \LaTeX to get the characters ‘LATEX’ plus the control codes for
all that shifting up and down and changes in font size.
\TeX => T\kern -.1667em\lower .5ex\hbox {E}\kern -.125emX
\LaTeX => L\kern -.36em {\sbox \z@ T\vbox to\ht \z@ {\hbox
{\check@mathfonts \fontsize \sf@size \z@ \math@fontsfalse
\selectfont A} \vss }}\kern -.15em\TeX
1.1.2
Macro packages
The old typesetting systems were limited in their control sequences: they had a fixed repertoire of commands related to typesetting, and there usually was some mechanism to defining ‘macros’ with replacement text. Formally, a macro is a piece of the input that gets
replaced by its definition text, which can be a combination of literal text and more macros
or typesetting commands.
TEX – LATEX – CS 594
1.1. DOCUMENT MARKUP
7
An important feature of many composition programs is the ability to
designate by suitable input instructions the use of specified formats. Previously stored sequences of commands or text replace the instructions,
and the expanded input is then processed. In more sophisticated systems, formats may summon other formats, including themselves [”System/360 Text Processor Pagination/360, Application Description Manual,” Form No. GE20-0328, IBM Corp., White Plains, New York.].
That was the situation with commercial systems by manufacturers of typesetting equipment such as Linotype. Systems developed by (and for!) computer scientists, such Scribe or
nroff/troff, were much more customizable. In fact, they sometimes would have the equivalent of a complete programming language on board. This makes it possible to take the
basic language, and design a new language of commands on top of it. Such a repertoire of
commands is called a macro package.
In our case, TEX is the basic package with the strange macro programming language, and
LATEX is the macro package1 . LATEX was designed for typesetting scientific articles and
books: it offers a number of styles, each with slightly different commands (for instance,
there are no chapters in the article style) and slightly different layout (books need a title
page, articles merely a title on the first page of the text). Styles can also easily be customized. For different purposes (art books with fancy designs) it is often better to write
new macros in TEX, rather than to bend the existing LATEX styles.
However, if you use an existing LATEX style, the whole of the underlying TEX programming
language is still available, so many extensions to LATEX have been written. The best place
to find them is through CTAN http://wwww.ctan.org/.
Exercise 1. Discuss the difference between a macro and a function or
procedure in a normal programming language. In a procedural language,
looping is implemented with goto instructions. How would you do that
in a macro language? Is there a difference in efficiency?
1.1.3
Logical markup
Macro packages are initially motivated as a labour-saving device: a macro abbreviates a
commonly used sequence of commands. However, they have another important use: a well
designed macro package lets you use commands that indicate the structure of a document
rather than the formatting. This means that you would write \section{Introduction}
and not worry about the layout. The layout would be determined by a statement elsewhere
as to what macros to load2 . In fact, you could take the same input and format it two different
ways. This is convenient in cases such as an article being reprinted as part of a collection,
or a book being written before the final design is commissioned.
In a well written document, there will be few explicit typesetting commands. Almost all
macros should be of the type that indicates the structure, and any typesetting is taken care
1.
In this tutorial I will say ‘TEX’ when a statement applies to the basic system, and ‘LATEX’ if it only applies
to the macro package.
2.
Compare this to the use of CSS versus HTML in web pages.
Victor Eijkhout
8
CHAPTER 1. TEX AND LATEX
of in the definition of these. Further control of the layout of the document should be done
through global parameter settings in the preamble.
1.2
The absolute basics of LATEX
Here is the absolute minimum you need to know to use LATEX.
1.2.1
Different kinds of characters
A TEX input file mostly contains characters that you want to typeset. That is, TEX passes
them on from input to output without any action other than placement on the page and font
choice. Now, in your text there can be commands of different sorts. So TEX has to know
how to recognize commands. It does that by making a number of characters special. In this
section you will learn which characters have special meaning.
•
•
•
•
•
•
3.
Anything that starts with a backslash is a command or ‘control sequence’. A control sequence consists of the backslash and the following sequence of letters – no
digits, no underscores allowed either – or one single non-letter character.
Spaces at the beginning and end of a line are ignored. Multiple spaces count as
one space.
Spaces are also ignored after control sequences, so writing \LaTeX is fun
comes out as ‘LATEXis fun’. To force a space there, write \LaTeX{} is fun
or \LaTeX\ is fun. Spaces are not ignored after control symbols such as \$,
but they are again after the ‘control space’ \ 3 .
A single newline or return in the input file has no meaning, other than giving a
space in the input. You can use newlines to improve legibility of the input. Two
newlines (leading to one empty line) or more cause a paragraph to end. You can
also attain this paragraph end by the \par command.
Braces {,} are mostly used for delimiting the arguments of a control sequence.
The other use is for grouping. Above you saw an example of the use of an empty
group; similarly \TeX{}ing is fun comes out as ‘TEXing is fun’.
Letters, digits, and most punctuation can be typed normally. However, a bunch of
characters mean something special to LATEX: %$&ˆ_#˜{}. Here are their functions:
% comment: anything to the end of line is ignored.
$,_,ˆ inline math (toggle), subscript, superscript. See section 1.6.
& column separator in tables.
˜ nonbreaking space. (This is called an ‘active character’)
{} Macro arguments and grouping.
In order to type these characters, you need to precede them with a backslash, for
instance \% to get ‘%’. This is called a ‘control symbol’. Exception: use $\backslash$
to get ‘\’.
The funny bucket character here is how we visualize the space character.
TEX – LATEX – CS 594
1.2. THE ABSOLUTE BASICS OF LATEX
•
9
Some letters do not exist in all styles. As the most commonly encountered example, angle brackets <> do not exist in the roman text font (note that they are
in the typewriter style here, in roman you would get ‘¡¿’), so you need to write,
somewhat laboriously $\langle$S$\rangle$ to get ‘hSi’4 .
Exercise 2. You read in a document ‘This happens only in 90rest of
the time it works fine.’ What happened here? There are articles in print
where the word ‘From’ has an upside down question mark in front of it.
Try to think of an explanation.
1.2.2
LATEX document structure
Every LATEX document has the following structure:
\documentclass[ <class options> ]{ <class name> }
<preamble>
\begin{document}
<text>
\end{document}
Typical document classes are article, report, book, and letter. As you may expect, that last one has rather different commands from the others. The class options are
optional; examples would be a4paper, twoside, or 11pt.
The preamble is where additional packages get loaded, for instance
\usepackage{times}
switches the whole document to the Times Roman typeface. This is also the place to define
new commands yourself (section 1.9.2).
1.2.2.1
Title
To list title and author, a document would start with
\title{My story}
\author{B.C. Dull}
\date{2004} %leave this line out to get today’s date
\maketitle
After the title of an article and such, there is often an abstract. This can be specified with
\begin{abstract}
... The abstract text ...
\end{abstract}
The stretch of input from \begin to \end is called an ‘environment’; see section 1.4.1.2.
1.2.2.2
Sectioning
The document text is usually segmented by calls
4.
That’s what macros are good for.
Victor Eijkhout
10
CHAPTER 1. TEX AND LATEX
\section{This}
\subsection{That}
\subsection{The other}
\paragraph{one}
\subparagraph{two}
\chapter{Ho}
\part{Hum}
which all get numbered automatically. Chapters only exist in the report and book styles.
Paragraphs and subparagraphs are not numbered. To prevent sections et cetera from being
numbered, use \section*{...} and such5 .
1.2.2.3
Frontmatter, backmatter
You can use commands \frontmatter, \mainmatter, \backmatter – in book
class only – to switch page numbering from roman to arabic, and, for the back matter,
section numbering from numbers to letters.
1.2.3
Running LATEX
With the last two sections you should know enough to write a LATEX document. Now how
do you get to see the final output? This takes basically two steps: formatting and viewing.
You need to know that TEX’s original output format is slightly unusual. It is called ‘DVI’
for DeVice Independent. There are viewers for this format, but usually you need another
step to print it.
Traditionally, you would run an executable called latex (or tex), which gives you a dvi
file, which you then view with a previewer such as xtex or xdvi. To print this file, you
would use
dvips -Pcmz foo.dvi -o foo.ps
to generate a ps file. This can be printed, or converted to pdf.
There are version of the latex executable that output to other formats, for instance
pdflatex (there is also a pdftex) goes straight to pdf, which you can view with the
Adobe Acrobat Reader, or xpdf. The big advantage of this approach is that you can get
hyperlinks in your pdf file; see section 1.10.3.
Exercise 3. Set up a document that will have the answers to your
homework exercises of the whole course.
1.3
The TEX conceptual model of typesetting
In TEX, the question ‘on what page does this character appear’ is hard to answer. That is
because TEX typesets all material for a page, sort of on a long scroll, before cutting a page
5.
This also makes the title not go into the table of contents. See section 1.7.2 on how to remedy that.
TEX – LATEX – CS 594
1.4. TEXT ELEMENTS
11
off that scroll. That means that when a piece of text is set, you do not know if it falls before
or after a certain page break.
A similar story holds for paragraph breaking, but the question on what line something
occurs is not usually interesting.
1.4
Text elements
Here are the main elements that make up a LATEX document.
1.4.1
Large scale text structure
We already discussed sectioning commands in section 1.2.2.2. Here are more major text
elements in a LATEX document.
1.4.1.1
Input files
Use \include{<file>} to input a file beginning on a new page, and \input for just
plain input. With
\includeonly{file1,file2}
you can save processing time – provided the files are \included to begin with.
The .tex extension can usually be left off; because of the way TEX works, be careful with
funny characters in the file name.
On Unix installations, input files can be in other directories, specified by the TEXINPUTS
environment variable.
1.4.1.2
Environments
If a certain section of text needs to be treated in a different way from the surrounding text,
it can be segmented off by
\begin{<environment name>}
... text ...
\end{<environment name>}
An environment defines a group, so you can have local definitions and changes of parameters.
Some predefined environments are
flushleft (flushright) for text that is left (right) aligned, but not right (left).
center for text that is centered.
quote, quotation for text that needs to be set apart, by indenting both margins. The quote
environment is for single paragraphs, the quotation for multiple.
abstract for the abstract that starts an article, report, or book. In the report and book style
it is set on a separate page. In the article style it is set in a smaller type size, and
with indented margins.
verbatim see section 1.4.1.3.
Victor Eijkhout
12
1.4.1.3
CHAPTER 1. TEX AND LATEX
Verbatim text
As we have observed already, TEX has a number of special characters, which can be printed
by prefixing them with a backslash, but that is a hassle. Good thing that there is a mechanism for printing input completely verbatim. For inline text, use \verb+&###\text+ to
get ‘&###\text’. The essential command here is \verb. Unlike with other commands
that have arguments, the argument is not delimited by braces, but by two occurrences of
a character that does not appear in the verbatim text. A plus sign is a popular choice. The
\verb* variant makes spaces visible: \verb*+{ }+ gives ‘{ }’.
For longer verbatim text there is a verbatim environment. The verbatim* version
prints each space as a symbol. To input whole files verbatim, use \verbatiminput{file},
which is defined in the verbatim package.
For TEXnical reasons, verbatim text can not appear in some locations such as footnotes or
command definitions.
Exercise 4. Why does the \verb command not have its argument in
braces?
1.4.1.4
Lists
Lists in LATEX are a special case of an environment; they are specified by
\begin{<list type>}
\item ...
\item ...
\end{<list type>}
The three main list types are unnumbered lists, itemize, numbered lists, enumerate,
and definition or description lists, description.
In the case of a description list, it is mandatory to give the item label:
\begin{description}
\item[Do] A deer, a female deer.
\item[Re] A drop of golden sun.
...
\end{description}
You can give item labels in the other list types too.
Putting a list inside a list item will change the style of the item labels and numbers in a way
that is dictated by the document class.
You can put a \label command after an item to be able to refer to the item number.
\begin{enumerate}
\item\label{first:item} One
\item Two comes after \ref{first:item}
\end{enumerate}
Output:
TEX – LATEX – CS 594
1.4. TEXT ELEMENTS
13
1.
One
2.
Two comes after 1
This only makes sense with enumerate environments.
1.4.1.5
Tabbing
The tabbing environment is useful for setting pieces of text, such as source code, that use
a small number of ‘tab stops’. Tab stops (a term deriving from old mechanical typewriters)
are locations on the line that one can ‘tab to’, no matter how much material is currently on
the line.
Example:
\begin{tabbing}
The first line sets this: \=point;\\
the second jumps\>there
\end{tabbing}
Output:
The first line sets this: point;
the second jumps
there
The \= command in the first line defines a tab stop; in every subsequent line a \> command
will jump to that position, if it has not been reached yet. There can be multiple tab stops,
not necessarily defined in the same line, and tab stops can be redefined.
A more interesting case is where the tab stop is used before the line that defines it. For this
case there is the \kill command, which prevents a line from being displayed. Example:
\begin{tabbing}
while \=\kill
do\>\{\\
\>$i_1\leftarrow{}$\=1\\
\>$\ldots$\>2\\
\>\}\\
while (1)
\end{tabbing}
Output:
do {
i1 ← 1
... 2
}
while (1)
1.4.1.6
Tabular material
The tabular environment generates a table. Tables are often placed independently of the
text, at the top or bottom of the page; see section 1.5 for details. The table itself is generated
by
Victor Eijkhout
14
CHAPTER 1. TEX AND LATEX
\begin{tabular}{<alignment>}
... material ...
\end{tabular}
Each line of the table has items separated by characters, and \\ at the end of each line but
the last.
In its simplest form, the alignment directions are a combination of the letters l,r,c:
\begin{tabular}{rl}
"Philly" Joe & Jones\\ Dizzie & Gillespie\\ Art&Tatum
\end{tabular}
Output:
”Philly” Joe
Dizzie
Art
Jones
Gillespie
Tatum
Vertical rules are inserted by placing a | character in the alignment specification; horizontal
lines you get from \hline.
\begin{tabular}{|r|rl|}
\hline
instrument&name&\\ \hline
drums: &"Philly" Joe & Jones\\
trumpet:& Dizzie & Gillespie\\
piano: &Art&Tatum\\ \hline
\end{tabular}
Output:
instrument
drums:
trumpet:
piano:
name
”Philly” Joe
Dizzie
Art
Jones
Gillespie
Tatum
Some more tricks:
•
•
In headings you often want to span one item over several columns. Use
\begin{tabular}{|r|rl|}
\hline
instrument&\multicolumn{2}{|c|}{name}\\ \hline
drums: &"Philly" Joe & Jones\\
trumpet:& Dizzie & Gillespie\\
piano: &Art&Tatum\\ \hline
\end{tabular}
Output:
instrument
name
drums: ”Philly” Joe Jones
trumpet:
Dizzie Gillespie
piano:
Art Tatum
LATEX inserts a standard amount of space between columns. You can override this
with @{<stuff>}:
TEX – LATEX – CS 594
1.4. TEXT ELEMENTS
•
1.4.1.7
15
\begin{tabular}{r@{.}l}
2&75\\ 800&1
\end{tabular}
2.75
gives
800.1
A column specification of p{<size>} (where <size> is something like 5.08cm
or 2in) makes the items in that column formatted as paragraphs with the width
as specified.
Footnotes
Use the command \footnote. The numbering style is determined by the document class.
The kinds of footnote to denote affiliation of the author of a paper and such (these often
use asterisk symbols and such, even if footnotes are numbered in the rest of the document)
are given by the command \thanks.
There are two common cases where want more detailed control over footnotes:
•
You want to dictate the label yourself, for instance using the same label again (this
often happens with author affiliations)
•
You want to place a footnote in a table or such; LATEX has trouble with that.
In such cases you can use \footnotemark to place the marker, and \footnotetext
for the text. You can also set or change the footnote counter explicitly with counter
functions (section 1.9.4), or use
\footnote[<label>]{<text>}
where the label is used instead, and the counter is not increased.
1.4.1.8
Text boxes
Sometimes you want a small amount of text to behave like one or more paragraphs, except
not as part of the main page. The main commands for that are
\parbox[pos]{width}{text}
\begin{minipage}[pos]{width} text \end{minipage}
The optional pos parameter specifies whether the top (t) or bottom (b) line of the box
should align with surrounding text: top-aligned box of text:
Chapter 1. \parbox[t]{2in}{\slshape Introduction. First easy
lessons. Exercises. More about things to come. Conclusions}
Output:
Chapter 1. Introduction. First easy lessons. Exercises. More about things to come.
Conclusions
The default is a vertically centered position.
The minipage environment is meant for longer pieces of text; it can also handle other
environments in the text.
The \mbox command is for text (or other objects) that need to stay on one line.
Victor Eijkhout
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CHAPTER 1. TEX AND LATEX
1.4.2
Minor text issues
1.4.2.1
Text styles
You can switch between roman (the style for this text), italic (also called ‘cursive’), slanted
(in some typefaces, italic and slanted may be identical), and bold with the commands
\texrm, \textit, \textsl, and \textbf respectively, used as
Text is stated \textbf{boldly} or \textsl{with a slant}.
These combinations are not independent: nesting the commands can give you bold slanted
text.
The above commands are mostly for short bits of text. See section 1.4.2.2 for commands
to change font parameters in a more global manner.
If you are using italics for emphasis, consider using \emph instead, which works better,
especially if you emphasize something in text that is already italic.
1.4.2.2
Fonts and typefaces
You already saw commands such as \textrm and \textit for switching from one type
style to another. These commands hide a more complicated reality: LATEX handles its fonts
as combination of three parameters. These individual switches can be used inside a group,
or as an environment:
{\ttfamily This is typewriter text}
\begin{mdseries}
This text is set in medium weight.
\end{mdseries}
Here are the categories and possible values.
family roman, sans serif, typewriter type: \rmfamily, \sffamily, \ttfamily.
series medium and bold: \mdseries, \bfseries.
shape upright, italic, slanted, and small caps: \upshape, \itshape, \slshape, \scshape.
1.4.2.3
Comments
Anything from % to the end of line is ignored. For multiline comments, load either
\usepackage{comment}
or
\usepackage{verbatim}
and in both cases surround text with
\begin{comment}
to be ignored
\end{comment}
where the closing line has to be on a line of its own.
TEX – LATEX – CS 594
1.4. TEXT ELEMENTS
1.4.2.4
17
Hyphenation
Sometimes TEX has a hard time finding a breakpoint in a word. When you are fixing the
final layout of a document, you can help it with helico\-pter. If TEX consistently
breaks your name wrong, do
\hyphenation{Eijk-hout}
in the preamble.
This is not the mechanism for telling TEX about a new language; see section 1.10.5.
To keep text together, write \mbox{do not break}. You could also write this with
a non-breaking space as do˜not˜break. (See also section 1.4.2.5.) It is a good idea to
write
A˜note on...
increase by˜$1$.
to prevent single characters at the beginning of a line (first example), or the end of a line
(second example). The second example could even give a line with a single character on it
if it were to occur at the end of a paragraph.
1.4.2.5
Tilde
The tilde character has special meaning as a nonbreaking space; see section 1.4.2.4. To get
a tilde accent, use \˜. To get a literal tilde, do \˜{}, $\sim$, or \char‘\˜. If you need
a tilde in URLs, consider using the url or hyperref package; see section 1.10.3.
1.4.2.6
Accents
In basic TEX, accents are formed by putting a control symbol of that accent in front of the
letter:
Sch\"on b\ˆet\’e
for ‘Sch¨on bˆet´e’. If you have an occasional foreign word in English text this works fine.
However, if your terminal allows you to input accented characters, you can use them in
LATEX with the inputenc package.
Standard TEX (or LATEX) does not understand Unicode encodings such as UTF-8.
1.4.2.7
Line/page breaking
In general, you should leave line and page breaking to TEX, at most adjusting parameters.
However,
should
you
really
need
it,
you can use the commands \linebreak[<num>] and \pagebreak[<num>], where
the number is 1,2,3,4, with 4 the highest
urgency. There is also \nolinebreak and \nopagebreak with a similar urgency parameter.
In this last paragraph there was a \linebreak after ‘need it’. You notice that TEX still
tried to fill out to the right margin, with ugly consequences. After ‘highest’ there was
a \newline, which really breaks then and there. Similarly, there is \newpage.
Victor Eijkhout
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CHAPTER 1. TEX AND LATEX
There is also \nolinebreak and \nopagebreak, both with optional numerical parameter, to discourage breaking.
1.4.2.8
Manual spacing
Most of the time you should leave spacing decisions to LATEX, but for custom designs it is
good to know the commands.
\hspace{1cm} \hspace*{1in} \hspace{\fill}
\vspace{1cm} \vspace*{1in} \vspace{\fill}
•
The *-variants give space that does not disappear at the beginning or end of a line
(for horizontal) or page (vertical).
A space of size \fill is infinite: this means it will stretch to take up however
much space is needed to the end of the line or page.
•
1.4.2.9
Drawing lines
Let us get one thing out of the way: underlining is a typewriter way of emphasizing text. It
looks bad in typeset text, and using italics or slanted text is a much better way. Use \emph.
Lines can be used a typographical decorations, for instance drawn between the regular text
and the footnotes on a page, or as part of chapter headings. The command is
\rule[lift]{width}{height}
Example
1\ \rule{2cm}{\fboxrule}\ The title
Output:
1
The title
You can draw a whole box around text: \fbox{text} gives text . The thickness of the
line is \fboxrule.
1.4.2.10 Horizontal and vertical mode
TEX is in horizontal or vertical mode while it is processing6 . In horizontal mode, elements
– typically letters – are aligned next to each other; in vertical mode elements are stacked on
top of one another. Most of the time you do not have to worry about that. When TEX sees
text, it switches to horizontal mode, and LATEX environments will briefly switch to vertical
mode so that they start on a new line.
In certain cases you want to force vertical mode; for that you can use \par. You can force
things into a line with \mbox (section 1.4.1.8). In rare cases, \leavevmode.
6.
The story is actually more complicated; for the whole truth see the notes about TEX.
TEX – LATEX – CS 594
1.5. TABLES AND FIGURES
1.5
19
Tables and figures
Tables and figures are objects that typically do not appear in the middle of the text. At the
very least they induce a paragraph break, and often they are placed at the top or bottom of
a page. Also, some publishers’ styles demand that a document have a list of tables and a
list of figures. LATEX deals with this by having environments
\begin{<table or figure>}[placement]
... table or figure material ...
\caption{Caption text}\label{tabfig:label}
\end{<table or figure>}
In this,
•
The ‘placement’ specifier is a combination of the letters htbp for ‘here’, ‘top’,
‘bottom’, and ‘page’, telling LATEX where to place the material, if possible. Suppose a placement of [ht] is given, then the material is placed ‘right here’, unless
there is not enough space on the page, in which case it will be placed on top of
the page (this one or the next).
•
Table material is given by a tabular environment; see section 1.4.1.6.
•
Figure material needs some extra mechanism, typically provided by another package; see section 1.10.4.
•
The caption goes into the list of tables/figures.
•
The label will refer to the number, which is automatically generated.
The list of tables/figures is generated by the command \listoftables or \listoffigures.
1.6
Math
TEX was designed by a computer scientist to typeset some books with lots of mathematics.
As a result, TEX, and with it LATEX’s, math capabilities are far better than those of other
typesetters.
1.6.1
Math mode
You can not just write formulas in the middle of the text. You have to surround them with
$<formula>$ or \(<stuff>\) for inline formulas, or
\begin{displaymath} ... \end{displaymath}
\[ ... \]
for unnumbered and
\begin{equation} ... \end{equation}
for numbered displayed equations respectively. You can refer to an equation number by
including a \label statement.
In math mode, all sorts of rules for text typesetting are changed. For instance, all letters
are considered variables, and set italic: $a$ gives ‘a’. Roman text is either for names
of functions, for which there are control sequences – \sin(x) gives ‘sin(x)’ – or for
connecting text, which has to be marked as such:
Victor Eijkhout
20
CHAPTER 1. TEX AND LATEX
\forall x \in \mathbf{R}
\quad \mathrm{(sufficiently large)} \quad: \qquad x>5
Output:
∀x ∈ R
(sufficientlylarge)
:
x>5
(1.1)
A formula is limited to one line; if you want to break it, or if you need several formulas
vertically after one another, you have to do it yourself. The eqnarray environment is
useful here. It acts as a three-column alignment.
\begin{eqnarray}
\sin x&=&x-\frac{xˆ3}{3!}+\frac{xˆ5}{5!}- \nonumber \\
&&{}-\frac{xˆ7}{7!}+\cdots
\end{eqnarray}
Output:
x5
x3
+
−
3!
5!
x7
+ ···
(1.2)
−
7!
Note the use of \nonumber here; with the eqnarray* all lines would be unnumbered
by default.
sin x = x −
In AMS LATEX there is an align environment which looks better than eqnarray.
1.6.2
Super and subscripts
In math mode, the character ˆ denotes a superscript, and _ denotes a subscript: x_iˆ2 is x2i .
(Outside of math mode these characters give an error.) Sub and superscripts of more than
one character have to be grouped.
1.6.3
Grouping
2
Grouping, in math mode as outside, is done with braces: x_{i-1}ˆ{nˆ2} looks like xni−1 .
1.6.4
Display math vs inline
Math looks different when used inline in a paragraph from that used as display math. This
is mostly clear for operators with ‘limits’:
∞
X
P∞
text mode: i=1 displaymode :
i=1
1.6.5
Delimiters, matrices
Delimiters are ()[]\{\}. You can prefix them with \big, \Big and such, but TEX can
resize them automatically:
TEX – LATEX – CS 594
1.7. REFERENCES
21
\left( \frac{1}{1-xˆ2} \right)
\left\{ \begin{array}{ccc}
\mathrm{(a)}&\Rightarrow&x>0\\
\mathrm{(b)}&\Rightarrow&x=0\\
\mathrm{(c)}&\Rightarrow&x<0
\end{array} \right.
Output:
(a) ⇒ x > 0
(b) ⇒ x = 0
(c) ⇒ x < 0
Note that with \right. you get a omitted right delimiter.
1
1 − x2
(1.3)
In the above example you also saw the array environment, which can be used for anything
tabular in math mode, in particular matrices. Here is a good example of a matrix. Note the
different kinds of dots:
A = \left( \begin{array}{cccccc}
a_{11}&0&&\ldots&0&a_{1n}\\
&a_{22}&0&\ldots&0&a_{2n}\\
&&\ddots&\ddots&\vdots&\vdots\\
&&&a_{n-2n-2}&0&a_{n-2n}\\
&\emptyset&&&a_{n-1n-1}&a_{n-1n}\\
&&&&&a_{nn}
\end{array} \right)
Output:
A=
1.6.6
a11
0
a22
0
..
.
...
...
..
.
0
0
..
.
a1n
a2n
..
.
an−2n−2
0
an−2n
an−1n
ann
∅
an−1n−1
(1.4)
There is more
See a good book for the whole repertoire of symbols. If what LATEX has is not enough, you
can also get AMS LATEX, which has even more fonts and tricky constructs.
1.7
References
1.7.1
Referring to document parts
One of the hard things in traditional typesetting is to keep references such as ‘see also
section 3’ in sync with the text. This is very easy in LATEX. You write
\section{Results}\label{section:results}
Victor Eijkhout
22
CHAPTER 1. TEX AND LATEX
after which you can use this as
see also section˜\ref{section:results}
on page˜\pageref{section:results}.
The \label command can appear after headings, or in general every time some counter
has been increased, whether that’s a section heading or a formula number.
LATEX implements this trick by writing the information to an auxiliary file – it has extension
.aux – and reading it in next run. This means that a LATEX document usually has to be
typeset twice for all references to be both defined and correct. You get a reminder after the
first run if a second one is needed, or if there are missing or duplicately defined labels.
Exercise 5. A document with references usually takes two passes to
get right. Explain why a table of contents can increase this number to
three.
1.7.2
Table of contents
Something that typically goes into the front or back matter is the table of contents. This
gets inserted automatically by the command \tableofcontents. No other actions required. You can add your own material to the contents with \addcontentsline or
\addtocontents.
1.7.3
Bibliography references
Another kind of the reference is that to external bibliographies. This needs a bit more work.
•
•
•
•
You write \cite{Knuth:1978} where you want the citation.
At the end of your document you write
\bibliographystyle{plain}
\bibliography{cs}
to get the bibliography included.
The bibliography references have to be defined in a file cs.bib.
After running LATEX once, you need to invoke bibtex <yourfile>, which
creates another auxiliary file, this time with .bbl extension, and run LATEX once
or twice more.
The bibliography files have a syntax of their own, but you can figure that out from looking
at some examples.
1.7.4
Index
Finally, a document can have an index. For this you need to have a statement \usepackage{makeidx}
in the preamble, \printindex wherever you want the index, and commands \index{<some term>}
throughout your document. Additionally, as with bibtex, you need to run the program
makeindex to generate the external .ind file.
Further indexing commands: \index{name!sub} for subentry; \index{foo@\textit{foo})
for sorting under ‘foo’ but formatted differently.
TEX – LATEX – CS 594
1.8. SOME TEXNICAL ISSUES
1.8
Some TEXnical issues
1.8.1
Commands inside other commands
23
For deep technical reasons you can get completely incomprehensible error messages by
writing things like
\section{My first section \footnote{and not my last}}
Remedy that by writing
\section{My first section \protect\footnote{and not my last}}
1.8.2
Grouping
Most modern programming languages have a block structure of some sort, where variables
can be declared inside a block, and are no longer known after the block ends. TEX has a
stronger mechanism, where assignments to a variable made inside a block are reverted at
the end of that block.
In LATEX you notice that only \newcommand and \newenvironment declarations are
local; \newcounters are global, as are \setcounter assignments. However, \savebox
assignments are local.
1.9
Customizing LATEX
LATEX offers a number of tools (on top of the possibility of doing straight TEX programming)
for customizing your document. The easiest customization is to change already defined
parameters. However, you can also define new commands and environments of your own.
In fact, several of the customization we will see in this section are not part of standard
LATEX, but have been written by other users. If they do not come with your installation, you
can download them from the Central TEX Archive Network; see section 1.10.1.
1.9.1
Page layout
1.9.1.1
Layout parameters
Page layout is controlled by parameters such as \textheight, \textwidth, \topmargin
(distance to the running head, not to the first text line), and \odd/evensidemargin
(distance to the ‘spine’ of the document). These are set with commands like
\setlength{\textwidth}{10in}
\addtolength{\oddsidemargin}{-1cm}
Some lengths are ‘rubber length’
\setlength{\parskip}{10pt plus 3pt minus 2pp}
Victor Eijkhout
24
1.9.1.2
CHAPTER 1. TEX AND LATEX
Page styles
Use the commands
\pagestyle{<style>}
and
\thispagestyle{<style>}
to change the style of all pages or one page. Available styles are empty (no page numbers), plain (the default), and headings (page numbers and running headers). See also
section 1.10.2 for many more options.
For two-sided printing, use the twoside option for the document class.
Exercise 6. Take a look at the headers and footers in Oetiker’s ‘Not
so short introduction’ and ‘TEX by Topic’ (the LATEX and TEX part of
the handout). Can you find a reason to prefer one over the other from a
point of usability? In both books, what is the rationale behind the header
on the odd pages? See in particular page 35 of the former and 77 of the
latter. Do you agree with this design?
1.9.1.3
Running page headers
The headings page style (section 1.9.1.2) uses running heads that can change through
the document. For instance it would have chapter titles in the left page head and section
titles in the right head. You can achieve this effect yourself by using the myheadings
page style, and using the
\markright{<right head>}
\markboth{<left>}{<right>}
You have access to these texts as \rightmark and \leftmark; this is needed in the
fancyhdr style.
1.9.1.4
Multicolumn text
Load
\usepackage{multicol}
and write
\begin{multicol}{3}
text in three column mode
\end{multicol}
1.9.2
New commands
You can define your own commands in LATEX. As example of a a simple command, consider
an often used piece of text
\newcommand{\IncrByOne}{increased by˜$1$}
The replacement text can have parameters:
TEX – LATEX – CS 594
1.9. CUSTOMIZING LATEX
25
\newcommand{\IncrDecrBy}[2]{#1creased by˜$#2$}
In this definition, the number of arguments is listed after the command name: [2], and
occurrences of the arguments in the replacement text are indicated by #1, #2 etc. Example
calls: \IncrDecrBy{in}{5}, \IncrDecrBy{de}{2}.
The declaration of a new command can specify an optional argument. If we define
\newcommand{\IncrDecrBy}[2][in]{#1creased by˜$#2$}
the [in] specification means that the first argument is optional (only the first argument
can ever be optional) with a default value of in. Example calls:
\newcommand{\IncrDecrBy}[2][in]{#1creased by˜$#2$}
\IncrDecrBy[de]{1}, \IncrDecrBy{5}.
Output:
decreased by 1, increased by 5.
To redefine an existing command, use \renewcommand.
1.9.3
New environments
It is possible to define environments, by specifying the commands to be used at their start
and end:
\newenvironment{example}%
{\begin{quote}\textbf{Example.}}%
{\end{quote}}
which, used as \begin{example}...\end{example} gives a quote environment
that starts with the word ‘Example’ in bold face. While defining that environment does not
save a lot of typing, it is a good idea nevertheless from a point of view of logical markup.
Using the example environment throughout ensures a uniform layout, and makes design
changes easy if you ever change your mind.
Special case: defining mathematical statements with
\newtheorem{majorfact}{Theorem}
\newtheorem{minorfact}[majorfact]{Lemma}
\begin{minorfact}Small fact\end{minorfact}
\begin{majorfact}Big fact\end{majorfact}
giving
Lemma 1 Small fact
Theorem 2 Big fact
The optional argument in the definition of lemma makes it use the theorem counter.
Exercise 7. Why does this not work:
\newenvironment{examplecode}%
{\textbf{Example code.}\begin{verbatim}}{\end{verbatimm}}
Victor Eijkhout
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CHAPTER 1. TEX AND LATEX
Exercise 8. Write macros for homework typesetting; make one master
document that will contain all your homework throughout this course.
1. Define an environment exercise so that
\begin{exercise}
My answer is...
\end{exercise}
gives
Problem 5. My answer is. . .
The counter should be incremented automatically. List your solution in your answer, and find a way that the listing is guaranteed to
be the code you actually use.
2. Write a macro \Homework that will go to a new page, and output
Answers to the exercises for chapter 3
at the top of the page. The exercise environment should now
take the question as argument:
\begin{exercise}{Here you paraphrase the question that was asked}
My answer is...
\end{exercise}
and this outputs
Problem 1.8 Here you paraphrase the question
that was asked
My answer is. . .
(Hint: read the section on text boxes. Also be sure to use \par to
get LATEX to go to a new line.) Allow for the question to be more
than one line long. Unfortunately you can not get verbatim text in
the question. Find a way around that.
1.9.4
Counters
LATEX has a number of counters defined, for sections and such. You can define your own
counters too. Here are the main commands:
create A new counter is created with
\newcounter{<name>}[<other counter>]
where the name does not have a backslash. The optional other counter indicates a counter (such as chapter) that resets the new counter every time
it is increased. (To do this reset for an already existing counter, check out the
chngcntr package.)
change values A counter can be explicitly set or changed as
\setcounter{<name>}{<value>}
\addtocounter{<name>}{<value>}
The command \refstepcounter also make the new value the target for a \label
command.
use To get a counter value numerically, use \value. To print the value, use
\arabic{<name>}, \roman{<name>}, \Roman{<name>}
et cetera.
TEX – LATEX – CS 594
1.10. EXTENSIONS TO LATEX
1.9.5
27
Lengths
Parameters such as \textwidth (section 1.9.1.1) are called ‘lengths’. You can define
your own with
\newlength{\mylength}
\setlength{\mylength}{5in}
These lengths can be used in horizontal or vertical space commands (section 1.4.2.8) for
your own designs.
1.9.6
The syntax of \new... commands
Have you noticed by now that the name you define starts with a backslash in \newcommand
and \newlength, but not in \newenvironment or \newcounter? Confusing.
1.10
Extensions to LATEX
LATEX may be a package on top of TEX, but that doesn’t mean that the programming power
of TEX is no longer available. Thus, many people have written small or large extensions to
be loaded in addition to LATEX. We will discuss a couple of popular ones here, but first we’ll
see how you can find them.
1.10.1
How to find a package, how to use it
Packages are typically loaded in the file preamble with
\usepackage{pack1,pack2,...}
(These course notes load about a dozen packages.)
Many popular packages are already part of the standard LATEX distribution, but you will
have to search to find where they are stored on your computer. Make a document that uses
a common package, say fancyhdr, and see in the log output on the screen or in the log
file where the file is loaded from. A typical location is /usr/share/texmf/.... With
a bit of searching you can also find7 the documentation, which can be a dvi, ps, or pdf
file.
If you have heard of a package and it is not on your system, go to the ‘Comprehensive TEX
Archive Network’ (CTAN for short) and download it from there: http://www.ctan.
org/.
1.10.2
Fancy page headers and footers
The fancyhdr8 package provides customized headers and footers. The simple interface
is
\lhead{<text>} \chead{<text>} \rhead{<text>}
7.
8.
For instance using the Unix command ‘find’.
This supersedes the fancyheadings package.
Victor Eijkhout
28
CHAPTER 1. TEX AND LATEX
for specifying text to get left, center, and right in the header. Likewise \lfoot and such
for the footer.
This is assuming that all pages are the same. If you format for two-sided printing (section 1.9.1.2), you can specify different text for odd and even pages:
\fancyhead[LE,RO]{<text>}
for text that is Left on Even and Right on Odd pages. Typically, you specify text for
[LE,RO] and [RE,LO], for instance
\fancyhead[EL,OR]{\textsl{\rightmark}}
(see section 1.9.1.3).
1.10.3
Pdf file generation
Making beautiful pdf documents, complete with hyperlinks and table of contents, from
your LATEX files is simplicity itself. Insert
\usepackage[pdftex]{hyperref}
in the preamble, and format with pdflatex. That’s it. Do see section 1.10.4.2 about
including pictures.
1.10.4
Graphics
Because of TEX’s ancient origins – and its desire to be machine-independent – graphics is not standard, and frankly a bit of a hassle. The basic TEX system knows nothing
about graphics, it just keeps some space open. An extension mechanism (‘specials’) then
puts information in the output file that the printer driver can use to place graphics. With
pdflatex this situation has become a bit less messy: now any graphics go straight into
the pdf output file.
1.10.4.1 The picture environment
There is a way to generate graphics from inside LATEX, using some graphics fonts rather
than a full drawing mode. While primitive and limited, the picture environment has two
advantages:
•
•
It is easier to get the fonts for labels to be the same as the text font.
Since it involves explicit drawing instructions, you can automatically draw bar
charts and such.
1.10.4.2 Including external graphics
Most of the time, you will have graphics to include that come from some drawing package.
Using the graphicx package, you write
\includegraphics[key=value,...]{<file name>}
TEX – LATEX – CS 594
1.10. EXTENSIONS TO LATEX
29
where the file name can refer to any format, but if you use pdflatex, Postscript can not be
used; if your picture is in Postscript, you can convert it with ps2pdf.
Commands such as \includegraphics, as well as similar commands in other packages, leave space in your document for the graphic. Now you have to be careful: you can
not leave space for a 3 inch picture, an inch from the bottom of the page. Here are two
approaches for placing a picture:
•
Just place it where you want it, and if it falls on a page break, deal with it later by
moving it.
•
Put the figure in a floating figure object (section 1.5) and let LATEX sort out the
placement.
You can also have text wrap around a figure, by using the wrapfig package.
There is a package color for colour output.
1.10.5
Other languages than English
The fact that TEX and LATEX were written by Americans becomes obvious in a couple of
places.
•
Various typographical conventions are geared towards American English.
•
Words like ‘Chapter’ are the default in the style files9 .
To address this and make LATEX easier to use with other languages, there is a package
babel.
9.
They used to be hard-wired, so the situation is improved.
Victor Eijkhout
30
CHAPTER 1. TEX AND LATEX
TEX programming.
No separate handout for this chapter; see the book ‘TEX by
Topic’.
Exercise 9. Write a macro \intt (‘in typewriter type’) such that
\intt{foo} and \intt{foo_bar} are output as foo and foo_bar,
in typewriter type.
Exercise 10. Write a macro that constructs another macro: \tees\three3
should be equivalent to \def\three{TTT}, \tees\five5 equivalent to \def\five{TTTTT} et cetera. In other words, the first argument of \tees is the name of the macro you are defining, the second
is the number of letters ‘T’ the defined macro expands to. To make sure
that your solution really expands to that string of ‘T’s, and not some
code that generates it when the macro is called, do \show\five and
check the screen output.
Exercise 11. TEX natively has addition, multiplication, and division
arithmetic. Write a square root routine in TEX. Hint: Use Newton’s method.
Exercise 12. Make this work:
\def\LeftDelim{(}\def\RightDelim{)}
\DefineWithDelims{foo}{My argument is ‘#1’.}
\def\LeftDelim{<}\def\RightDelim{>}
\DefineWithDelims{bar}{But my argument is ‘#1’.}
\foo(one)\par
\bar<two>
Output:
My argument is ‘one’.
But my argument is ‘two’.
In other words, \DefineWithDelims defines a macro – in this case
\foo – and this macro has one argument, delimited by custom delimiters. The delimiters can be specified for each macro separately.
Hint: \DefineWithDelims is actually a macro with only one argument. Consider this code snippet:
\Define{foo}{ ... #1 ...}
\def\Define#1{
\expandafter\def\csname #1\endcsname##1}
TEX – LATEX – CS 594
1.10. EXTENSIONS TO LATEX
31
TEX visuals.
Exercise 13. Use the \everypar command so that the first paragraph after a heading (define your own heading command) will have a
bullet ($\bullet$) in the left margin.
Exercise 14. Set TEX up so that every paragraph starts in mediaeval
‘initial’ style: the first letter of the paragraph is set in a large type size,
and takes the first two or three lines. Use the following auxiliary macro:
\def\Hang#1{\hbox to 0pt
{\raise 1.2ex \vbox to 0pt
{\hbox{#1}\vss}\hss}}
% small test:
A \Hang{$\bullet$} B \Hang{\Huge B} C. \bigskip
Output:
A •B C.
B
Also, set \parindent=0pt. The result should look like this. Input:
This is an old-fashioned mediaeval paragraph that has lots
of text and...
Also, the second paragraph is an old-fashioned mediaeval
paragraph that...
with output:
his is an old-fashioned mediaeval paragraph that
has lots of text and a very long first sentence. The
second sentence is also long, and only serves the purpose to make this more than 2 or so lines long. For
good measure we throw in a third line which should
make this four lines long, if not five with a little luck.
T
the second paragraph is an old-fashioned meA lso,
diaeval paragraph that has lots of text and a very
long first sentence. The second sentence is also long,
and only serves the purpose to make this more than 2
or so lines long. For good measure we throw in a third
line which should make this four lines long, if not five
with a little luck.
Victor Eijkhout
32
CHAPTER 1. TEX AND LATEX
Projects for this chapter.
Project 1.1. TEX has a syntax that can to a large extent be altered, dynamically or statically. This has had the effect that macro packages typically use a syntax that is
somewhere between ad hoc and plain unsystematic. Explore how it would be possible to set up a macro package with object-oriented syntax and design. It would
probably be a good idea to read [3, 4, 5], and to look at macro package such as
Lollipop and ConTeXt.
Project 1.2. The web site http://wwww.cookingforengineers.com uses a tablelike layout for notating recipes. Design an easy syntax for inputting these diagrams, and write a LATEX package that implements them.
TEX – LATEX – CS 594
Chapter 2
Parsing
The programming language part of TEX is rather unusual. In this chapter we will learn the
basics of language theory and parsing, and apply this to parsing TEX and LATEX. Although
TEXcan not be treated like other programming languages, it is interesting to see how far we
can get with existing tools.
Handouts and further reading for this chapter
The theory of languages and automata is discussed in any number of books, such as the
Hopcroft and Ulman one. For a discussion that is more specific to compilers, see the compilers book by Aho and Ulman or Aho, Seti, and Ulman.
The tutorials on lex and yacc should suffice you for most applications. The O’Reilly book
by Levine, Mason, and Brown is probably the best reference on lex and yacc. A copy of it
is on reserve in the library, QA76.76.U84M37 .
The definitive reference on hashing is Knuth’s volume 3 of The Art of Computer Programming [14], section 6.4. This is on reserve, QA76.5.K57 .
33
34
CHAPTER 2. PARSING
Parsing theory.
2.1
Levels of parsing
A compiler, or other translation software, has two main tasks: checking the input for validity, and if it is valid, understanding its meaning and transforming it into an executable
that realizes this meaning. We will not go into the generation of the executable code here,
but focus on the validity check and the analysis of the meaning, both of which are parsing
tasks.
A parser needs to look at the input on all sorts of levels:
•
•
•
•
Are all characters valid – no 8-bit ascii?
Are names, or identifiers, well-formed? In most programming languages a1 is a
valid name, but 1a is not. By contrast, in TEX a name can only have letters, while
in certain Lisp dialects !!important_name!! is allowed.
Are expressions well-formed? An arithmetic expression like 5/*6- does not
make sense, nor does CALL )FOO( in Fortran.
If the input is well-formed, are constraints satisfied such as that every name that
is used is defined first?
These different levels are best handled by several different software components. In this
chapter we will look at the two initial stages of most translators1 .
1.
2.
2.2
First of all there is the lexical analysis. Here a file of characters is turned into a
stream of tokens. The software that performs this task is called a tokenizer, and
it can be formalized. The theoretical construct on which the tokenizer is based is
called a ‘Finite State Automaton’.
Next, we need to check if the tokens produced by the tokenizer come in a legal
sequence. For instance, opening and closing parentheses need to come in matched
pairs. This stage is called the syntactical analysis, and the software doing this is
called a parser.
Very short introduction
A language is a set of words (strings) over an alphabet, that satisfies certain properties. It
is also possible to define a language as the output of a certain type of grammar, or as the
strings accepted by a certain type of automaton. We then need to prove the equivalences of
the various formulations. In this section we briefly introduce the relevant concepts.
1.
I will use the terms ‘translating’ and ‘translater’ as informal concepts that cover both compilers and interpreters and all sorts of mixed forms. This is not the place to get philosophical about the differences.
TEX – LATEX – CS 594
2.2. VERY SHORT INTRODUCTION
2.2.1
35
Languages
A language is a set of words that are constructed from an alphabet. The alphabet is finite
in size, and words are finite in length, but languages can have an infinite number of words.
The alphabet is often not specified explicitly.
Languages are often described with set notation and regular expressions, for example ‘L =
{an b∗ cn |n > 0}’, which says that the language is all strings of equal number of as and cs
with an arbitrary number of bs in between.
Regular expressions are built up from the following ingredients:
α|β either the expression α or β
αβ the expression α followed by the expression β
α∗ zero or more occurrences of α
α+ one or more occurrences of α
α? zero or one occurrences of α
We will see more complicated expressions in the lex utility.
2.2.2
Automata
A description of a language is not very constructive. To know how to generate a language
we need a grammar. A grammar is a set of rules or productions α → β that state that,
in deriving a word in the language, the intermediate string α can be replaced by β. These
strings can be a combination of
•
•
•
•
A start symbol S,
‘Terminal’ symbols, which are letters from the alphabet; these are traditionally
rendered with lowercase letters.
‘Non-terminal’ symbols, which are not in the alphabet, and which have to be
replaced at some point in the derivation; these are traditionally rendered with
uppercase letters.
The empty symbol .
Languages can be categorized according to the types of rules in their grammar:
type 0 These are called ‘recursive languages’, and their grammar rules can be of any form:
both the left and right side can have any combination of terminals, non-terminals,
and .
type 1 ‘Context-sensitive languages’ are limited in that can not appear in the left side of
a production. A typical type 1 rule would look like
αAβ → γ
which states that A, in the context of αAβ, is replaced by γ. Hence the name of
this class of languages.
type 2 ‘Context-free languages’ are limited in that the left side of a production can only
consist of single non-terminal, as in A → γ. This means that replacement of the
non-terminal is done regardless of context; hence the name.
type 3 ‘Regular languages’ can additionally have only a single non-terminal in each righthand side.
Victor Eijkhout
36
CHAPTER 2. PARSING
In the context of grammars, we use the notation α ⇒ β to indicate that the string β as
derived from α by a single application of a grammar rule; α ⇒∗ β indicates multiple
rules. For example, αAβ ⇒ αBγ indicates that the rhs string was derived from the lhs by
replacing Aβ with Bγ.
2.2.3
Automata
Corresponding to these four types of formal languages, there are four types of ‘automata’:
formal machines that can recognize these languages. All these machines have a starting
state, they go from one state to another depending on the input symbols they encounter, and
if they reach the end state, the string is accepted as being in the language. The difference
between the different types of automata lies in the amount of memory they have to store
information. Very briefly the classes of automaton are:
for type 3 Finite State Automata. These machines have no memory. They can only make
transitions.
for type 2 Pushdown Automata. These machines have a stack where they can store information; only the top of the stack can be inspected.
for type 1 Linear Bounded Automata. These have random-access memory, the size of
which is equal to (a linear function of) the size of the input.
for type 0 Turing machines. These have an unbounded tape for storing intermediate calculations.
TEX – LATEX – CS 594
2.3. FINITE STATE AUTOMATA AND REGULAR LANGUAGES
37
Lexical analysis.
The lexical analysis phase of program translation takes in a stream of characters and outputs
a stream of tokens.
A token is a way of recognizing that certain characters belong together, and form an object
that we can classify somehow. In some cases all that is necessary is knowing the class, for
instance if the class has only one member. However, in general a token is a pair consisting
of its type and its value. For instance, in 1/234 the lexical analysis recognizes that 234 is
a number, with the value 234. In an assignment abc = 456, the characters abc are recognized as a variable. In this case the value is not the numeric value, but rather something
like the index of where this variable is stored in an internal table.
Lexical analysis is relatively simple; it is performed by software that uses the theory of
Finite State Automata and Regular Languages; see section 2.3.
Remark. It might be tempting to consider the input stream to consist of lines, each of
which consist of characters, but this does not always make sense. Programming languages
such as Fortran do look at the source, one line at a time; C does not. TEX is even more
complicated: the interpretation of the line end is programmable.2
2.3
Finite state automata and regular languages
Regular languages are the strings accepted by a particularly simple kind of automaton.
However, we initially define these languages – non-constructively – by so-called ‘regular
expressions’.
2.3.1
Definition of regular languages
A regular language over some alphabet can be described by a ‘regular expression’.
•
•
•
•
denotes the empty language: the language with no words in it.
If a is a letter in the alphabet, then a denotes the language {a}.
If α and β are expressions denoting regular languages A and B, then
–
αβ or α · β denotes the language {xy|x ∈ A, y ∈ B}.
–
α|β denotes the language A ∪ B.
–
α∗ denotes the language ∪n≥0 An .
Parentheses can be used to indicate grouping: (α) simply denotes the language A.
Any regular expression built up this way describes a regular language.
2.
Ok, if we want to be precise, TEX does look at the input source on a line-by-line basis. There is something
of a preprocessor before the lexical analysis which throws away the machine-dependent line end, and replaces it
with the TEX-defined one.
Victor Eijkhout
38
2.3.2
CHAPTER 2. PARSING
Non-deterministic automata
A Finite State Automaton is an abstract machine that recognizes (‘accepts’) words from a
language:
•
•
•
The automaton is initially in a beginning state;
every letter or ‘symbol’ from the input word causes unambiguously a transition
to the same or to a next state; if no transition is defined for a given combination
of current state and input symbol, then the word is not in the language;
a word is accepted if the last symbol causes a transition to a state that is marked
as an accepting state.
Formally, we can define a FSA as the combination of
•
•
•
A set S of states, with a starting state S0 and a set of final states.
A finite input alphabet I.
A transition diagram I × S → S that specifies how the combination of a state and
an input symbol effects a transition to a new state.
This kind of automaton is deterministic in the sense that every transition from one state
to the next is deterministically made by accepting an input symbol. However, in the context of lexical analysis, the so-called ‘non-deterministic FSA’ is more convenient. A nondeterministic FSA (also NFA) differs in two ways from the deterministic type:
•
•
An NFA can make spontaneous transitions from one state to another. If an automaton has such a transition, we can say that this is caused by the symbol , and
this is called an -transition.
An NFA can be ambiguous in that there can be more than one possible transition
for a given state and input symbol.
Exercise 15. Show that the second condition in the definition of an
NFA can be reduced to the first. Is a reduction the other way possible?
2.3.3
The NFA of a given language
We now construct a nondeterministic automaton that accepts a regular language.
•
The automaton that accepts the expression has a single transition from the starting state to the accepting state.
0
•
The automaton that accepts the expression a has a single transition from the starting state to the accepting state.
a
0
•
1
1
If A and B are automata accepting the languages A and B with expressions α
and β, then
–
the language AB is accepted by the automaton that has the states and transition of both automata combined, with the initial state of A as the new initial
state, the accepting state of B as the new accepting state, and an -transition
TEX – LATEX – CS 594
2.3. FINITE STATE AUTOMATA AND REGULAR LANGUAGES
39
from the accepting state of A to the initial state of B;
A
1
0
1
B
the language A ∪ B is accepted by an automaton with a new starting state
that has -transitions to the initial states of A and B;
A
1
0
0
–
s
s
1
B
the expression α∗ is accepted by A modified such that the initial state is
also the accepting state, or equivalently by adding an -transition from the
starting to the accepting state, and one the other way around.
0
–
2.3.4
Examples and characterization
Any language that can be described by the above constructs of repetition, grouping, concatenation, and choice, is a regular language. It is only slightly harder to take a transition
diagram and write up the regular expression for the language that it accepts.
An informal way of characterizing regular languages is to say that FSAs ‘do not have
memory’. That means that any language where parts of words are related, such as {an bm |
m ≥ n}, can not be recognized by a FSA. Proof: suppose there is a recognizing FSA.
When it first accepts a b, it can come from only a fixed number of states, so that limits the
information it can carry with it.
We can give a slightly more rigorous proof if we first characterize regular languages:
Theorem 1 Let L be a regular language, then there is an n so that all strings α in L longer
than n can be written as α = uvw, such that for any k uv k w is also in the language.
Using this theorem it is easy to see that the above language can not be regular.
This theorem is proved by observing that in order to accept a sufficiently long string the
same state must have been encountered twice. The symbols accepted in between these
encounters can then be repeated arbitrarily many times.
2.3.5
Deterministic automata
Non-deterministic automata, as defined above, are easy to define. However, from a practical
point of view they do not look very constructive: a string in the language is accepted by
the automaton if there is any sequence of transitions that accepts it. Fortunately, for every
NFSA, there is a DFSA that accepts the same language.
Victor Eijkhout
40
CHAPTER 2. PARSING
Sometimes it is easy to derive the DFSA. Consider the language a∗ |b∗ and the automaton
a
1
0
b
2
The following automaton is derived by splitting off one a and one b:
a
1
a
0
b
b
2
This next example leads up to what happens in the lexical analysis of a compiler:
B
E–G–I
...
5
1
2
N
0
7
A–Z
a–z
6
The resulting DFA is a bit more messy:
E
G
B
1
2
¬E
0
3
...
text
¬G
¬B
6
A–Z
8
7
A–Z
...
A–Z
(and we can collapse states 6 . . . to one.)
Sketch of the proof: the states of the DFSA are sets of states of the NFSA. The states
we are actually interested in are defined inductively, and they satisfy the property that
TEX – LATEX – CS 594
2.4. LEXICAL ANALYSIS WITH FSAS
41
they are closed under -transitions of the original NFSA. The starting state contains the
original starting state plus everything reachable with -transitions from it. Given a state
of the DFSA, we then define more states by considering all transitions from the states
contained in this state: if there is a transition based on a symbol x, the next state has all
states reachable from this state by accepting x, plus any subsequent -transitions.
Since the number of subsets of a finite set of states is finite, this will define a finite number
of states for the DFSA, and it is not hard to see that an accepting sequence in the one
automaton corresponds to an accepting sequence in the other.
2.3.6
Equivalences
Above, we saw how the NFA of a regular language is constructed. Does every NFA correspond to a regular language, and if so, how can that be derived? We make a detour by first
talking about the equivalence of automata and grammars.
Let X be a string in the language L of a DFA, and suppose that after t transitions state i
is reached. That means we can split X = Xi (t)Yi . This is merely one of the strings that
is in state i at time t; let us call the set of all these strings Li (t). Let us call the set of all
strings that, given a state i, bring the automaton to an accepting state Ri . This set is clearly
m
not dependent on t. Defining Li = ∪∞
t=0 Li (t), we have that L = ∪i=1 Li Ri where m is the
number of states.
This inspires us to tackle the derivation of a grammar by describing the production of
the remainder strings Ri . Suppose the automaton is in state i; we will derive the productions Ni → . . .. If state i is an accepting state, there will be a production Ni → ; for all
other transitions by a symbol x to a state Ni0 we add a production Ni → xNi0 . It is easy
to see the equivalence of strings accepted by the DFA and derivations of the grammar thus
constructed.
Going the other way, constructing an automaton from a grammar runs into a snag. If there
are productions Ni → aNi0 and Ni → aNi00 , we can of necessity only construct an NFA.
However, we know that these are equivalent to DFAs.
We note that the grammars used and constructed in this – informal – proof are rightrecursive, so they generate precisely the regular languages.
Exercise 16. Show how this proof can be modified to use left-recursive
grammars, that is, grammars that have productions of the form Ni →
Ni0 a.
2.4
Lexical analysis with FSAs
A FSA will recognize a sequence of language elements. However, it’s not enough to simply
say ‘yes, this was a legal sequence of elements’: we need to pass information on to the next
stage of the translation. This can be done by having some executable code attached to
the accepting state; if that state is reached, the code snippet tells the next stage what kind
Victor Eijkhout
42
CHAPTER 2. PARSING
of element has been recognized, and its value. This value can be a numerical value for
numbers recognized, but more generally it will be an index into some table or other.
Formally, we can extend the definition of a FSA (section 2.3.2) by the addition of an output
alphabet O and an output table I × S → O. This models the output of a symbol, possibly ,
at each transition.
Exercise 17. One could also define the output with a mapping S → O.
Show that the definitions are equivalent.
An FSA is not enough to recognize a whole language, but it can recognize elements from a
language. For instance, we can build multiple FSAs for each of the keywords of a language
(‘begin’ or ‘void’), or for things like numbers and identifiers. We can then make one big
FSA for all the language elements by combining the multiple small FSAs into one that has
•
a starting state with -transitions to the start states of the element automata, and
•
from each of the accepting states an -transition back to the start state.
B
EGIN
s
s. . . . . . . . . . . . s
s E s N s D s
0–9
s
s
s
a-z, 0-9
s
a-z
s
Exercise 18. Write a DFA that can parse Fortran arithmetic expressions. In Fortran, exponentiation is written like 2**n. It is also not allowed to have two operators in a row, so 2 × −3 is notated 2*(-3).
There is a problem with the -transition from the final state to the initial state in the above
NFA. This transition should only be taken if no other transitions can be taken, in other
words, if the maximal string is recognized. For instance, most programming languages allow quote characters inside a quoted string by doubling them: ‘"And then he said ""Boo!"""’.
The final state is reached three times in the course of this string; only the last time should
the jump back be taken.
However, sometimes finding the maximum matched string is not the right strategy. For
instance, in most languages, 4.E3 is a floating point number, but matching the E after the
decimal point is not necessarily right. In Fortran, the statement IF (4.EQ.VAR) ...
would then be misinterpreted. What is needed here is one token ‘look-ahead’: the parser
needs to see what character follows the E.
At this point it would be a good idea to learn the Unix tool lex.
TEX – LATEX – CS 594
2.5. CONTEXT-FREE LANGUAGES
43
Syntax parsing.
Programming languages have for decades been described using formal grammars. One
popular way of notating those grammars is Backus Naur Form, but most formalisms are
pretty much interchangable. The essential point is that the grammars are almost invariably
of the context-free type. That is, they have rules like
hfunction calli −→ hfunction namei ( hoptargsi )
hoptargsi −→ empty | hargsi
hargsi −→ word | word , hargsi
The second and third rule in this example can be generated by a regular grammar, but the
first rule is different: when the opening parenthesis is matched, the parser has to wait an
unlimited time for the closing parenthesis. This rule is of context-free typeCHECKTHIS.
It is important to keep some distinctions straight:
•
A grammar has a set of rules, each indicating possible replacements during a
derivation of a string in the language. Each rule looks like A → α.
•
A derivation is a specific sequence of applications of rules; we denote each step
in a derivation as α ⇒ β, where β can be derived from α by application of some
rule. The derivation of some string α is a sequence of step such that S ⇒ · · · ⇒ α;
we abbreviate this as S ⇒∗ α.
•
Ultimately, we are interested in the reverse of a derivation: we have a string that
we suspect is in the language, and we want to reconstruct whether and how it
could be derived. This reconstruction process is called ‘parsing’, and the result
often takes the form of a ‘parse tree’.
We will first give some properties of context-free languages, then in section 2.6 we will
discuss the practical parsing of context-free languages.
2.5
Context-free languages
Context-free languages can be defined as the output of a particular kind of grammar (the left
side can only consist of a single nonterminal), or as the set of string accepted by a certain
kind of automaton. For languages of this type, we use a Pushdown Automaton (PDA) to
recognize them. A PDA is a finite-state automaton, with some scratch memory that takes
the form of a stack: one can only push items on it, and inspect or remove the top item. Here
we will not give an equivalence proof.
An example of a language that is context-free but not regular is {an bn }. To parse this, the
automaton pushes as on the stack, then pops them when it finds a b in the input, and the
string is accepted if the stack is empty when the input string is fully read.
2.5.1
Pumping lemma
As with regular languages (section 2.3.4), there is a way to characterize the strings of a
context-free language.
Victor Eijkhout
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CHAPTER 2. PARSING
Theorem 2 Let L be a context-free language, then there is an n so that all strings α in L
longer than n can be written as α = uvwxy, such that for any k the string uv k wxk y is also
in the language.
The proof is as before: the derivation of a sufficiently long string must have used the same
production twice.
S
HH
H
u
A
HH
v A x
H
y
w
2.5.2
Deterministic and non-deterministic PDAs
As with Finite State Automata, there are deterministic and non-deterministic pushdown
automata. However, in this case they are not equivalent. As before, any DPA is also a NPA,
so any language accepted by a DPA is also accepted by a NPA. The question is whether
there are languages that are accepted by a NPA, and that are not accepted by a DPA.
A similar example to the language {an bn } above is the language over an alphabet of at
least two symbols L = {ααR }, where αR stands for the reverse of α. To recognize this
language, the automaton pushes the string α on the stack, and pops it to match the reverse
part. However, the problem is knowing when to start popping the stack.
Let the alphabet have at least three letters, then the language Lc = {αcαR |c 6∈ α} can
deterministically be recognized. However, in absence of the middle symbol, the automaton
needs an -transition to know when to start popping the stack.
2.5.3
Normal form
Context-free grammars have rules of the form A → α with A a single nonterminal and α
any combination of terminals and nonterminals. However, for purposes of parsing it is
convenient to have the rules in a ‘normal form’. For context-free grammars that is the form
A → aα where a is a terminal symbol.
One proof that grammars can always be rewritten this way uses ‘expression equations’. If x
and y stand for sets of expressions, then x + y, xy, and x∗ stand for union, concatenation,
and repetition respectively.
Consider an example of expression equations. The scalar equation x = a + xb states that
x contains the expressions in a. But then it also contains ab, abb, et cetera. One can verify
that x = ab∗ .
The equation in this example had a regular language as solution; the expression x = a +
bxc does not have a regular solution.
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2.6. PARSING CONTEXT-FREE LANGUAGES
45
Now let x be a vector of all non-terminals in the grammar of a context-free language, and
let f be the vector of righthandsides of rules in the grammar that are of normal form. We
can then write the grammar as
xt = xt A + f t
where the multiplication with A describes all rules not of normal form.
Example:
S
X
Y
→ aSb|XY |c
→ Y Xc|b
→ XS
⇒
φ
[S, X, Y ] = [S, X, Y ] Y
φ
φ φ
φ S + [aSb + c, b, φ]
Xc φ
The solution to this equation is
xt = f t A∗
which describes rules on normal form. However, we need to find a more explicit expression
for A∗ .
Noting that A∗ = λ + AA∗ we get
xt = f t + f t AA∗ = f t + f t B
(2.1)
∗
where B = AA . This is a grammar on normal form. It remains to work out the rules
for B. We have
B = AA∗ = A + AAA∗ = A + AB
These rules need not be of normal form. However, any elements of A that start with a
nonterminal, can only start with nonterminals in x. Hence we can substitute a rule from
equation (2.1).
2.6
Parsing context-free languages
The problem of parsing is this:
Given a grammar G and a string α, determine whether the string is in
the language of G, and through what sequence of rule applications it
was derived.
We will discuss the LL and LR type parser, which correspond to a top-down and bottom-up
way of parsing respectively, then go into the problem of ambiguity
2.6.1
Top-down parsing: LL
One easy parsing strategy starts from the fact that the expression has to come from the start
symbol. Consider the expression 2*5+3, which is produced by the grammar
Expr −→ number Tail
Tail −→ | + number Tail | * number Tail
In the following analysis the stack has its bottom at the right
Victor Eijkhout
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CHAPTER 2. PARSING
initial queue:
2∗5+3
start symbol on stack:
Expr
replace
number Tail
match
∗5+3
Tail
replace
* number Tail
match
5+3
number Tail
match
+3
Tail
replace
+ number Tail
match
3
number Tail
match
Tail
match
The derivation that we constructed here is
E ⇒ nT ⇒ n ∗ nT ⇒ n ∗ n + nT ⇒ n ∗ n + n
that is, we are replacing symbols from the left. Therefore this kind of parsing is called
LL parsing: read from left to right, replace from left to right. Because we only need to look
at the first symbol in the queue to do the replacement, without need for further ‘look ahead’
tokens, this is LL(1) parsing.
But this grammar was a bit strange. Normally we would write
Expr −→ number | number + Expr | number * Expr
If our parser can now see the first two symbols in the queue, it can form
initial queue:
2∗5+3
start symbol on stack:
Expr
replace
number * Expr
match
5+3
Tail
replace
number + Expr
match
3
Expr
replace
3
number
match
This is called LL(2) parsing: we need one token look ahead.
2.6.1.1
Problems with LL parsing
If our grammar had been written
Expr −→ number | Expr + number | Expr * number
an LL(k) parser, no matter the value of k, would have gone into an infinite loop.
In another way too, there are many constructs that can not be parsed with an LL(k) parser
for any k. For instance if both A<B and A<B> are legal expressions, where B can be of
arbitrary length, then no finite amount of look-ahead will allow this to be parsed.
2.6.1.2 LL and recursive descent
The advantages of LL(k) parsers are their simplicity. To see which rule applies at a given
point is a recursive-descent search, which is easily implemented. The code for finding
which rule to apply can broadly be sketched as follows:
TEX – LATEX – CS 594
2.6. PARSING CONTEXT-FREE LANGUAGES
47
define FindIn(Sym,NonTerm)
for all expansions of NonTerm:
if leftmost symbol == Sym
then found
else if leftmost symbol is nonterminal
then FindIn(Sym,that leftmost symbol)
This implies that a grammar is LL-parsable if distinct rules for some non-terminal can not
lead to different terminals. In other words, by looking at a terminal, it should be clear what
production was used.
The LR parsers we will study next are more powerful, but much more complicated to
program. The above problems with LL(k) are largely non-existent in languages where
statements start with unique keywords.
2.6.2
Bottom-up parsing: shift-reduce
In this section we will look at the ‘bottom-up’ parsing strategy, where terminal symbols are
gradually replaced by non-terminals.
One easily implemented bottom-up parsing strategy is called ‘shift-reduce parsing’. The
basic idea here is to move symbols from the input queue to a stack, and every time the
symbols on top of the stack form a right hand size of a production, reduce them to the left
hand side.
For example, consider the grammar
E −→ number | E + E | E * E
and the expression 2 ∗ 5 + 3. We proceed by moving symbols from the left side of the queue
to the top of the stack, which is now to the right.
stack
initial state:
shift
reduce
shift
shift
reduce
reduce
shift, shift, reduce
reduce
2
E
E*
E*5
E*E
E
E+E
E
queue
2∗5+3
*5+3
*5+3
5+3
+3
+3
+3
(Can you tell that we have ignored something important here?)
The derivation we have reconstructed here is
E ⇒E+E ⇒E+3⇒E∗E+3⇒E∗5+3⇒2∗5+3
which proceeds by each time replacing the right-most nonterminal. This is therefore called
a ‘rightmost derivation’. Analogously we can define a ‘leftmost derivation’ as one that
proceeds by replacing the leftmost nonterminal.
Victor Eijkhout
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CHAPTER 2. PARSING
For a formal definition of shift-reduce parsing, we should also define an ‘accept’ and ‘error’
action.
2.6.3
Handles
Finding the derivation of a legal string is not trivial. Sometimes we have a choice between
shifting and reducing, and reducing ‘as soon as possible’ may not be the right solution.
Consider the grammar
S −→ aAcBe
A −→ bA | b
B −→ d
and the string abbcde. This string can be derived (writing the derivation backwards for a
change) as
abbcde ⇐ abAcde ⇐ aAcde ⇐ aAcBe ⇐ S.
However, if we had started
abbcde ⇐ aAbcde ⇐ aAAcde ⇐?
we would be stuck because no further reductions would be applicable.
The problem then is to know where to start replacing terminal symbols and, later in the
derivation, non-terminals. The shift-reduce strategy of the previous section is here seen to
lead to problems, so some extra power is needed. We introduce the concept of ‘handle’
as a formal definition of ‘the right production and place to start reducing’. The following
definition is totally unhelpful:
If S ⇒∗ αAw ⇒ αβw is a right-most derivation, then A → β at the
position after α is a handle of αAw.
Clearly, if we can identify handles, we can derive a parse tree for a given string. However,
the story so far has not been constructive. Next we will look at ways of actually finding
handles.
2.6.4
Operator-precedence grammars
It is easy to find handles if a grammar is of an ‘operator grammar’ form. Loosely, by this
we mean that expressions in the language look like expression-operator-expression. More
strictly, we look at grammars where there are never two adjacent nonterminals, and where
no right hand side is . We also assume that precedence relations between operators and
terminals are known.
Let us look again at arithmetic expressions; we will introduce relations op1 l op2 if the
first operator has lower precedence, and op1 m op2 if it has higher precedence. If the two
operators are the same, we use predence to force associativity rules. For instance, right
associativity corresponds to definitions such as + m +.
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2.6. PARSING CONTEXT-FREE LANGUAGES
49
For the + and ∗ operators we then have the following table:
number + ×
number
m m
l
m l
+
×
l
m m
Now we can find a handle by scanning left-to-right for the first m character, then scanning
back for the matching l. After reducing handles thus found, we have a string of operators
and nonterminals. Ignoring the nonterminals, we insert again the comparisons; this allows
us to find handles again.
For example, 5 + 2 ∗ 3 becomes l5 m + l 2 m ∗ l 3m; replacing handles this becomes
E + E ∗ E. Without the nonterminals, the precedence structure is l + l ∗ m, in which we
find lE ∗ Em as the handle. Reducing this leaves us with E + E, and we find that we have
parsed the string correctly.
This description sounds as if the whole expression is repeatedly scanned to insert precedence relations and find/reduce handle. This is not true, since we only need to scan as far
as the right edge of the first handle. Thus, a shift/reduce strategy will still work for operator
grammars.
2.6.5
LR parsers
We will now consider LR parsers in more detail. These are the parsers that scan the input
from the left, and construct a rightmost derivation, as in the examples we have seen in
section 2.6.2. Most constructs in programming languages can be parsed in an LR fashion.
An LR parser has the following components
•
•
•
A stack and an input queue as in the shift-reduce examples you have already seen
in section 2.6.2. The difference is that we now also push state symbols on the
stack.
Actions ‘shift’, ‘reduce’, ‘accept’, ‘error’, again as before.
An Action and Goto function that work as follows:
–
Suppose the current input symbol is a and the state on top of the stack is s.
–
If Action(a, s) is ‘shift’, then a and a new state s0 = Goto(a, s) are
pushed on the stack.
–
If Action(a, s) is ‘reduce A → β’ where |β| = r, then 2r symbols are
popped from the stack, a new state s0 = Goto(a, s00 ) is computed based on
the newly exposed state on the top of the stack, and A and s0 are pushed.
The input symbol a stays in the queue.
An LR parser that looks at the first k tokens in the queue is called an LR(k) parser. We will
not discuss this issue of look-ahead any further.
It is clear that LR parser are more powerful than a simple shift-reduce parser. The latter
has to reduce when the top of the stack is the right hand side of a production; an LR parser
additionally has states that indicate whether and when the top of the stack is a handle.
Victor Eijkhout
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CHAPTER 2. PARSING
2.6.5.1
A simple example of LR parsing
It is instructive to see how LR parsers can deal with cases for which simple shift/reduce
parsing is insufficient. Consider again the grammar
E −→ E + E | E * E
and the input string 1 + 2 ∗ 3 + 4. Give the + operator precedence 1, and the * operator
precedence 2. In addition to moving tokens onto the stack, we also push the highest precedence seen so far. In the beginning we declare precedence 0, and pushing a non-operator
does not change the precedence.
Shift/reduce conflicts are now resolved with this rule: if we encounter at the front of the
queue a lower precedence than the value on top of the stack, we reduce the elements on top
of the stack.
1 S0
1 S0
1 S0
1 S0
1 S0
1 S0
1 S0
1 S0
1 S0
11
+ S1
+ S1
+ S1
+ S1
+ S1
+ S1
+ S1
+ S1
2 S1
2 S1 * S2
2 S1 * S2 3 S2
6 S1
6 S1 + S1
6 S1 + S1 4 S1
10 S1
1 + 2 ∗ 3 + 4 push symbol; highest precedence is 0
+2 ∗ 3 + 4 highest precedence now becomes 1
2∗3+4
∗3 + 4
highest precedence becoming 2
3+4
+4
reduce because P(+) < 2
+4
the highest exposed precedence is 1
4
at the end of the queue we reduce
Even though this example is phrased very informally, we see the key points:
•
•
only the top of the stack and the front of the queue are inspected;
we have a finite set of rules governing shift/reduce behaviour.
As we shall see, this mechanism can also identify handles.
2.6.5.2
States of an LR parser
An LR parser is constructed automatically from the grammar. Its states are somewhat complicated, and to explain them we need a couple of auxiliary constructs.
item An ‘item’ is a grammar rule with a location indicated. From the rule A → B C we
get the items A → • B C, A → B • C, A → B C• . The interpretation of an item
will be that the symbols left of the dot are on the stack, while the right ones are
still in the queue. This way, an item describes a stage of the parsing process.
closure The closure of an item is defined as the smallest set that
•
Contains that item;
•
If the closure contains an item A → α • B β with B a nonterminal symbol,
then it contains all items B → • γ. This is a recursive notion: if γ starts with
a non-terminal, the closure would also contain the items from the rules of γ.
The states of our LR parser will now be closures of items of the grammar. We motivate
this by an example.
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2.6. PARSING CONTEXT-FREE LANGUAGES
51
Consider now an item A → β1• β2 in the case that we have recognized αβ1 so far. The item
is called valid for that string, if a rightmost derivation S ⇒∗ αAw ⇒ αβ1 β2 w exists. If
β2 = , then A → β1 is a handle and we can reduce. On the other hand, if β2 6= , we have
not encountered the full handle yet, so we shift β2 .
As an example, take the grammar
E −→ E+T | T
T −→ T*F | F
F −→ (E) | id
and consider the partially parsed string E+T*. The (rightmost) derivation
E ⇒E+T ⇒E+T ∗F
shows that T → T*• F is a valid item,
E ⇒ E + T ⇒ E + T ∗ F ⇒ E + T ∗ (E)
gives F → • (E) as a valid item, and
E ⇒ E + T ⇒ E + T ∗ F ⇒ E + T ∗ id
gives F → • id as a valid item.
2.6.5.3
States and transitions
We now construct the actual states of our parser.
•
•
•
•
We add a new start symbol S’, and a production S0 → S.
The starting state is the closure of S0 → • S.
The transition function d(s, X) of a state s and a symbol X is defined as the closure
of
{A → α X• β|A → α • X β is in s}
The ‘follow’ of a symbol A is the set of all terminal symbols that can follow its
possible expansions. This set is easy to derive from a grammar.
Here is an example
We construct the states and transition for the grammar
S −→ (S)S |
which consists of all strings of properly matched left and right parentheses.
Solution: we add the production S0 → • S. We now find the states
1. {S0 → • S, S → • (S)S, S → • }
2. {S0 → S• }
3. {S → (• S)S, S → • (S)S, S → • }
4. {S → (S• )S}
5. {S → (S)• S, S → • (S)S, S → • }
6. {S → (S)S• }
Victor Eijkhout
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CHAPTER 2. PARSING
with transitions
d(0, S) = 1
d(0,0 (0 ) = 2
d(2, S) = 3
d(2,0 (0 ) = 2
d(3,0 )0 ) = 4
d(4, S) = 5
d(4,0 (0 ) = 2
The only thing missing in our parser is the function that describes the stack handling. The
parsing stack consists of states and grammar symbols (alternating). Initially, push the start
state onto the stack. The current state is always the state on the top of the stack. Also, add
a special endmarker symbol to the end of the input string.
Loop:
(1) if the current state contains S0 → S•
accept the string
(2) else if the current state contains any other final item A → α•
pop all the tokens in α from the stack, along with the corresponding states;
let s be the state left on top of the stack: push A, push d(s,A)
(3) else if the current state contains any item A → α • x β,
where x is the next input token
let s be the state on top of the stack: push x, push d(s,x)
else report failure
Explanation:
1.
2.
3.
If we have recognized the initial production, the bottom-up parse process was
successful.
If we have a string of terminals on the stack, that is the right hand side of a
production, replace by the left hand side non-terminal.
If we have a string of terminals on the stack that is the start of a right hand side,
we push the current input symbol.
Exercise 19. Give the states and transitions for the grammar
S −→ x
S −→ (L)
L −→ S
L −→ L S
Apply the above parsing algorithm to the string (x,x,(x)).
The parsers derived by the above algorithm can only handle cases where there is no ambiguity in condition (3). The class of grammars recognized by this type of parser is called LR(0)
and it is not very interesting. We get the more interesting class of SLR(1) by adding to condition (2) the clause that the following symbol is in the follow of A. Another similar class,
which is the one recognized by yacc, is LALR(1).
TEX – LATEX – CS 594
2.6. PARSING CONTEXT-FREE LANGUAGES
2.6.6
53
Ambiguity and conflicts
The problem of finding out how a string was derived is often important. For instance, with
a grammar
hexpri −→ hnumberi | hexpri + hexpri | hexpri × hexpri
the expression 2 + 5 ∗ 3 is ambiguous: it can mean either (2 + 5) ∗ 3 or 2 + (5 ∗ 3).
+
H
H
H
2
*
+
3
H
H
H
H
3
5
2
5
An LR parser would report a ‘shift/reduce conflict’ here: after 2 + 5 has been reduced to
<expr> + <expr>, do we reduce that further to <expr>, or do we shift the minus,
since <expr> - is the start of a legitimate reducible sequence?
*
H
Another example of ambiguity is the ‘dangling else’ problem. Consider the grammar
hstatementi −→ if hclausei then hstatementi | if hclausei
then hstatementi else hstatementi
and the string
if c1 then if c2 then s1 else s2
This can be parsed two ways:
S
P
P
@
@ PPP
PP
@
Then
S
Else
If
H
H
If Then
S
H
HH
HH
H
If
Then
S
H
H
H
If
Then Else
Does the else clause belong to the first if or the second?
Let us investigate the first example. We can solve the ambiguity problem in two ways:
•
Reformulate the grammar as
hexpri −→ hmulexi | hmulexi + hmulexi
hmulexi −→ htermi | htermi × htermi
htermi −→ number
so that the parser can unambiguously reconstruct the derivation,
Victor Eijkhout
54
CHAPTER 2. PARSING
expr
HH
HH
H
mulex
+
mulex
H
HH
term
term * term
•
2
5
3
or
Teach the parser about precedence of operators. This second option may be easier
to implement if the number of operators is large: the first option would require a
large number of rules, with probably a slower parser.
Exercise 20. Rewrite the grammar of the second example to eliminate
the dangling else problem.
Since we are not used to thinking of keywords such as then in terms of precedence, it is
a better solution to eliminate the dangling else problem by introducing a fi keyword to
close the conditional. Often, however, ambiguity is not so easy to eliminate.
Exercise 21. In case of a shift-reduce conflict, yacc shifts. Write an example that proves this. Show what this strategy implies for the dangling
else problem.
Another type of conflict is the ‘reduce/reduce conflict’. Consider this grammar:
A −→ B c d | E c f
B −→ x y
E −→ x y
and the input string that starts x y c.
•
•
•
An LR(1) parser will shift x y, but can not decide whether to reduce that to B
or E on the basis of the look-ahead token c.
An LR(2) parser can see the subsequent d or f and make the right decision.
An LL parser would also be confused, but already at the x. Up to three tokens (x
y c) is unsufficient, but an LL(4) parser can again see the subsequent d or f.
The following grammar would confuse any LR(n) or LL(n) parser with a fixed amount of
look-ahead:
A
B
E
C
−→
−→
−→
−→
BCd | ECf
xy
xy
c | Cc
which generates x y cn {d|f }.
As usual, the simplest solution is to rewrite the grammar to remove the confusion e.g.:
A −→ BorE c d | BorE c f
BorE −→ x y
TEX – LATEX – CS 594
2.6. PARSING CONTEXT-FREE LANGUAGES
55
or assuming we left-factorise the grammar for an LL(n) parser:
A −→ BorE c tail
tail −→ d | f
BorE −→ x y
Another example of a construct that is not LR parsable, consider languages such as Fortran,
where function calls and array indexing both look like A(B,C):
hexpressioni −→ hfunction calli | harray elementi
hfunction calli −→ name ( hparameter listi )
harray elementi −→ name ( hexpression listi )
hparameter listi −→ name | name , hparameter listi
hexpression listi −→ name | name, hexpression listi
After we push B on the stack, it is not clear whether to reduce it to the head of a parameter list or of an expression list, and no amount of lookahead will help. This problem can
be solved by letting the lexical analyzer have access to the symbol table, so that it can
distinguish between function names and array names.
Victor Eijkhout
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CHAPTER 2. PARSING
Lex.
2.7
Introduction
The unix utility lex parses a file of characters. It uses regular expression matching; typically
it is used to ‘tokenize’ the contents of the file. In that context, it is often used together
with the yacc utility. However, there are many other applications possible. By itself, lex is
powerful enough to build interesting programs with, as you will see in a few examples.
2.8
Structure of a lex file
A lex file looks like
...definitions...
%%
...rules...
%%
...code...
Here is a simple example:
%{
int charcount=0,linecount=0;
%}
%%
. charcount++;
\n {linecount++; charcount++;}
%%
int main()
{
yylex();
printf("There were %d characters in %d lines\n",
charcount,linecount);
return 0;
}
In this example, all three sections are present:
definitions All code between %{ and %} is copied to the beginning of the resulting C file.
rules A number of combinations of pattern and action: if the action is more than a single
command it needs to be in braces.
code This can be very elaborate, but the main ingredient is the call to yylex, the lexical
analyser. If the code segment is left out, a default main is used which only calls
yylex.
TEX – LATEX – CS 594
2.9. DEFINITIONS SECTION
2.8.1
57
Running lex
If you store your lex code in a file count.l, you can build an executable from it by
lex -t count.l > count.c
cc -c -o count.o count.c
cc -o counter count.o -ll
You see that the lex file is first turned into a normal C file, which is then compiled and
linked.
If you use the make utility (highly recommended!) you can save a few steps because make
knows about lex:
counter: count.o
cc -o counter count.o -ll
2.9
Definitions section
There are three things that can go in the definitions section:
C code Any indented code between %{ and %} is copied to the C file. This is typically
used for defining file variables, and for prototypes of routines that are defined in
the code segment.
definitions A definition is very much like a #define cpp directive. For example
letter [a-zA-Z]
digit [0-9]
punct [,.:;!?]
nonblank [ˆ \t]
These definitions can be used in the rules section: one could start a rule
{letter}+ {...
state definitions If a rule depends on context, it’s possible to introduce states and incorporate those in the rules. A state definition looks like %s STATE, and by default
a state INITIAL is already given. See section 2.10.2 for more info.
2.10
Rules section
The rules section has a number of pattern-action pairs. The patterns are regular expressions
(see section 2.11, and the actions are either a single C command, or a sequence enclosed in
braces.
If more than one rule matches the input, the longer match is taken. If two matches are the
same length, the earlier one in the list is taken.
It is possible to associate one action with more than one pattern:
[0-9]+
[0-9]+\.[0-9]*
\.[0.9]+
Victor Eijkhout
process_integer();
|
process_real();
58
2.10.1
CHAPTER 2. PARSING
Matched text
When a rule matches part of the input, the matched text is available to the programmer as
a variable char* yytext of length int yyleng.
To extend the example from the introduction to be able to count words, we would write
%{
int charcount=0,linecount=0,wordcount=0;
%}
letter [ˆ \t\n]
%%
{letter}+ {wordcount++; charcount+=yyleng;}
. charcount++;
\n {linecount++; charcount++;}
Exercise 22. Write an integer postfix calculator in lex: expression such
as 1 2 + and 1 2 3 4/*- should be evaluated to 3 and -.5 respectively. White space only serves to separate number, but is otherwise
optional; the line end denotes the end of an expression. You will probably need the C function int atoi(char*) which converts strings to
ints.
2.10.2
Context
If the application of a rule depends on context, there are a couple of ways of dealing with
this. We distinguish between ‘left context’ and ‘right context’, basically letting a rule depend on what comes before or after the matched token.
See section 2.13.1 for an elaborate example of the use of context.
2.10.2.1 Left context
Sometimes, using regular expression as we have seen so far is not powerful enough. For
example:
%%
"/*".*"*/"
.
\n
;
|
ECHO;
works to filter out comments in
This line /* has a */ comment
but not in
This /* line has */ a /* comment */
TEX – LATEX – CS 594
2.10. RULES SECTION
59
What we want is, after the /* string to change the behaviour of lex to throw away all
characters until */ is encountered. In other words, we want lex to switch between two
states, and there is indeed a state mechanism available.
We can consider states to implement implement a dependence on the left context of a rule,
since it changes the behaviour depending on what came earlier. To use a state, a rule is
prefixed as
<STATE>(some pattern) {...
meaning that the rule will only be evaluated if the specified state holds. Switching between
states is done in the action part of the rule:
<STATE>(some pattern) {some action; BEGIN OTHERSTATE;}
where all state names have been defined with %s SOMESTATE statements, as described in
section 2.9. The initial state of lex is INITIAL.
Here is the solution to the comment filtering problem:
%x COMM
%%
.
\n
"/*"
<COMM>"*/"
<COMM>.
<COMM>\n
|
ECHO;
BEGIN COMM;
BEGIN INITIAL;
|
;
%%
We see that the state is defined with %x COMM rather than as indicated above with %s. This
is called an ‘exclusive state’. If an exclusive state is active, rules without state prefix will
not be matched if there is a match in a rule with the prefix of the current state.
2.10.2.2 Right context
It is also possible to let a rule depend on what follows the matched text. For instance
abc/de {some action}
means ‘match abc but only when followed by de. This is different from matching on
abcde because the de tokens are still in the input stream, and they will be submitted to
matching next.
It is in fact possible to match on the longer string; in that case the command
abcde {yyless(3); .....}
pushes back everything after the first 3 characters. The difference with the slash approach
is that now the right context tokens are actually in yytext so they can be inspected.
Victor Eijkhout
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2.11
CHAPTER 2. PARSING
Regular expressions
Many Unix utilities have regular expressions of some sort, but unfortunately they don’t all
have the same power. Here are the basics:
. Match any character except newlines.
\n A newline character.
\t A tab character.
ˆ The beginning of the line.
$ The end of the line.
<expr>* Zero or more occurrences of the expression.
<expr>+ One or more occurrences of the expression.
(<expr1>|<expr2>) One expression of another.
[<set>] A set of characters or ranges, such as [,.:;] or [a-zA-Z].
[ˆ<set>] The complement of the set, for instance [ˆ \t].
Exercise 23. It is possible to have ] and - in a character range. The
] character has to be first, and - has to be either first or last. Why?
Exercise 24. Write regular expressions that match from the beginning
of the line to the first letter ‘a’; to the last letter ‘a’. Also expressions
that match from the first and last ‘a’ to the end of the line. Include representative input and output in your answer.
2.12
Remarks
2.12.1
User code section
If the lex program is to be used on its own, this section will contain a main program. If
you leave this section empty you will get the default main:
int main()
{
yylex();
return 0;
}
where yylex is the parser that is built from the rules.
2.12.2
Input and output to lex
Normally, the executable produced from the lex file will read from standard in and write to
standard out. However, its exact behaviour is that it has two variables
FILE *yyin,*yyout;
that are by default set that way. You can open your own files and assign the file pointer to
these variables.
TEX – LATEX – CS 594
2.13. EXAMPLES
2.12.3
61
Lex and Yacc
The integration of lex and yacc will be discussed in the yacctutorial; here are just a few
general comments.
2.12.3.1 Definition section
In the section of literal C code, you will most likely have an include statement:
#include "mylexyaccprog.h"
as well as prototypes of yacc routines such as yyerror that you may be using. In some
yacc implementations declarations like
extern int yylval;
are put in the .h file that the yacc program generates. If this is not the case, you need to
include that here too if you use yylval.
2.12.3.2 Rules section
If you lexprogrammer is supplying a tokenizer, the yacc program will repeatedly call the
yylex routine. The rules will probably function by calling return everytime they have
constructed a token.
2.12.3.3 User code section
If the lex program is used coupled to a yacc program, you obviously do not want a main
program: that one will be in the yacc code. In that case, leave this section empty; thanks to
some cleverness you will not get the default main if the compiled lex and yacc programs
are linked together.
2.13
Examples
2.13.1
Text spacing cleanup
(This section illustrates the use of contexts; see section 2.10.2.)
Suppose we want to clean up sloppy spacing and punctuation in typed text. For example,
in this text:
This
text (all of it )has occasional lapses , in
punctuation(sometimes pretty bad) ,( sometimes not so).
(Ha! ) Is this : fun?Or what!
We have
•
Multiple consecutive blank lines: those should be compacted.
•
Multiple consecutive spaces, also to be compacted.
•
Space before punctuation and after opening parentheses, and
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•
CHAPTER 2. PARSING
Missing spaces before opening and after closing parentheses.
That last item is a good illustration of where context comes in: a closing paren followed by
punctuation is allowed, but followed by a letter it is an error to be corrected.
We can solve this problem without using context, but the lex code would be longer and
more complicated. To see this, consider that we need to deal with spacing before and after
a parenthesis. Suppose that there are m cases of material before, and n of material after,
to be handled. A lex code without context would then likely have m × n rules. However,
using context, we can reduce this to m + m.
2.13.1.1 Right context solution
Let us first try a solution that uses ‘right context’: it basically describes all cases and corrects the spacing.
punct [,.;:!?]
text [a-zA-Z]
%%
")"" "+/{punct}
")"/{text}
{text}+" "+/")"
{printf(")");}
{printf(") ");}
{while (yytext[yyleng-1]==’ ’) yyleng--; ECHO;}
({punct}|{text}+)/"(" {ECHO; printf(" ");}
"("" "+/{text}
{while (yytext[yyleng-1]==’ ’) yyleng--; ECHO;}
{text}+" "+/{punct}
{while (yytext[yyleng-1]==’ ’) yyleng--; ECHO;}
ˆ" "+
" "+
.
\n/\n\n
\n
;
{printf(" ");}
{ECHO;}
;
{ECHO;}
In the cases where we match superfluous white space, we manipulate yyleng to remove
the spaces.
2.13.1.2 Left context solution
Using left context, we implement a finite state automaton in lex, and specify how to treat
spacing in the various state transitions. Somewhat surprisingly, we discard spaces entirely,
and reinsert them when appropriate.
We recognise that there are four categories, corresponding to having just encountered an
open or close parenthesis, text, or punctuation. The rules for punctuation and closing parentheses are easy, since we discard spaces: these symbols are inserted regardless the state. For
text and opening parentheses we need to write rules for the various states.
TEX – LATEX – CS 594
2.13. EXAMPLES
punct [,.;:!?]
text [a-zA-Z]
%s
%s
%s
%s
OPEN
CLOSE
TEXT
PUNCT
%%
" "+ ;
<INITIAL>"(" {ECHO; BEGIN OPEN;}
<TEXT>"(" {printf(" "); ECHO; BEGIN OPEN;}
<PUNCT>"(" {printf(" "); ECHO; BEGIN OPEN;}
")" {ECHO ; BEGIN CLOSE;}
<INITIAL>{text}+ {ECHO; BEGIN TEXT;}
<OPEN>{text}+ {ECHO; BEGIN TEXT;}
<CLOSE>{text}+ {printf(" "); ECHO; BEGIN TEXT;}
<TEXT>{text}+ {printf(" "); ECHO; BEGIN TEXT;}
<PUNCT>{text}+ {printf(" "); ECHO; BEGIN TEXT;}
{punct}+ {ECHO; BEGIN PUNCT;}
\n {ECHO; BEGIN INITIAL;}
%%
Exercise 25. Write a lex parser that analyzes text the way the TEX
input processor does with the normal category code values. It should
print its output with
•
<space> denoting any space that is not ignored or skipped, and
•
<cs: command> for recognizing a control sequence \command;
•
open and close braces should also be marked as <{>, <}>.
Here is some sample input:
this is {a line} of text.
handle \control sequences \andsuch
with \arg{uments}.
Aha!
this line has %a comment
x
y%
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64
CHAPTER 2. PARSING
z
\comm%
and
TEX – LATEX – CS 594
2.14. INTRODUCTION
65
Yacc.
2.14
Introduction
The unix utility yacc (Yet Another Compiler Compiler) parses a stream of token, typically
generated by lex, according to a user-specified grammar.
2.15
Structure of a yacc file
A yacc file looks much like a lex file:
...definitions...
%%
...rules...
%%
...code...
definitions As with lex, all code between %{ and %} is copied to the beginning of the
resulting C file. There can also be various definitions; see section 2.17.
rules As with lex, a number of combinations of pattern and action. The patterns are now
those of a context-free grammar, rather than of a regular grammar as was the case
with lex.
code This can be very elaborate, but the main ingredient is the call to yyparse, the
grammatical parse.
2.16
Motivating example
It is harder to give a small example of yacc programming than it was for lex. Here is a
program that counts the number of different words in a text. (We could have written this
particular example in lex too.)
First consider the lex program that matches words:
%{
#include "words.h"
int find_word(char*);
extern int yylval;
%}
%%
[a-zA-Z]+ {yylval = find_word(yytext);
return WORD;}
.
;
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66
\n
CHAPTER 2. PARSING
;
%%
The lexer now no longer has a main program, but instead returns a WORD return code. It
also calls a routine find_word, which inserts the matched word in a list if it is not already
there.
The routine find_word is defined in the yacc code:
%{
#include <stdlib.h>
#include <string.h>
int yylex(void);
#include "words.h"
int nwords=0;
#define MAXWORDS 100
char *words[MAXWORDS];
%}
%token WORD
%%
text : ;
| text WORD ; {
if ($2<0) printf("new word\n");
else printf("matched word %d\n",$2);
}
%%
int find_word(char *w)
{
int i;
for (i=0; i<nwords; i++)
if (strcmp(w,words[i])==0) {
return i;
}
words[nwords++] = strdup(w);
return -1;
}
int main(void)
{
yyparse();
printf("there were %d unique words\n",nwords);
TEX – LATEX – CS 594
2.17. DEFINITIONS SECTION
67
}
Other things to note:
•
The WORD token that was used in the lex code is defined here in the definitions
section; lex knows about it through including the words.h file.
•
The lex rule also sets a variable yylval; this puts a value on the stack top, where
yacc can find it with $1, $2, et cetera.
All of this will be explained in detail below.
2.17
Definitions section
There are three things that can go in the definitions section:
C code Any code between %{ and %} is copied to the C file. This is typically used for
defining file variables, and for prototypes of routines that are defined in the code
segment.
definitions The definitions section of a lex file was concerned with characters; in yacc this
is tokens. These token definitions are written to a .h file when yacc compiles this
file.
associativity rules These handle associativity and priority of operators; see section 2.20.
2.18
Lex Yacc interaction
Conceptually, lex parses a file of characters and outputs a stream of tokens; yacc accepts a
stream of tokens and parses it, performing actions as appropriate. In practice, they are more
tightly coupled.
If your lex program is supplying a tokenizer, the yacc program will repeatedly call the
yylex routine. The lex rules will probably function by calling return every time they
have parsed a token. We will now see the way lex returns information in such a way that
yacc can use it for parsing.
2.18.1
The shared header file of return codes
If lex is to return tokens that yacc will process, they have to agree on what tokens there are.
This is done as follows.
•
The yacc file will have token definitions
%token NUMBER
in the definitions section.
•
When the yacc file is translated with yacc -d -o, a header file <file>.h3 is
created that has definitions like
#define NUMBER 258
This file can then be included in both the lex and yacc program.
3.
If you leave out the -o option to yacc, the file is called y.tab.h.
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CHAPTER 2. PARSING
•
The lex file can then call return NUMBER, and the yacc program can match on
this token.
The return codes that are defined from %TOKEN definitions typically start at around 258,
so that single characters can simply be returned as their integer value:
/* in the lex program */
[0-9]+ {return NUMBER}
[-+*/] {return *yytext}
/* in the yacc program */
sum : NUMBER ’+’ NUMBER
The yacc code now recognizes a sum if lex returns in sequence a NUMBER token, a plus
character, and another NUMBER token.
See example 2.22.1 for a worked out code.
2.18.2
Return values
In the above, very sketchy example, lex only returned the information that there was a number, not the actual number. For this we need a further mechanism. In addition to specifying
the return code, the lex parser can return a value that is put on top of the stack, so that yacc
can access it. This symbol is returned in the variable yylval. By default, this is defined
as an int, so the lex program would have
extern int yylval;
%%
[0-9]+ {yylval=atoi(yytext); return NUMBER;}
See section 2.19.1 for how the stack values are used by yacc.
If more than just integers need to be returned, the specifications in the yacc code become
more complicated. Suppose we are writing a calculator with variables, so we want to return
double values, and integer indices in a table. The following three actions are needed.
1.
The possible return values need to be stated:
%union {int ival; double dval;}
2.
These types need to be connected to the possible return tokens:
%token <ival> INDEX
%token <dval> NUMBER
3.
The types of non-terminals need to be given:
%type <dval> expr
%type <dval> mulex
%type <dval> term
The generated .h file will now have
#define INDEX 258
#define NUMBER 259
typedef union {int ival; double dval;} YYSTYPE;
extern YYSTYPE yylval;
This is illustrated in example 2.22.2.
TEX – LATEX – CS 594
2.19. RULES SECTION
2.19
69
Rules section
The rules section contains the grammar of the language you want to parse. This looks like
name1 :
THING something OTHERTHING {action}
| othersomething THING
{other action}
name2 :
.....
This is the general form of context-free grammars, with a set of actions associated with
each matching right-hand side. It is a good convention to keep non-terminals (names that
can be expanded further) in lower case and terminals (the symbols that are finally matched)
in upper case.
The terminal symbols get matched with return codes from the lex tokenizer. They are typically defines coming from %token definitions in the yacc program or character values;
see section 2.18.1.
A simple example illustrating the ideas in this section can be found in section 2.22.1.
2.19.1
Rule actions
The example in section 2.22.1 had such rules as:
expr:
expr ’+’ mulex
| expr ’-’ mulex
| mulex
{ $$ = $1 + $3; }
{ $$ = $1 - $3; }
{ $$ = $1; }
The action belonging to the different right hand sides refer to $n quantities and to $$. The
latter refers to the stack top, so by assigning to it a new item is put on the stack top. The
former variables are assigned the values on the top of the stack: if the right hand side has
three terms, terminal or nonterminal, then $1 through $3 are assigned and the three values
are removed from the stack top.
2.20
Operators; precedence and associativity
The example in section 2.22.1 had separate rules for addition/subtraction and multiplication/division. We could simplify the grammar by writing
expr:
expr ’+’
expr ’-’
expr ’*’
expr ’/’
expr ’ˆ’
number ;
expr
expr
expr
expr
expr
;
;
;
;
;
but this would have 1+2*3 evaluate to 9. In order to indicate operator precedence, we can
have lines
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CHAPTER 2. PARSING
%left ’+’ ’-’
%left ’*’ ’/’
%right ’ˆ’
The sequence of lines indicates increasing operator precedence and the keyword sets the
associativity type: we want 5-1-2 to be 2, so minus is left associative; we want 2ˆ2ˆ3 to
be 256, not 64, so exponentiation is right associative.
Operators that can be both unary and binary are handled by declaring a non-associative
token, and explicitly indicating its precedence.
%left ’-’ ’+’
%nonassoc UMINUS
%
expression : expression ’+’ expression
| expression ’-’ expression
| ’-’ expression %prec UMINUS
2.21
Further remarks
2.21.1
User code section
The minimal main program is
int main()
{
yyparse();
return 0;
}
Extensions to more ambitious programs should be self-evident.
In addition to the main program, the code section will usually also contain subroutines, to
be used either in the yacc or the lex program. See for instance example 2.22.3.
Exercise 26. Try to write lex or yacc programs for the following languages:
an bm , an bn , an bn cn
Discuss the theoretical power of lex and yacc.
2.21.2
Errors and tracing
So far we have assumed that the input to yacc is syntactically correct, and yacc need only
discover its structure. However, occasionally input will be incorrect.
2.21.2.1 Tracing
If you assign yydebug=1;, yacc will produce trace output. While its states may not make
sense to you, at least you will see which tokens it matches, and which rules it can reduce.
TEX – LATEX – CS 594
2.21. FURTHER REMARKS
71
2.21.2.2 Syntax errors
Sometimes, yacc reports ‘syntax error’ and stops processing. This means that an unexpected symbol is found. A common source for this is the case that you have made a typo
in your grammar, and the symbol it is trying to match is not defined. Example: suppose we
have just matched an open token:
group : open body close
bodytext : ;
| character bodytext
If you are tracing yacc’s workings, you will probably see it matching the character,
then giving the syntax error message.
The ‘syntax error’ message is actually yacc’s default implementation of the yyerror
routine, but it can be redefined at will. For example, suppose we have a declaration
int lineno=1;
/* in yacc */
extern int lineno; /* in lex */
and every line with \n in lex increases this variable. We could then define
void yyerror(char *s)
{
printf("Parsing failed in line %d because of %s\n",
lineno,s);
}
2.21.2.3 Error recovery
Error recovery in yacc is possible through the error token. In the rule
foo : bar baz ;
| error baz printf("Hope for the best\n");
recognizing any token but bar will make yacc start skipping tokens, hoping to find baz
and recover from that point. This is not guaranteed to work.
2.21.2.4 Semantical errors
Both lex and yacc are stronger than simple finite-state or pushdown automata, for instance
if they are endowed with a symbol table. This can be used to detect semantic errors. For
instance, while you would like to write
array_slice : array_name ’[’ int_expr ’]’
you may be limited to
array_slice : ident ’[’ int_expr ’]’
{if (!is_array($1)) { ....
There are a couple of tools here:
yyerror(char*) is a default write to stderr; you can redefine it.
YYABORT is a macro that halts parsing.
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2.21.3
CHAPTER 2. PARSING
Makefile rules for yacc
The make utility knows about lex and yacc, but if you want to do things yourself, here are
some good rules:
# disable normal rules
.SUFFIXES:
.SUFFIXES: .l .y .o
# lex rules
.l.o :
lex -t $*.l > $*.c
cc -c $*.c -o $*.o
# yacc rules
.y.o :
if [ ! -f $*.h ] ; then touch $*.h ; fi
yacc -d -t -o $*.c $*.y
cc -c -o $*.o $*.c ;
rm $*.c
# link lines
lexprogram : $(LEXFILE).o
cc $(LEXFILE).o -o $(LEXFILE) -ll
yaccprogram : $(YACCFILE).o $(LEXFILE).o
cc $(YACCFILE).o $(LEXFILE).o -o $(YACCFILE) -ly -ll
2.21.4
The power of yacc
Theoretically, yacc implements an LALR(1) parser, which is essentially an LR parser with
one token look-ahead. This describes a large class of useful grammars. As an example of a
grammar with two tokens look-ahead, consider
phrase −→ CART ANIMAL and cart | WORK ANIMAL and plow
CART ANIMAL −→ horse | goat
WORK ANIMAL −→ horse | ex
Now to distinguish between horse and cart and horse and plow from the word
horse takes two tokens look-ahead.
Exercise 27. Use the TEX parser you wrote in lex to parse LATEX documents. The parser should
•
Report the documentclass used;
•
Check that \begin{document} and \end{document} are
used, with no text before the begin command;
•
Know about some commands with one argument, such as \textbf,
and properly recognize that argument
•
Recognize proper matching of begin/end of an environment.
TEX – LATEX – CS 594
2.22. EXAMPLES
73
Bonus: let your parser interpret \newcommand correctly. You can limit
yourself to the the case of commands with one argument, that is
\newcommand{\foo}[1]{ ...}
2.22
Examples
2.22.1
Simple calculator
This calculator evaluates simple arithmetic expressions. The lex program matches numbers
and operators and returns them; it ignores white space, returns newlines, and gives an error
message on anything else.
%{
#include <stdlib.h>
#include <stdio.h>
#include "calc1.h"
void yyerror(char*);
extern int yylval;
%}
%%
[ \t]+ ;
[0-9]+
[-+*/]
"("
")"
\n
.
{yylval = atoi(yytext);
return INTEGER;}
{return *yytext;}
{return *yytext;}
{return *yytext;}
{return *yytext;}
{char msg[25];
sprintf(msg,"%s <%s>","invalid character",yytext);
yyerror(msg);}
Accepting the lex output, the following yacc program has rules that parse the stream of
numbers and operators, and perform the corresponding calculations.
%{
#include <stdlib.h>
#include <stdio.h>
int yylex(void);
#include "calc1.h"
%}
%token INTEGER
%%
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CHAPTER 2. PARSING
program:
line program
| line
line:
expr ’\n’
| ’n’
expr:
expr ’+’ mulex
| expr ’-’ mulex
| mulex
mulex:
mulex ’*’ term
| mulex ’/’ term
| term
term:
’(’ expr ’)’
| INTEGER
{ printf("%d\n",$1); }
{ $$ = $1 + $3; }
{ $$ = $1 - $3; }
{ $$ = $1; }
{ $$ = $1 * $3; }
{ $$ = $1 / $3; }
{ $$ = $1; }
{ $$ = $2; }
{ $$ = $1; }
%%
void yyerror(char *s)
{
fprintf(stderr,"%s\n",s);
return;
}
int main(void)
{
/*yydebug=1;*/
yyparse();
return 0;
}
Here we have realized operator precedence by having separate rules for the different priorities. The rule for plus/minus comes first, which means that its terms, the mulex expressions involving multiplication, are evaluated first.
2.22.2
Calculator with simple variables
In this example the return variables have been declared of type double. Furthermore, there
can now be single-character variables that can be assigned and used. There now are two
different return tokens: double values and integer variable indices. This necessitates the
%union statement, as well as %token statements for the various return tokens and %type
statements for the non-terminals.
TEX – LATEX – CS 594
2.22. EXAMPLES
This is all in the yacc file:
%{
#include <stdlib.h>
#include <stdio.h>
int yylex(void);
double var[26];
%}
%union { double dval; int ivar; }
%token <dval> DOUBLE
%token <ivar> NAME
%type <dval> expr
%type <dval> mulex
%type <dval> term
%%
program:
line program
| line
line:
expr ’\n’
{ printf("%g\n",$1); }
| NAME ’=’ expr ’\n’ { var[$1] = $3; }
expr:
expr ’+’ mulex
{ $$ = $1 + $3; }
| expr ’-’ mulex
{ $$ = $1 - $3; }
| mulex
{ $$ = $1; }
mulex:
mulex ’*’ term
{ $$ = $1 * $3; }
| mulex ’/’ term
{ $$ = $1 / $3; }
| term
{ $$ = $1; }
term:
’(’ expr ’)’
{ $$ = $2; }
| NAME
{ $$ = var[$1]; }
| DOUBLE
{ $$ = $1; }
%%
void yyerror(char *s)
{
fprintf(stderr,"%s\n",s);
return;
}
int main(void)
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75
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CHAPTER 2. PARSING
{
/*yydebug=1;*/
yyparse();
return 0;
}
The lex file is not all that different; note how return values are now assigned to a component
of yylval rather than yylval itself.
%{
#include <stdlib.h>
#include <stdio.h>
#include "calc2.h"
void yyerror(char*);
%}
%%
[ \t]+ ;
(([0-9]+(\.[0-9]*)?)|([0-9]*\.[0-9]+)) {
yylval.dval = atof(yytext);
return DOUBLE;}
[-+*/=]
{return *yytext;}
"("
{return *yytext;}
")"
{return *yytext;}
[a-z]
{yylval.ivar = *yytext;
return NAME;}
\n
{return *yytext;}
.
{char msg[25];
sprintf(msg,"%s <%s>","invalid character",yytext);
yyerror(msg);}
2.22.3
Calculator with dynamic variables
Basically the same as the previous example, but now variable names can have regular
names, and they are inserted into a names table dynamically. The yacc file defines a routine
for getting a variable index:
%{
#include <stdlib.h>
#include <stdio.h>
#include <string.h>
int yylex(void);
#define NVARS 100
char *vars[NVARS]; double vals[NVARS]; int nvars=0;
%}
TEX – LATEX – CS 594
2.22. EXAMPLES
77
%union { double dval; int ivar; }
%token <dval> DOUBLE
%token <ivar> NAME
%type <dval> expr
%type <dval> mulex
%type <dval> term
%%
program:
line program
| line
line:
expr ’\n’
| NAME ’=’ expr ’\n’
expr:
expr ’+’ mulex
{
| expr ’-’ mulex
{
| mulex
{
mulex:
mulex ’*’ term
{
| mulex ’/’ term
{
| term
{
term:
’(’ expr ’)’
{
| NAME
{
| DOUBLE
{
{ printf("%g\n",$1); }
{ vals[$1] = $3; }
$$ = $1 + $3; }
$$ = $1 - $3; }
$$ = $1; }
$$ = $1 * $3; }
$$ = $1 / $3; }
$$ = $1; }
$$ = $2; }
$$ = vals[$1]; }
$$ = $1; }
%%
int varindex(char *var)
{
int i;
for (i=0; i<nvars; i++)
if (strcmp(var,vars[i])==0) return i;
vars[nvars] = strdup(var);
return nvars++;
}
int main(void)
{
/*yydebug=1;*/
yyparse();
return 0;
}
Victor Eijkhout
78
CHAPTER 2. PARSING
The lex file is largely unchanged, except for the rule that recognises variable names:
%{
#include <stdlib.h>
#include <stdio.h>
#include "calc3.h"
void yyerror(char*);
int varindex(char *var);
%}
%%
[ \t]+ ;
(([0-9]+(\.[0-9]*)?)|([0-9]*\.[0-9]+)) {
yylval.dval = atof(yytext);
return DOUBLE;}
[-+*/=]
{return *yytext;}
"("
{return *yytext;}
")"
{return *yytext;}
[a-z][a-z0-9]* {
yylval.ivar = varindex(yytext);
return NAME;}
\n
{return *yytext;}
.
{char msg[25];
sprintf(msg,"%s <%s>","invalid character",yytext);
yyerror(msg);}
TEX – LATEX – CS 594
2.23. INTRODUCTION
Hashing.
79
Hashing, hash functions, hash tables, come into play when a compiler, and
in particular its parser, needs to store names (of identifiers) and further information about
the object of that name.
2.23
Introduction
A compiler, and in particular its parser, needs to store variables and information about them.
The data structure for this is in effect addressed by the name of the variable, rather than by
any numerical index. This sort of storage is sometimes called ‘associative’. The design of
a data structure for this is a good example of trade-offs between efficiency and expediency.
•
•
•
•
If variables are stored in the order in which they are encountered, storage is very
fast, but searching will take time linear in the number of items stored.
If the list if kept sorted, searching for an item will take logarithmic time. However,
insertion is now more expensive, taking linear time because all elements following
have to be copied up.
A naively implemented linked list would give both insertion and retrieval time
linearly in the number of stored items. In the insertion phase we avoid the copying,
but finding the place to insert is harder, and in fact we can not use bisection here.
Items can be stored in a treewise fashion:
•
HH
HH
B
E
H
H
H
H
H
LSE ND
ART
E
H
H
GIN LL
The cost of inserting and retrieving is then linear in the length of the string, at
least for as far as is necessary to disambiguate from other strings.
These are then the issues we need to address:
•
•
•
What is the cost of inserting a new item?
What is the cost of finding and retrieving an item?
What is the cost of deleting an item?
2.24
Hash functions
A hash function is function that maps non-numeric keys into a range of integers, interpreted
as the locations in a table. It would be nice if this function was injective, to avoid mapping
two keys to the same location but surprisingly hard, as is clear from the ‘birthday paradox’:
it takes only 23 random picks from a 365-entry table to have even chances of a collision.
If we know all keys in advance, it is possible to design a function that maps them uniquely
into a table of precisely the right size, but this is unrealistic, since the number of possible
keys in a programming language is very large, indeed unbounded.
Victor Eijkhout
80
CHAPTER 2. PARSING
Figure 2.1: A hash function without conflicts
A ‘hash function’ is then a function that maps keys in some space to a range of integers 0 . . . M − 1. A good hash function has the following properties:
•
•
•
•
The hash value is fully determined by the data being hashed. For instance, it
should not have a ‘memory’.
The hash function uses as much as possible of the input data. Program variables
often have names such as ikey, so emphasis on the first letter is a bad idea.
The hash function ”uniformly” distributes the data across the entire set of possible
hash values.
The hash function generates very different hash values for similar strings. Variables like key1, key2, et cetera should not be mapped into a cluster.
Figure 2.1 illustrates a hash function without conflicts.
Let us assume that we have found a way of mapping the names onto a large integer space,
for instance by interpreting the bit pattern of the name as an integer. A simple hash function
would be
h(K) = K mod M,
(2.2)
where M is the size of the hash table.
Certain values of M are less desirable. For instance, if M is even, say M = 2M 0 , then the
statement r = K mod M (say K = nM + r for some n) implies
K = 2K 0 ⇒ r = 2(nM 0 − K 0 )
K = 2K 0 + 1 ⇒ r = 2(nM 0 − K 0 ) + 1
so the key is even, iff the original number is. This places an undue influence on the last
digit. If M is a multiple of 3, we find for numbers stored decimally or in bytes that keys
that are a permutation of each other get mapped to numbers that differ by a multiple of 3,
since both 10n mod 3 = 1 and 28 mod 1 = 1.
TEX – LATEX – CS 594
2.24. HASH FUNCTIONS
2.24.1
81
Multiplication and division strategies
A good strategy is to take M prime, and such that rk 6= ±a mod M , where r the radix
of the number system, and a, k small. (There is a list of suitable primes on http://
planetmath.org/encyclopedia/GoodHashTablePrimes.html.)
Equation (2.2) requires you to perform a division. The same equation based on multiplication would use an integer A ≈ w/M , where w is the maxint value. Then 1/M = A/w,
which is simply A with an imaginary decimal point to its left. Observing that
K mod M = M (K/M mod 1)
we define
A
h(K) = bM
K mod 1 c.
w
As an example of the value of using a prime table size, consider hashing the Bible, which
consists of 42,829 unique words, into an open hash table with 30,241 elements (a prime
number). In the end, 76.6 percent of the slots were used and that the average chain was
1.85 words in length (with a maximum of 6). The same file run into a hash table of 30,240
elements (evenly divisible by integers 2 through 9) fills only 60.7 percent of the slots and
the average chain is 2.33 words long (maximum: 10).
2.24.2
String addition strategies
One could Derive a hash key by adding or XORing together all bytes in a string.
h = <some value>
for (i=0; i<len(var); i++)
h = h + <byte i of string>;
This runs into the problem that anagrams map into the same key, and nearby strings into
nearby keys. This could be remedied by introducing a table of random numbers:
h = <some value>
for (i=0; i<len(var); i++)
h = Rand( h XOR <byte i of string> );
Exercise 28. This algorithm only gives a one-byte key. How would
you derive longer keys? Give pseudo-code for the algorithm.
2.24.3
Examples
Here are a couple of published hash functions:
/* UNIX ELF hash
* Published hash algorithm used in the UNIX ELF format for object files
*/
unsigned long hash(char *name)
{
unsigned long h = 0, g;
while ( *name ) {
Victor Eijkhout
82
CHAPTER 2. PARSING
h = ( h << 4 ) + *name++;
if ( g = h & 0xF0000000 )
h ˆ= g >> 24;
h &= ˜g;
}
}
This hash key is then reduced to an index in the hash table by
#define HASHSIZE 997
static int M = HASHSIZE;
return h % M;
Another hash function:
/* djb2
* This algorithm was first reported by Dan Bernstein
* many years ago in comp.lang.c
*/
unsigned long hash(unsigned char *str)
{
unsigned long hash = 5381;
int c;
while (c = *str++) hash = ((hash << 5) + hash) + c;
return hash;
}
Note the use of bit shifts to implement multiplication.
2.25
Collisions
The above techniques for generating randomly spread out addresses are generally sufficient. The problem to worry about is how to handle collisions, that is, if h(k1 ) = h(k2 ) for
different keys k1 , k2 . We will investigate several techniques for dealing with this.
For all of the strategies below, any performance analysis is statistical in nature. The average
expected behaviour is often excellent, but the worst case behaviour is always very bad. In
the worst case, all hash addresses map to the same location, and search time is propertional
to the number of elements in the table.
The question is now how to find the storage locations for the elements with conflicting hash
keys. We will look at one strategy that allocates space outside the hash table (‘open hash
table’), and two that resolve the conflict by finding different locations in the table (‘closed
hash table’).
2.25.1
Separate chaining
A simple solution to hash conflicts is the create a linked list from each table entry, as shown
in figure 2.5. This way of implementing a hash table is called ‘separate chaining’ or ‘open
TEX – LATEX – CS 594
2.25. COLLISIONS
83
Figure 2.2: Separate chaining as a solution for hash conflicts
hashing’. One problem with this approach is that we need to maintain two different kinds
of storage: static in the table and dynamic for the linked lists.
The linked lists can be created by malloc (and released by free) calls. However, these
are very expensive compared to using a freespace pointer in a list. To amortize the cost, one
could allocate a block of cells, and dole them out gradually. The problem with this strategy
is dealing with deletions. If a cell is deleted, rerouting the pointers is easy, but additionally
we have to keep track of which cells in the allocated chunk are free. This makes the open
hash table somewhat complicated to implement.
Another problem is that, while this strategy is fairly efficient if the number of collisions is
low, in order to get the number of collisions low, one typically chooses the table size fairly
large, and then this solution is wasteful of storage. It is then better to store all elements in
the hash table and maintain links in the table.
Exercise 29. Discuss the value of keeping the lists ordered by key:
how does this influence the run time of insertion and retrieval? Programming languages like C have local variables. Does this change your
argument?
2.25.2
Linear probing
The easiest solution is to store a conflicting element in the location immediately after the
computed hash address.
struct { ... } node;
node Table[M]; int Free;
/* insert K */
addr = Hash(K);
if (IsEmpty(addr)) Insert(K,addr);
else {
Victor Eijkhout
84
CHAPTER 2. PARSING
Figure 2.3: Linear probing as a solution for hash conflicts
I
I
J3
J
J2
J
I
J3
J
K
J2
J
K
K
I
I
L
J3
J
J2
J
K
L
I
Figure 2.4: Coalescing of blocks in linear probing
/* see if already stored */
test:
if (Table[addr].key == K) return;
else {
addr = Table[addr].link; goto test;}
/* find free cell */
Free = addr;
do { Free--; if (Free<0) Free=M-1; }
while (!IsEmpty(Free) && Free!=addr)
if (!IsEmpty(Free)) abort;
else {
Insert(K,Free); Table[addr].link = Free;}
}
However, this is not the best solution. Suppose that the blocks of size N is occupied, then
the free pointer will search N/2 locations on average for an address that maps into this
block. While this is acceptable, if two blocks coalesce, this makes the search time double.
Note that the chance of the cell between two blocks filling up is much larger than the
chance of that exact address being generated as hash: each hash in the top block will cause
the address to be filled.
TEX – LATEX – CS 594
2.25. COLLISIONS
85
This is illustrated in figure 2.4. There is a gap of size one between h(I) and a block starting
at h(J). When a conflict h(K) = h(I) occurs, the free space pointer fills the gap. A subsequent conflict h(L) = h(I) (or h(L) = h(K)) needs the free space pointer to traverse
the whole J block to find the next location.
With α = N/M the ratio between occupied cells and total table size, the expected search
time with this algorithm is
2
1
1 1+
unsuccessful
2
1−α
T ≈
1 1+ 1
successful
2
1−α
It is clear that when α approaches 1, this time will go up unbounded.
The clumping behaviour of this algorithm makes it sensitive to the hash algorithm used.
Care has to be taken that successive keys, such as Ptr1, Ptr2. . . , do not get mapped to
successive hash values K, K + 1, . . ..
2.25.3
Chaining
The problems with linear probing can be prevented by storing conflicting elements at the
start or end of the table.
struct { ... } node;
node Table[M]; int Free = M;
/* insert K */
addr = Hash(K);
if (IsEmpty(addr)) Insert(K,addr);
else {
/* see if already stored */
test:
if (Table[addr].key == K) return;
else {
addr = Table[addr].link; goto test;}
/* find free cell */
do { Free--; }
while (!IsEmpty(Free)
if (Free<0) abort;
else {
Insert(K,Free); Table[addr].link = Free;}
}
This algorithm does the same list traversal as linear probing in the case a search is ultimately successful. However, for an unsuccessful search the Free pointer will usually be
decreased by one, and only occasionally by two or more, when it runs into already occupied positions. Since these are likely to be spread out, having to search more than two steps
will be rare.
In this algorithm, occasionally a hash address will be an address that has further links. We
say that we have lists coalescing. This increases search time slightly, but not by much, and
preventing this increases insertion time, because we would have to move cells.
Victor Eijkhout
86
CHAPTER 2. PARSING
Figure 2.5: Chaining as a solution for hash conflicts
With α = N/M the fraction of used to total table entries, find that the number of entries
searched is
1 + (e2α − 1 − 2α)/4
unsuccessful
T ≈
1 + (e2α − 1 − 2α)/8α + α/4 successful
The hash algorithm of TEX is a variant of this chaining algorithm.
2.25.4
Other solutions
The solutions to the conflict problem given so far can be called ‘linear rehashing’. The
following strategies are called ‘nonlinear rehashing’.
Random probing Try (h(m) + pi ) mod s, where pi is a sequence of random numbers.
This requires either reproducible random numbers, or storing these numbers. In
order to prevent colliding keys to collide on the next try too, the random number
needs to depend on the key.
Add the hash Try (i × h(m)) mod s. This requires s to be a prime number; with this
approach clumping is prevented.
They have the advantage that occupied locations in the table remain fairly scattered. On the
other hand, they require further hash calculations. Also, because of the irregular memory
access pattern, the cost of memory operations may become significant here.
2.25.5
Deletion
A surprising aspect of closed hash table algorithms is that generally it is hard to delete
elements. Since most algorithms give coalescing lists, we can not mark a cell empty if its
key is to be removed. Instead, we mark a cell ‘deleted’, which removes the key, but leaves
the link intact. There are algorithms that can deal with deletions, but they have a fairly high
complexity.
On the other hand, deleting in an open hash table algorithm is simple. The complication
there is the freeing of the memory. If the cells are allocated in chunks, the decision to free
a chunk becomes complicated.
2.26
Other applications of hashing
The foremost application of hashing is in compilers. However, there are other uses.
TEX – LATEX – CS 594
2.27. DISCUSSION
2.26.1
87
Truncating searches
In applications such as chess programs, you want to avoid evaluating a configuration twice
if it’s arrived at two different ways. This can be done by storing evaluations in a table. This
table can be addressed by the configuration as a key itself, but these keys are long and span
a large space, so searching will probably be expensive. Instead, one can use a hash table.
If two configurations generate the same hash key, they can be, but need not be the same,
so further decision might be needed. To avoid this second stage work, a good quality hash
function is essential.
(This example comes from http://www.seanet.com/˜brucemo/topics/hashing.
htm.)
2.26.2
String searching
The question ‘does a string of length M appear anywhere in a document of length N ’ can
be answered in O(N M ) time by a sequence of string comparisons. However, we can do
considerably better, reducing the complexity to O(N + M ).
A hash function that adds characters together will give the same hash key for strings that
are anagrams of each other. This means that instead of doing a string comparison we can
compare hash keys, and only if they are equal resort to a full string comparison. To get the
complexity down, we note that if the hash function is of the form
(
)
X
h(k) =
k[i] mod K,
i
where k is a character string, then (for a text t long enough)
h(t[2 . . . n + 1]) = h(t[1 . . . n]) + t[n + 1] − t[1]
(with addition/subtraction modulo K) so we can cheaply update the hash key in O(1) time.
2.27
Discussion
In a comparison between hash tables and, for instance, tree-based storage, there is no clear
preference. Hash tables can be faster, because, until the table fills up, access is O(1). A similar complexity can be obtained for trees, but
•
memory access in trees is more chaotic, leading to worse cache or paging behaviour;
•
memory is allocated and freed dynamically; circumventing that takes considerable programming effort;
•
trees can become unbalanced, and balancing them is tricky to program, and takes
time;
•
the optimal search time can be made approximately equal, but that is again hard
to code.
Closed hash tables have the advantage of simpler storage management, and, until they fill
up, no worse performance than open tables.
Victor Eijkhout
88
CHAPTER 2. PARSING
Projects for this chapter.
Project 2.1. Use the lex and yacc programs you have written for LATEX to write a full
LATEX-to-HTML translator.
Project 2.2. A number of projects involve parsers or code generators for (parts of) TEX or
LATEX.
formulas Reinhold Heckmann and Reinhard Wilhelm. 1997. A functional description of TeX’s formula layout. Journal of Functional Programming 7(5):451485. Available online at http://rw4.cs.uni-sb.de/users/heckmann/
doc.html. For software, see ftp://ftp.cs.uni-sb.de/formulae/.
also formulas Preview-LaTeX (http://preview-latex.sourceforge.
net/) displays formulas right in the emacs edit buffer.
math on web pages see http://www.forkosh.com/mimetex.html.
LATEX to HTML HeVeA, TtH, TeX4ht and LaTeX2HTML.
front end for LATEX http://www.lyx.org/ Ages ago there was ‘griff’. Scientific Word still exists but is commercial.
reimplementation of TEX TEX in Python: http://www.pytex.org/
Investigate these parsers: what is their basic approach and theoretical power, what
are they good at, what are they bad at.
Project 2.3. Develop the full theory of the compound NFA that does lexical analysis.
•
This automaton basically parses the whole file, rather than small chunks; every once in a while it will report that it has recognized an identifier, number,
special symbol &c. This means that the definition of the output alphabet has
to be expanded. Analyze the structure of this output language.
•
As explained, the returning -transition only has to be taken if a maximal
string is recognized. Formalyze this notion.
•
The automaton may need look-ahead of one or more tokens. Formalize this
notion and give equivalence proofs.
Project 2.4. Do an experimental study of hashing. Implement open and closed hash table
algorithms, and do a search on the web for hash functions. Compare to storing
the names in a tree. Find some documents to hash: the source of some programs,
books. Construct some input data sets to trip up the various algorithms. Measure
statistics and do timings.
TEX – LATEX – CS 594
Chapter 3
Breaking things into pieces
The line breaking algorithm of TEX is interesting, in that it produces an aesthetically optimal solution in very little time.
Handouts and further reading for this chapter
If you still have the book ‘Introduction to Algorithms’ by Cormen et al., you can find a
discussion of Dynamic Programming and NP-completeness there. The books by Bellman
are the standard works in this field. Bellman’s ‘Applied Dynamic Programming’ [1] has
been put on reserve, QA264.B353 . The TEX line breaking algorithm is described in an
article by Knuth and Plass [13], reprinted in [9].
The standard work on Complexity Theory, including NP-completeness, is Garey and Johnson ‘Computers and intractibility’ [7]. There is excellent online material about this subject
on Wikipedia, for instance http://en.wikipedia.org/wiki/Complexity_classes_
P_and_NP. Issues in page breaking are discussed in Plass’ thesis [18].
89
90
CHAPTER 3. BREAKING THINGS INTO PIECES
Dynamic Programming.
Dynamic programming is an optimization tech-
nique, that is, a way of solving problems where a yield function is to be maximized, or a
cost function minimized, under certain constraints. Certain optimization problems can be
solved using calculus – unconstrained optimization being a prime example – and others by
such linear algebra techniques as the simplex method. However, such continuous methods
have a hard time dealing with integer constraints. For instance, in computing the yield of a
car factory, the number of cars has to be integer.
The type of problems that dynamic programming is suited for is the type where the problem
can be formulated as a series of decisions. For instance, in work allocation problems, there
is a separate decision for the amount of work to be allocated in each month.
We will start with some examples that show the essential features of the dynamic programming approach.
3.1
Some examples
3.1.1
Decision timing
Our first example concerns the problem of when to make a one-time decision, giving a
number of opportunities to do so. This example illustrates the concept of a series of decisions, and of starting at the final stage and working backward from that.
The surprise menu in a new restaurant works as follows. You will be shown 5 dishes in
sequence, and you can pick one that you like, but if you turn one down, you can not reconsider. Let us say that each dish scores between 0 and 1 on your attractiveness scale. How
do you maximize your choice of dish?
Call the scores you give the dishes xi , and N the number of dishes.
•
If you wait till the last dish, you have no choice.
•
The last dish can be anything, so the best you can say is that it will have an
expected attractiveness of 0.5. Therefore, if xN −1 > 0.5, you take that one, since
it is better than what you can expect in the next step.
•
Now, you will take dish N −1 in half the cases, giving you on average a .75 score,
and the other half of the cases you take dish N , with a score of .5. Therefore, you
are expecting to score .625, and you will take dish N − 2 if it scores more than
that.
•
In .375 of the cases, dish N − 3 will score higher than that.
•
Et cetera.
From this example we see some characteristics:
Stages The optimization problem involves a sequence of stages, each involving a choice.
Principle of optimality Once you arrive at a certain stage, the optimal solution for the rest
of the path is independent of how you got to that stage.
Stepwise solution The solution (here: solution strategy) is arrived at by starting at the final
and working backward. We will also examples that are solved forward; in general
it is a characteristic that dynamic programming problems are solved stage-bystage.
TEX – LATEX – CS 594
3.1. SOME EXAMPLES
91
Often, the fact that the problem is solved starting at the last stage is considered an essential feature. We will not do so: many dynamic programming problems can also be solved
forward.
Exercise 30. For this example, draw up the recurrence relation between the expected scores. Prove that the relation is monotonically increasing (for decreasing index), and bounded above by 1, in fact with
limit 1. Bonus: solve this relation explicitly.
3.1.2
A manufacturing problem
Suppose a factory has N months time to produce a quantity S of their product, which we
will for now assume to be bulk. Because of seasonal variations, in month k the cost of
producing an amount pk is wk p2k . The problem is to produce the requested amount in the
given time, at minimal cost, that is
X
wk p2k .
Pmin
pk =S
We break down the problem by looking at the cost for producing the remaining amount in
the remaining time. Define the minimum cost as
X
wk p2k
v(s|n) = P min
k>N −n
pk =s
and p(s|n) as the work that has to be done n months from the end, given that s work is left,
then, by splitting this into the cost this month plus the cost for the then remaining time, we
get
X
v(s|n) = min wn p2n +
wk p2k
pn ≤s
k>N −n+1
P
pk =s−pn
= min wn p2n + v(s − pn |n − 1)
pn ≤s
That is, we get a recurrence where the remaining work for n months is expressed in that
for n − 1 months.
Starting off is easy: p(s|1) = s, and v(s|1) = w1 s2 . By the above rule then
v(s|2) = min{w2 p22 + v(s − p2 |1)} = min c(s, p2 )
p2
p2
w2 p22 + w1 (s − p2 )2 .
where c(s, p2 ) =
We find the minimum by taking δc(s, p2 )/δp2 = 0,
which gives us p(s|2) = w1 s/(w1 + w2 ) and v(s|2) = w1 w2 s2 /(w1 + w2 ).
Solving one or two more steps like this, we notice the general form:
n
X
1/wn
s,
v(s|n) = s2
1/wi .
p(s|n) = Pn
i=1 1/wi
i=1
This solution can in fact be derived by a variational approach to the constraint minimization
problem
X
X
wk p2k + λ(
pk − S)
k
Victor Eijkhout
k
92
CHAPTER 3. BREAKING THINGS INTO PIECES
for which we set the derivatives to both pn and λ to zero.
This problem shows another characteristic of dynamic programming:
State The cost (yield) function that we define as a function of the stage, also has a state
parameter. It typically describes the amount of some constrained quantity that is
still left. In this case, that is the amount still to be produced.
We see that for this problem there is both an analytical solution, and one found by dynamic programming, using analytical techniques for local optimization. However, these
techniques would become increasingly cumbersome, if we imposed restrictions such as
that the factory does not have unlimited production capacity, or even impossible, if the
product can only be made in discrete units, meaning that the pn have to be integers.
The next problem is a good example of the sort of discrete choices that dynamic programming is well suited for.
3.1.3
The stagecoach problem
A business man in the Old West needs to travel from city 0 to city 8. For this, he has to go
through 3 other cities, but in each stage of the trip there are choices. This being the Wild
West, he decides to get travel insurance. However, not all routes are equally safe, so the
cost of insurance varies. The problem is to find the trip that minimizes the total cost of the
insurance.
3
1
5
1
1
3
4
4
0
6
5
2
2 4
8
2
4
4
2
7
3
2
5
We will look at various ways of solving this problem. First let us define the data.
table = [ [0, 5, 4,
[0, 0, 0,
[0, 0, 0,
[0, 0, 0,
[0, 0, 0,
[0, 0, 0,
[0, 0, 0,
[0, 0, 0,
]
final = len(table);
0,
1,
4,
0,
0,
0,
0,
0,
0,
3,
2,
0,
0,
0,
0,
0,
0,
4,
2,
0,
0,
0,
0,
0,
0,
0,
0,
5,
2,
4,
0,
0,
0,
0,
0,
1,
4,
3,
0,
0,
0],
0],
0],
0],
0],
0],
5],
2]
# first stage: 0
# second: 1 & #2
# third: 3, #4, #5
# fourth: 6 & #7
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3.1. SOME EXAMPLES
3.1.3.1
93
The wrong way to solve this
The solution to the stage coach problem is easy to formulate recursively, given that you are
in some city:
•
If you are in the final city, the cost is zero;
•
Otherwise it is the minimum – over all cities reachable from here – of the cost of
the next leg plus the minimum cost from that city.
# the wrong way
def cost_from(n):
# if you’re at the end, it’s free
if n==final: return 0
# otherwise range over cities you can reach
# and keep minimum value
val = 0
for m in range(n+1,final+1):
# for all later cities
local_cost = table[n][m]
if local_cost==0: continue
# if there is a connection from here,
# compute the minimum cost
local_cost += cost_from(m)
if val==0 or local_cost<val:
val = local_cost
return val
print "recursive minimum cost is",cost_from(0)
If there are N cities, divided into S stages of L cities in each stage, and assuming that in
each stage all cities of the next stage are reachable, the cost of this algorithm is O(LS ).
3.1.3.2
Dynamic programming solution
The implementation strategy of the previous section is wasteful. Consider some city 1, and
cities 2 and 3 in the stage before that. The cost computations from 2 and 3 both compute
the cost from 1, which is clearly redundant. We call this characteristic
Overlapping subproblems: A straightforward (recursive) solution to the problem would
revisit a subproblem, meaning that different solution attempts have a common
subproblem.
Recognizing this leads us to a better solution strategy: once the minimum cost from one
city has been computed, we store it for future reference.
An equivalent interpretation of this strategy is that we compute in each stage a cost function fn (xn ), describing the cost of reaching the end starting at the nth step, giving that we
start there in city xn . This is in fact a dynamic programming solution, with xn the state
variable in the nth stage.
Formally, fk (s) is the minimum cost for traveling the from stage k given that your are in
city s in that stage. Then
fk−1 (s) = min{cst + fk (t)
t
Victor Eijkhout
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CHAPTER 3. BREAKING THINGS INTO PIECES
where cst is the cost of traveling from city s to t.
Initially, the cost from every city till the final one (in particular from the final one itself) is
zero:
# initialization
cost = (final+1)*[0]
Now we loop backwards over the stages, computing in each stage the cost of all city we
can go through. These two loops – the stages, and the cities in each stage – can actually be
collapsed into one loop:
# compute cost backwards
for t in range(final-1,-1,-1):
# computing cost from t
For each city t we consider the ones i that are reachable from it. By induction, for these
later ones we already know the cost of going from them to the final one, so the cost from t
to the final one is the cost from t to i plus the cost from i:
for i in range(final+1):
local_cost = table[t][i]
if local_cost==0: continue
local_cost += cost[i]
If there was no cost yet associated with t, we set it now, otherwise we compare the cost
from t over i with the cost of an earlier evaluated route:
if cost[t]==0 or local_cost<cost[t]:
cost[t] = local_cost
In the end, the minimum cost is given in cost[0].
We see that the main difference between this solution and the recursive one given above, is
that the recursive function call has been replaced by a lookup in a table.
The running time of this algorithm is O(N · L) or O(L2 S), which is a considerable improvement over LS for the straightforward implementation. This solution carries an extra
cost of N memory locations; on the other hand, it does not build up a recursion stack.
3.1.3.3
Forward dynamic programming solution
This problem was solved by starting at the end point. We can also work our way to the
solution forwards, with a code that is essentially the same. Instead of computing the cost
of reaching the final city from an intermediate, we now compute the cost of reaching the
intermediate city from the initial one.
We loop over all cities and all their connections:
cost = (final+1)*[0]
for t in range(final):
for i in range(final+1):
local_cost = table[t][i]
if local_cost == 0: continue
TEX – LATEX – CS 594
3.1. SOME EXAMPLES
95
Now we can compute the cost to the connecting city as the transition cost, plus the known
minumum cost to get where we are:
cost_to_here = cost[t]
newcost = cost_to_here+local_cost
if cost[i]==0 or newcost<cost[i]:
cost[i] = newcost
The result is now in cost[final].
The minimization problem corresponding to this algorithm concerns fk s, the cost to get to
city s in stage k. Then
fk+1 (t) = min{cst + fk (s)
s
which is equivalent to the earlier problem.
Exercise 31. A ‘sparse matrix’ is a matrix where a number of matrix
elements are zero, sufficiently many that you do not want to store them.
To compute the matrix
P vector product y = Ax you then do not compute
the full sum yi = j aij xj , but only those terms for which aij 6= 0.
This sort of operation is easy enough to code, but is pretty inefficient in
execution.
Suppose that for small k the product with k consecutive matrix elements
(that is aij xj + aij+ xj+1 + · · · + aij+k−1 xj+k−1 can be executed more
efficiently than doing it as k separate operations. For instance, suppose
that with k = 3 the time per ai· x· reduced to .4 of the normal multiply
time, that is, doing three consecutive multiplications as a block takes
time 1.2, while doing them separately takes time 3.
Now, if a11 6= 0, a12 = 0, a13 6= 0, the multiplication a11 x1 + a13 x3
takes time 2, but if we store the intermediate zero at a12 , the size 3 block
multiplication takes time 1.2. This means that doing some superfluous
operations (multiplying by zero) we can actually speed up the matrixvector product.
Let a pattern of nonzeros and reduction factors be given. The pattern
stands for the locations of the nonzeros in a matrix row, for instance
row = [1,0,0,1,0,1,1,0,0,1,0,0,0,1,1,0,1,1]
redux = [1, .75, .4, .28]
Formulate a principle of optimality for this problem, and write a dynamic programming solution for finding the covering of the sparse row
that will lead to the shortest multiplication time. Tinker with the redux
times (but make sure the n-th is more than 1/n in base-1 indexing) and
see how the solution changes.
3.1.4
Traveling salesman
The above problems all had dynamic programming solutions with a cost slightly more than
linear in the input problem size. Dynamic programming does not always give that low a
complexity.
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CHAPTER 3. BREAKING THINGS INTO PIECES
The traveling salesman problem looks a bit like the stagecoach problem above. However,
now the traveler does not only need to go from a starting to a final city, he also has to visit
every city on his travel.
This problem can be solved by dynamic programming, but the concept of stage is now
more complicated. We can no longer map the cities into a linear ordered set of stages since
they can be visited in any sequence.
The solution is to let stage n correspond to picking the nth city, and to define the current
state as the last visited city, plus the set of the remaining ones. Initially we loop over all
possible last cities, so the cost is the sum of the single leg trip to the end point, plus the
minimum cost through remaining cities. unvisited cities.
To be precise: a state is a pair (S, f ) of a set of cities left to be visited, and the current
city f ∈ S.
We can now construct a cost function that depends on the stage and the current state.
C({1}, 1) = 0
C({f }, f ) = a1f for f = 2, 3, . . .
C(S, f ) =
min [C(S − f, m)] + amf ]
m∈S−f
This is easily enough implemented:
def shortest_path(start,through,lev):
if len(through)==0:
return table[start][0]
l = 0
for dest in through:
left = through[:]; left.remove(dest)
ll = table[start][dest]+shortest_path(dest,left,lev+1)
if l==0 or ll<l:
l = ll
return l
to_visit = range(1,ntowns);
s = shortest_path(0,to_visit,0)
This solution has factorial complexity.
3.2
Discussion
In the example above we saw various properties that a problem can have that make it
amenable to dynamic programming.
Stages The optimization problem involves a sequence of stages, each involving a choice,
or a discrete or continuous parameter to be determined.
Stepwise solution The solution is arrived at by solving the subproblems in the stages one
by one. Often this is done starting at the final stage and working backward.
TEX – LATEX – CS 594
3.2. DISCUSSION
97
State The cost (yield) function that we define as a function of the stage, also has a state
parameter. It typically describes the amount of some constrained quantity that is
still left to be consumed or produced.
Overlapping subproblems This is the property that a straightforward (recursive) solution to the problem would revisit a subproblem, meaning that different solution
attempts have a common subproblem.
Principle of optimality This is the property that the restriction of a global solution to a
subset of the stages is also an optimal solution for that subproblem.
The principle of optimality is often phrased thus:
An optimal policy has the property that whatever the initial state and
initial decision are, the remaining decisions must be an optimal policy
with regard to the state resulting from the first decision.
It is easy to show that the principle of optimality holds for P
certain functions and conN
straints.PFor example, consider the problem of maximizing i gi (xi ) under the constraint i xi = X where xi ≥ 0. Call this maximal sum fN (X), then
fN (X)
=
max
PN
i xi =X
N
X
gi (xi )
i
(
=
=
max
xN <X
gN (xN ) + P
max
N −1
i
N
−1
X
xi =X−xN
)
gi (xi )
i
max {gN (xN ) + fN −1 (X − xN )}
xN <X
We see here that the form of the gi functions does not enter the argument, but the fact that
the total utility is a sum of gi s does. Utility functions that are not symmetric in the component gi functions clearly can not straightforwardly be solved with dynamic programming.
Victor Eijkhout
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CHAPTER 3. BREAKING THINGS INTO PIECES
TEX paragraph breaking.
Breaking a paragraph into lines is the problem
of, given a string of words and other material with intervening spaces, breaking that string
into chunks (lines) of approximately equal length, and doing so in a visually attractive way.
Simple strategies (see the ‘first fit’ algorithm below) give a result that is easy to compute,
but that can be visually very unappealing. While the problem of finding globally optimal
line breaks sounds very hard – with n words there are 2n ways of breaking the paragraph;
also, this problem resembles the bin-packing problem which is NP-complete – it can actually be solved fairly efficiently.
TEX’s basic strategy is to calculate the badness of breaking the lines at certain points, and
to minimize the badness over the whole paragraph.
3.3
The elements of a paragraph
TEX’s paragraph breaking algorithm is based around the concepts of
•
Boxes: this comprises letters, formulas, TEX boxes, and other material of a fixed
with.
•
Glue: this is white space; a glue item has a natural width, stretchability, and
shrinkability.
•
Penalties: these are items that express the desirability or undesirability of breaking
a line at a particular point.
The same elements are also present in a vertical list; in both cases there are some other,
more rare items, that we will ignore here.
3.3.1
Some details
3.3.1.1
Boxes
The boxes in a paragraph are such things as words, rules, math formulas, and actual TEX
\boxes. A box can not be broken: it is completely described by its height, depth, width.
Actually, looking at words as boxes is too simplistic, since words can often be hyphenated.
This means that a word is a sequence of boxes alternating with penalties.
3.3.1.2
Penalties
A penalty item describes the desirability or undesirability of breaking a list at some point.
Penalties can be inserted by hand (most often as the \nobreak macro, which is equivalent
to \penalty10000), or in a macro, or are inserted by TEX itself. An example of the latter
case is the \hyphenpenalty which discourages breaking at a hyphen in a word.
Hyphenating a word can be simulated by having a penalty of zero at the hyphenation location. Since this usually introduces a hyphen character, TEX internally pretends that penalties
can have a width if the list is broken at that place.
The most common types of penalty are the infinite penalty that starts a non-breaking space,
and the penalty associated with breaking by hyphenating a word. The latter kind is called
TEX – LATEX – CS 594
3.3. THE ELEMENTS OF A PARAGRAPH
99
a ‘flagged penalty’, and TEX has an extra amount of demerits associated with them, for
instance to prevent two consecutive lines ending in a hyphen.
Penalties can have positive and negative values, to discourage or encourage breaking at a
specific place respectively. The values +∞ and −∞ are also allowed, corresponding to a
forbidden and forced break respectively.
3.3.1.3
Glue
A ‘glue’ is a dimension with possible stretch and/or shrink. There are glue denotations,
such as 2cm plus .5cm minus .1cm, or glue parameters, such as \leftskip or
\abovedisplayskip. The parameters are inserted automatically by the various TEX
mechanisms.
Glue can be measured in points pt, centimeters cm, millimeters mm, inches in. There is
also infinite glue: fil, fill, and filll. Presence of TEX’s infite glue (fill) causes
all other glue to be set at natural width, that is, with zero stretch and shrink.
If there is more than one glue item in a list, the natural widths and the stretch and shrink
amounts are added together. This means that a list with both 2cm plus 1cm and 2cm
plus -1cm has no stretch since the stretch amounts add up to zero. On the other hand,
with 2cm plus 1cm and 2cm minus 1cm it has both stretch and shrink.
The stretch and shrink components of glue are not treated symmetrically. While in a pinch
we can allow a line to be stretched further than the indicated maximum, we can not allow
spaces to be shrunk to zero, or even close to that.
3.3.1.4
Stretch and shrink
Each space can have stretch and shrink. When we consider a line, we add up all the stretch
and shrink and compute an ‘adjustment ratio’ as the ratio of the shortfall or excess space to
the available stretch or shrink repectively. This ratio r is negative for lines that need to be
shrunk.
A simple algorithm would be to impose a limit of |r| ≤ 1 (and then to minimize the number
of hyphenations under that restriction), but that might be too restrictive. Instead, TEX uses
a concept of ‘badness’. The badness of a line break is infinite if r < −1; otherwise it is
cubic in the absolute size of r.
3.3.1.5
Line break locations
Here are the main (but not the only) locations where TEX can decide to break a line.
•
•
•
•
At a penalty
At a glue, if it is preceeded by a non-discardable item, meaning, not a penalty or
other glue
At a hyphen in a word
At a place where TEX knows how to hyphenate the word. (There is actually a
mechanism, called ‘discretionaries’ that handles these last two cases.)
Victor Eijkhout
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3.3.2
CHAPTER 3. BREAKING THINGS INTO PIECES
Examples
Here are a few examples of the things that the boxes/penalties/glue mechanism is capable
of.
3.3.3
Centered text
By using \leftskip and \rightskip we can get centered text.
\begin{minipage}{4cm}
\leftskip=0pt plus 1fil \rightskip=0pt plus 1fil
\parfillskip=0pt
This paragraph puts infinitely stretchable glue at
the left and right of each line.
The effect is that the lines will be centered.
\end{minipage}
Output:
This paragraph puts
infinitely stretchable glue at
the left and right of each
line. The effect is that the
lines will be centered.
The following centers only the last line. This is done by letting the \leftskip and
\rightskip cancel each other out, except on the last line.
\begin{minipage}{5cm}
\leftskip=0pt plus 1fil \rightskip=0pt plus -1fil
\parfillskip=0pt plus 2fil
This paragraph puts infinitely stretchable glue at
the left and right of each line, but the amounts cancel out.
The parfillskip on the last line changes that.
\end{minipage}
Output:
This paragraph puts infinitely
stretchable glue at the left and right
of each line, but the amounts cancel
out. The parfillskip on the last line
changes that.
3.3.3.1
Hanging punctuation
Hanging punctuation is a style of typesetting where punctuation that would wind up against
the right margin is actually set in the right margin. Optically, this makes the margin look
straighter.
\newbox\pbox \newbox\cbox
\setbox\pbox\hbox{.} \wd\pbox=0pt
\setbox\cbox\hbox{,} \wd\cbox=0pt
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3.3. THE ELEMENTS OF A PARAGRAPH
101
\newdimen\csize \csize=\wd\cbox
\newdimen\psize \psize=\wd\pbox
\catcode‘,=13 \catcode‘.=13
\def,{\copy\cbox \penalty0 \hskip\csize\relax}
\def.{\copy\pbox \penalty0 \hskip\psize\relax}
3.3.3.2
Mathematical Reviews
In ‘Mathematical Reviews’ the name of the reviewer should be separated sufficiently from
the review, but fit on the same line if space allows.
We do this by having two separate infinite glues with a break in between, and with a total
natural width equal to the minimum separation. The trick is to make sure that the second
glue is not discarded after a break, which we do by putting an invisible box at the beginning.
\def\signed#1{\unskip
\penalty10000 \hskip 40pt plus 1fill
\penalty0
\hbox{}\penalty10000
\hskip 0pt plus 1fill
\hbox{#1}%
\par
}
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CHAPTER 3. BREAKING THINGS INTO PIECES
3.4
TEX’s line breaking algorithm
3.4.1
Elements
3.4.1.1
Glue setting and badness
In order to make a list fit a prescribed dimension, there is a process called ‘glue setting’.
The natural size of the list and the desired size are compared. Let ρ be the ratio of the
amount stretched (shrunk) to the possible amount of stretch (shrink). The exact definition
is such that the ratio is positive for stretch and negative for shrink: let ` be the desired length
of the line, L the natural width, X the available stretch and Y the available shrink, then
0
`=L
(` − L)/X (stretch:) ` > L and X > 0
ρ=
(` − L)/Y (shrink:) ` < L and Y > 0
undefined otherwise
Then the ‘badness’ of the needed glue setting is
10 000
ρ < 1 or undefined
b=
min 10 000, 100|ρ|3
otherwise
Since 10 000 is considered infinite in glue arithmetic, this algorithm allows glue to be
stretched further than the indicated amount, but not shrunk beyond what is available.
A list that is stretched or shrunk is put in one of the following four categories:
tight (3) if it has shrunk with b ≥ 13
decent (2) if b ≤ 12
loose (1) if it has stretched with 100 > b ≥ 13
very loose (0) if it has stretched with b ≥ 100
Note that 100×(1/2)3 = 12.5, so the crossover values denote that half the stretch or shrink
is used.
Lines that differ by more than one in their classifications are called ‘visually incompatible’.
3.4.1.2
Demerits
Breaking a line at a certain points gives the penalty p associated with that point, and the
badness b of the resulting stretch/shrink. These are combined into a ‘demerits’ figure:
b2 + p2 0 ≤ p < 10 000
d=
b2 − p2 −10 000 < p < 0
The demerits for breaking a paragraph along a certain sequence of break points is then the
sum of the demerits of the lines, plus \adjdemerits for every two lines that are not
visually compatible (section 3.4.1.1), \doublehyphendemerits for pairs of lines that
end in a hyphen, and \finalhyphendemerits if the last full line ends in a hyphen.
TEX acts as if before the first line there is a line that is ‘decent’; the last line will typically
contain infinite glue, so all spaces are set at natural width.
For full generality, the last line of a paragraph is handled like any other. Filling out the
line to the margin is realized by added infinite glue plus a trailing penalty that forces a line
break at the end of the paragraph.
TEX – LATEX – CS 594
3.4. TEX’S LINE BREAKING ALGORITHM
3.4.2
103
Breaking strategies
We will now look at a couple of line breaking strategies. The first two will be strictly local;
the third – TEX’s algorithm – is able to optimize in a more global sense.
The problem with local algorithms is that they can not take a slightly worse solution in
one line to prevent much worse from happening later. This will for instance allow tight and
very loose lines to occur next to each other.
3.4.2.1
First fit
The traditional algorithm for line breaking only looks at the current line. Once a word is
starting to exceed the right margin, the following cases are investigated.
1.
2.
3.
4.
If the spaces in the line can be compressed without exceeding some maximum
shrinkage, break after this word.
Otherwise, if the spaces can be stretched short of some maximum, break before
this word.
Otherwise, try hyphenating this word.
If no hyphenation point can be found, accept a line with spaces stretched to whatever degree is needed to break before this word.
If you have set text with TEX, you will have noticed that TEX’s solution to the last point
is slightly different. It will let a word protrude into the right margin as a visual indicator
that no good breakpoint could be found. (TEX’s tolerance to bad breaks can be increased
by setting the \emergencystretch parameter.)
This method can be called ‘first fit’, because it will the first option (compress), without
comparing if later options (stretching) look better. This is remedied in TEX by, instead of
having an all-or-nothing it fits / it doesn’t fit distinction, there is a continuous scale of
evaluation.
3.4.2.2
Best fit
A slight improvement on the first fit algorithm results from deciding between the possibilities 1–3 based on badness calculations. This can be termed ‘best fit’, and while it may work
slightly better than fit, it is still a local decision process.
3.4.2.3
Total fit
TEX’s actual algorithm calculates the ‘demerits’ of a line break as a compound of badness,
the breakpoint penalty, plus any flagged penalties. It then adds together the demerits of the
whole paragraph, and minimizes this number. This makes it possible to use a slightly worse
line break early in the paragraph, to prevent a much worse one later.
Exercise 32. In dynamic programming, many solutions start from a
final stage and work backwards. Why is this approach inappropriate for
TEX’s line breaking algorithm? Why would it be even less appropriate
for a page breaking algorithm?
Victor Eijkhout
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3.4.3
CHAPTER 3. BREAKING THINGS INTO PIECES
Model implementation
We will here only discuss implementations of solutions based on dynamic programming.
The line breaking algorithm goes linearly through the items in the horizontal list, and for
each considers whether there is a valid breakpoint after it, and with what cost. For the latter
point, it needs to know what the beginning of the line is, so there is an inner loop over
all preceeding words. This would make the running time of the algorithm quadratic in the
number of words, which is much better than the initial estimate of 2n .
However, the situation is better than that. The number of words that can fit on a line is
limited by what can fit when all spaces are sqeezed to zero. The easiest way to implement
this is by maintaining an ‘active list’ of possible words to begin the line with. A word can
be removed from the active list if the material from it to the current word does not fit: if it
does not fit now, it will certainly not fit when the next word is considered.
This is then the main program; we will mainly vary the function that computes the breakpoint cost.
active = [0]
nwords = len(paragraph)
for w in range(1,nwords):
# compute the cost of breaking after word w
for a in active:
line = paragraph[a:w+1]
ratio = compute_ratio(line)
if w==nwords-1 and ratio>0:
ratio = 0 # last line will be set perfect
print "..line=",line,"; ratio=",ratio
if ratio<-1:
active.remove(a)
print "active point",a,"removed"
else:
update_cost(a,w,ratio)
report_cost(w)
active.append(w)
The only thing different between various strategies is how the cost of a line break is computed by update_cost(a,w,ratio).
Exercise 33. Not every word should be added to the active list. For
instance, for any realistic line length, the second word in the paragraph
will not have a valid breakpoint after it, so we need not consider it.
Take the model implementation and add this modification. Measure the
improvement in running time, for instance by counting the number of
calls to some inner routine. Give a theoretical argument for how this
reduces the complexity of the algorithm.
TEX – LATEX – CS 594
3.4. TEX’S LINE BREAKING ALGORITHM
3.4.3.1
105
First fit implementation
Since at first we are looking only locally, for each breakpoint we only keep track of the
cost and the previous breakpoint that the cost was computed from. Here we set up the
data structure cost. Element cost[w] describes the cost of breaking after word w; the
’from’ component is the number of the first word of the line.
def init_costs():
global cost
cost = len(paragraph)*[0]
for i in range(len(paragraph)):
cost[i] = {’cost’:0, ’from’:0}
cost[0] = {’cost’:10000, ’from’:-1}
The essential function is the cost computation. In first fit we accept any stretch or shrink
that is |ρ| < 1.
def update_cost(a,w,ratio):
global cost
if a>0 and cost[a-1][’cost’]<10000:
if ratio<=1 and ratio>=-1:
to_here = abs(ratio)
else: to_here = 10000
if cost[w][’cost’]==0 or to_here<cost[w][’cost’]:
cost[w][’cost’] = to_here; cost[w][’from’] = a-1
(The first test serves to make sure that the previous point being considered is in fact a valid
breakpoint.)
Here is the remaining function that constructs the chain of breakpoints:
def final_report():
global cost,nwords,paragraph
print "Can break this paragraph at cost",\
cost[nwords-1][’cost’]
cur = len(paragraph)-1; broken = []
while cur!=-1:
prev = cost[cur][’from’]
line = paragraph[prev+1:cur+1]
broken.insert(0,line)
cur = prev;
set_paragraph(broken)
A small example text, faked in monospace:
You may never have thought of it, but fonts (better:
typefaces) usually have a mathematical definition somehow.
If
a
font
is
given
as
bitmap,
this
is often
a result originating from a more compact description.
Imagine the situation that you have bitmaps at 300dpi, and
you
buy
a 600dpi printer. It wouldn’t look pretty.
There
is
then
a
need
for a mathematical way of
describing arbitrary shapes. These shapes can also be
three-dimensional; in fact, a˜lot of the mathematics in
Victor Eijkhout
-0.111111111111
-0.666666666667
0.888888888889
0.0
-0.777777777778
0.25
0.555555555556
0.0
-0.285714285714
106
CHAPTER 3. BREAKING THINGS INTO PIECES
this chapter was developed by a car manufacturer for
modeling
car
body shapes. But let us for now only
look
in two dimensions, which means that the curves
are lines, rather than planes.
0.0
0.222222222222
0.125
We see ugly stretched break in line 3, especially after the compressed line 2. However, both
of them fit the test.
It is in fact simple to turn this into a dynamic programming solution that considers a global
minimum:
def update_cost(a,w,ratio):
global cost
if ratio<=1 and ratio>=-1:
to_here = abs(ratio)
else: to_here = 10000
if a>0:
from_there = cost[a-1][’cost’]
to_here = to_here+from_there
else: from_there = 0
if cost[w][’cost’]==0 or to_here<cost[w][’cost’]:
cost[w][’cost’] = to_here; cost[w][’from’] = a-1
3.4.3.2
Best fit
In the best fit strategy, we compute badness from the stretch/shrink ratio. This involves only
a slight change in the cost computation function:
def update_cost(a,w,ratio):
global cost
to_here = 100*abs(ratio)**2
if a>0:
from_there = cost[a-1][’cost’]
to_here = to_here+from_there
else: from_there = 0
if cost[w][’cost’]==0 or to_here<cost[w][’cost’]:
cost[w][’cost’] = to_here; cost[w][’from’] = a-1
The same text:
You may never have thought of it, but fonts (better:
typefaces) usually have a mathematical definition somehow.
If
a
font
is
given
as bitmap, this is often a
result
originating
from
a more compact description.
Imagine the situation that you have bitmaps at 300dpi, and
you
buy
a 600dpi printer. It wouldn’t look pretty.
There
is
then
a
need
for a mathematical way of
describing arbitrary shapes. These shapes can also be
three-dimensional; in fact, a˜lot of the mathematics in
this chapter was developed by a car manufacturer for
modeling
car
body shapes. But let us for now only
look
in two dimensions, which means that the curves
are lines, rather than planes.
-0.111111111111
-0.666666666667
0.5
0.5
-0.777777777778
0.25
0.555555555556
0.0
-0.285714285714
0.0
0.222222222222
0.125
TEX – LATEX – CS 594
3.4. TEX’S LINE BREAKING ALGORITHM
107
While there are no lines stretched with ρ >, the quadratic function has improved the break
in line 3.
3.4.3.3
Total fit
For the algorithm that TEX uses, we have to distinguish between lines that are tight, decent,
loose. This makes our datastructure more complicated:
def init_costs():
global cost
nul = [0,0,0]
cost = len(paragraph)*[ 0 ]
for i in range(len(paragraph)):
cost[i] = nul[:]
for j in range(3):
cost[i][j] = {’cost’:10000, ’from’:-2}
for j in range(3):
cost[0][j] = {’cost’:10000, ’from’:-1}
An element cost[i] is now an array of three possible breakpoints, one in each of the
classifications. An actual breakpoint is now in cost[word][type][’from’] and
cost[word][type][’cost’].
The cost computation becomes more complicated:
def minimum_cost_and_type(w):
global cost
c = 10000; t = 0
for type in range(3):
nc = cost[w][type][’cost’]
if nc<c:
c = nc; t = type
return [c,t]
def update_cost(a,w,ratio):
global cost
type = stretch_type(ratio)
to_here = 100*abs(ratio)**2
if a>0:
[from_there,from_type] = minimum_cost_and_type(a-1)
to_here += from_there
else: from_there = 0
if cost[w][type][’cost’]==0 or\
to_here<cost[w][type][’cost’]:
cost[w][type][’cost’] = to_here;
cost[w][type][’from’] = a-1
Exercise 34. The total fit code does not yet contain the equivalent of
TEX’s \adjdemerits. Add that.
Let us look at the same test text again:
Victor Eijkhout
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CHAPTER 3. BREAKING THINGS INTO PIECES
You may never have thought of it, but fonts (better:
typefaces)
usually
have
a
mathematical
definition
somehow. If a font is given as bitmap, this is often a
result
originating
from
a more compact description.
Imagine
the
situation
that
you
have
bitmaps
at
300dpi, and you buy a 600dpi printer. It wouldn’t look
pretty. There is then a need for a mathematical way of
describing arbitrary shapes. These shapes can also be
three-dimensional; in fact, a˜lot of the mathematics in
this chapter was developed by a car manufacturer for
modeling
car
body shapes. But let us for now only
look
in two dimensions, which means that the curves
are lines, rather than planes.
-0.111111111111
1.2
-0.454545454545
0.5
1.0
-0.333333333333
-0.4
0.0
-0.285714285714
0.0
0.222222222222
0.125
In this output, line 2 is stretched further than before, to prevent lower badnesses later.
Exercise 35. Add the functionality for hanging indentation to this code.
Exercise 36. (bonus point exercise) TEX has the possibility of forcing
a paragraph to be a line longer or shorter than optimal. Implement that.
3.4.3.4
Utility parts
File header: we read a text and store it.
#! /usr/bin/env python
import sys
max_line_length = 60
paragraph = []
while 1:
try:
a = raw_input()
paragraph.extend(a.split())
except (EOFError):
break
In order to fake stretch and shrink with a monospace font, we let a ‘space’ be two spaces
by default.
def line_length(words):
l = 2*(len(words)-1)
for w in words:
l += len(w)
return l
#
# ratio = -1 : shrink each double space to one
# ratio = 1 : stretch each double space to three
#
def compute_ratio(line):
spaces = len(line)-1
TEX – LATEX – CS 594
3.4. TEX’S LINE BREAKING ALGORITHM
109
need = 1.*(max_line_length-line_length(line))
#print "ratio:",need,spaces
if spaces==0: return 10000
else: return need/spaces
Output formatting with this idea:
def set_paragraph(para):
for l in range(len(para)-1):
line = para[l]
set_line(line)
set_last_line(para[len(para)-1])
def set_line(line):
shortfall = max_line_length-line_length(line)
for w in range(len(line)-1):
sys.stdout.write(line[w])
if shortfall>0:
sys.stdout.write(’
’); shortfall = shortfall-1
elif shortfall<0:
sys.stdout.write(’ ’); shortfall = shortfall+1
else:
sys.stdout.write(’ ’)
sys.stdout.write(line[len(line)-1])
print " ",compute_ratio(line)
def set_last_line(line):
for w in range(len(line)-1):
sys.stdout.write(line[w])
sys.stdout.write(’ ’)
sys.stdout.write(line[len(line)-1])
Victor Eijkhout
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CHAPTER 3. BREAKING THINGS INTO PIECES
NP completeness.
3.5
Introduction
The ‘NP’ stands for ‘nondeterministic polynomial time’, which stands for the fact that
a solution can be checked (not: found) in polynomial time. This class of algorithms is
informally characterized by the fact there is polymomial time for checking their solution.
However, it is also true that there is no polynomial algorithm known for solving them.
The fact that there is no efficient algorithms known would not be bad if it could be proved
that no efficient algorithm exists.
TEX – LATEX – CS 594
3.6. BASICS
111
However, also there exists no non-polynomial lower bound on the solution time. Thus, the
question whether they can be solved in polynomial time is still open. Since methods in this
class can all be translated into each other, having a solution method for one implies that
methods exist for all of them. This also means that none of the problems in this class have
been solved so far.
3.6
Basics
3.6.1
Optimization versus decision problems
Many problems that are said to be NP-complete are optimization problems. For instance, in
the traveling salesman problem the shortest route through a set of cities is asked. However,
it is more convenient to look at decision problems, that is, problems that have a yes or no
answer for each input.
It is not hard to transform an optimization problem into a decision problem. Rather than
asking for an optimal solution, we determine a bound B, and ask whether there is a solution
that is within that bound.
Exercise 37. Finish this argument. Show that, if we can solve the optimization problem, we can solve the decision problem. Now, supposing
we can solve the decision problem, how does that give a solution of
the optimization problem? Assume that the outcome of the optimization
problem is an integer quantity. Do you have to make other assumptions;
discuss? What is the complexity of the one solution method given a certain complexity for the other?
Victor Eijkhout
112
3.6.2
CHAPTER 3. BREAKING THINGS INTO PIECES
Language of a problem
For each optimization or decision problem we can defined ‘instances’, which are ways of
setting all the free variables of the problem. Since these variables are in sets of types that
depend on the problem, we can not be more precise here. A problem can then be phrased as
a question over this set of instances: which instances optimize the cost function, or which
give a yes answer. That last set we will denote YΠ .
Again depending on the problem, we can encode instances of a problem. For instance, in
the traveling salesman problem, an instance would be encoded as the ordered list of cities
to pass through.
With this, we can define the language of a problem:
L[Π, e] = {the instances in YΠ encoded under e}
3.6.3
Turing machines
A Turing machine, given some input, can halt with the yes state qY , the no state qN , or can
never halt. We say that a string is accepted if it leads the Turing machine to halt with qY .
The language LM of a Turing machine M is the set of strings that are accepted.
A deterministic Turing machine (DTM) M is said to solve a problem Π (under some encoding e), or equivalently to recognize L[Π, e], if
•
•
it halts for all strings over its input alphabet, and
its language LM is L[Π, e].
Note that ‘solving a problem’ here actually means ‘recognizing a solution of the problem’.
This DTM is a solution checker, not a solution generator.
As an example, consider the recast the traveling salesman problem ‘does a route, shorter
than B, exist?’. The set of purported solutions are then lists of cities, and the DTM gives
for each list a verdict ‘yes, this route is shorter than B’ or ‘no, this route is not shorter
than B’.
3.7
Complexity classes
3.7.1
Class P
This allows us to define class P :
P = {L : there is DTM that recognizes L in polynomial time}
and with this
Π∈P
≡ L[Π, e] ∈ P for some encoding e
≡ there is a polynomial time DTM that recognizes L[Π, e]
What this means is that for problems in P there is a polynomial time Turing machine that
recognizes strings in YΠ as valid, and that on strings not in YΠ it both halts, and gives a
negative verdict.
TEX – LATEX – CS 594
3.7. COMPLEXITY CLASSES
3.7.2
113
Class NP
Solving the traveling salesman problem may be hard, but if we have a network and some
bound, and someone gives us an itinerary with the claim that it satisfies that bound, we can
check this with a DTM in polynomial time. We can now build a non-deterministic Turing
machine (NDTM) which ‘solves’ such problems: given a decision problem it ‘guesses’
some purported solution, and then checks (with a DTM) whether that solves the decision
problem. The guessed solution is called a ‘certificate’.
Clearly, only if the decision problem has an answer of ‘true’ can the NDTM guess a solution, so the YΠ of this problem is precisely the set of decision problem instances with a yes
answer.
For instance, for the traveling salesman problem, the instances in YΠ are a combination
of a cities network plus a feasible bound on the travel time. The non-deterministic Turing
machine would then guess an itinerary, and it can be checked in polynomial time that that
is indeed a valid route, and that is satisfies the bound.
We can write this whole story compactly: a problem Π is in NP if there is a polynomial
time function A(·, ·) such that
w ∈ YΠ ⇔ ∃C : A(w, C) = true
and C itself can be polynomially generated.
The final condition on the generation of the certificate is needed for a total polynomial
runtime, but in practice it is not much of a limitation. For instance, in the traveling salesman
problem, a list of cities can be guessed in linear time.
Exercise 38. Prove that NP is closed under union and intersection.
What difficulty is there in showing that it is closed under complement
taking?
3.7.3
Examples
As was indicated above, while finding a solution to a problem is often hard, checking that
something is a solution is often fairly simply and doable in polynomial time. A nontrivial example of a polynomial time problem is checking whether a number is prime. This
question was only settled in 2002. Before that there were polynomial time probabilistic
testers, which would test a number and return a verdict with high reliability, or with a high
probability of polynomial running time.
Exercise 39. Why is the following algorithm not a linear time solution
to the P RIME√problem?
for i = 0 . . . n:
if mod(n, i) ≡ 0
return true
Other algorithms have provably an exponential running time. Examples here are finding
the best move in chess, or checking statements in Pressburger arithmetic.
It is possible to find levels in between polynomial and exponential. The problem of factoring an integer (note that this is more powerful than primality testing) has a runtime of
Victor Eijkhout
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CHAPTER 3. BREAKING THINGS INTO PIECES
O(exp((n · 64/9)1/3 )(log n)2/3 ). Interestingly, on a quantum computer, a polymial algorithm is known; see http://en.wikipedia.org/wiki/Shors_algorithm.
In the next section we will go further into the middle ground, of algorithms for which no
polymomial time algorithm is known, but for which no exponential bound is known either.
3.8
NP-completeness
3.8.1
Transformations
Often it is possible to transform one problem into another. This has the advantage that, if
you prove that one problem is in a certain class, the other one is also in that class. Of course
this depends on the nature of the transformation.
We will here consider ‘polynomial
transformations’. Let L1 and L2 be the languages of two
P∗
P
∗
problems over alphabets 1 and 2 respectively, then f is a polymomial transformation
of problem 1 into problem 2 if
•
•
There is a DTM that computes f (x) in time Tf (x) ≤ pf (|x|) for some polynomial pf , andP
∗
For all x ∈ 1 , x ∈ L1 iff f (x1 ) ∈ L2 .
The transformation does not have to be a one-to-one mapping, so it is sometimes explicitly
terms a ‘many-to-one polynomial transformation’.
Lemma 1 Suppose f is a polynomial transformation from L1 to L2 , then
L2 ∈ P ⇒ L1 ∈ P
Proof: assume that M2 : L2 → {0, 1} is a DTM that recognizes L2 , then M2 ◦ f is a
DTM that recognizes L1 , and this composition still runs in polynomial time TM2 ◦f (x) ≤
pT2 (|pf (|x|)|).
If L1 transforms to L2 (in polynomial time), we notate that as L1 ≤ L2 . This notation also
suggests the idea that L1 is easier than L2 .
It is easy to see that
L1 ≤ L2 ∧ L2 ≤ L3 ⇒ L1 ≤ L3 ,
that is, the ‘transforms into’ relation is transitive.
3.8.2
NP-complete
A language L is said to be NP-complete if
•
•
L ∈ N P , and
for all L0 ∈ N P : Ł0 ≤ L
(Languages that satisfy the second clause but not the first are called ‘NP-hard’. One example is the halting problem, which is known not to be decidable. In other words, the DTM
that should recogize the language does not always halt with yes or no.)
TEX – LATEX – CS 594
3.8. NP-COMPLETENESS
115
Informally, the class NP-complete is that of the problems where a solution can be verified
quickly (meaning, in polynomial time). On the other hand, P is the class of problems where
the solution can be computed quickly. The question whether these classes are disjoint is
open. In fact, you can win a million dollars by settling it one way or another.
Lemma 2 If L1 , L2 ∈ N P , L1 is NP-complete, and L1 ≤ L2 , then L2 is NP-complete.
Proof: the only thing we need to check is that every L0 ≤ L2 for all L2 ∈ N P . Since L1 is
NP-complete, L0 ≤ L1 . Now use the transitivity of the transform relation.
3.8.3
Proving NP-completeness
Above we saw that, given one NP-complete problem, others can easily be proved NPcomplete by constructing a mapping between the one problem and the other. This raises a
bootstrapping question.
Stephen Cook was the first to prove the NP-completeness of any problem (1971), in his
case the satisfiability problem. This is the problem of, given boolean variables x1 . . . xn
and a logical formula F (x1 , . . . , xn ), deciding whether there is a way of specifying the
variables such that the result is true.
Examples: the formula x1 ∨ =
6 x1 is always true; x1 ∧ =
6 x1 is always false, and x1 ∧ =
6 x2
is only true for the pair (x1 = T, x2 = F ). For the first and third examples, there are values
of x1 , x2 for which the formula is true, so these are satisfiable. The second example is not
satisfiable.
The Boolean satisfiability problem is in NP because a non-deterministic Turing machine
can guess an assignment of truth values to the variables, determine the value of the expression under that assignment, and accept if the assignment makes the entire expression
true.
Now, to prove that the satisfiability problem is NP-complete, it remains to show that any
language L ∈ N P can polynomially be transformed into it. This is done by assuming a
NDPT Turing machine for L, and transforming it into a logical formula, so that there is
a correspondence between successful computation in the NDTM, and a satisfied logical
formula.
Let the Turing machine be
M = hQ, s, Σ, F, δi
where
Q is the set of states, and s ∈ Q the initial state,
Σ the alphabet of tape symbols,
F ⊂ Q the set of accepting states, and
δ ⊂ Q × Σ × Q × Σ × {−1, +1} the set of transitions,
and that M accepts or rejects an instance of the problem in time p(n) where n is the size
of the instance and p(·) is a polynomial function.
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CHAPTER 3. BREAKING THINGS INTO PIECES
We describe for each instance I a Boolean expression which is satisfiable if and only if the
machine M accepts I.
The Boolean expression uses the variables set out in the following table, where q ∈ Q,
−p(n) ≤ i ≤ p(n), j ∈ Σ, and 0 ≤ k ≤ p(n):
Variables
Tijk
Hik
Qqk
Intended interpretation
True iff tape cell i contains symbol j
at step k of the computation
True iff the M ’s read/write head is
at tape cell i at step k of the computation.
True iff M is in state q at step k of
the computation.
How many
O(p(n)2 )
O(p(n)2 )
O(p(n))
Define the Boolean expression B to be the conjunction of the clauses in table ??, for all
−p(n) ≤ i ≤ p(n), j ∈ Σ, and 0 ≤ k ≤ p(n).
For all:
initial conditions
Tape cell i of the
input I contains
symbol j.
Add the clauses
Interpretation
How many
clauses?
Tij0
Initial contents of the tape.
O(p(n))
Qs0
H00
Initial state of M
Initial
position
read/write head.
O(1)
O(1)
physical constraints
symbols j 6= j 0
Tijk → ¬Tij 0 k
0
states q 6= q
Qqk → ¬Qq0 k
0
cells i 6= i
Hik → ¬Hi0 k
Turing machine basics
i, j, k
Tijk = Tij(k+1) ∨ Hik
f ∈F
transition table
(q, σ, q 0 , σ 0 , d) ∈
δ
The disjunction
clauses Qf,p(n)
of
the
The disjunction of the
clauses
(Hik ∧ Qqk ∧ Tiσk ) →
(H(i+d)(k+1) ∧ Qq0 (k+1) ∧
Tiσ0 (k+1) )
of
One symbol per tape cell.
Only one state at a time.
Only one head position at a
time.
O(p(n)2 )
O(p(n))
O(p(n))
Tape remains unchanged
unless written.
Must finish in an accepting
state.
O(p(n)2 )
Possible transitions at
computation step k when
head is at position i.
O(p(n)2 )
O(1)
Table 3.1: Translation table from a NDPT Turing machine to a logic formula
This table describes how to construct a logical formula in the variables Tijk , Hik , Qqk
(describing tape contents, head positions, and states, respectively) that corresponds to the
TEX – LATEX – CS 594
3.8. NP-COMPLETENESS
117
Turing machine. If there is an accepting computation for M on input I, then B is satisfiable,
by assigning Tijk , Hik and Qik their intended interpretations. On the other hand, if B is
satisfiable, then there is an accepting computation for M on input I that follows the steps
indicated by the assignments to the variables.
How large is B? There are O(p(n)2 ) Boolean variables, each of which may be encoded in
space O(log p(n)). The number of clauses is O(p(n)2 ). So the size of B is O((log p(n))p(n)2 ).
This is polynomial in n, the size of the input, so the transformation is certainly a polynomialtime reduction, as required.
Victor Eijkhout
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CHAPTER 3. BREAKING THINGS INTO PIECES
Page breaking.
3.9
Introduction
TEX’s page breaking algorithm is simpler than the line breaking one. The reason for this
is probably that global optimization of breakpoints, the way it is done in the paragraph
algorithm, would take prohibitively much memory, at least, for the amount of memory that
computers around 1980 had. The algorithm used is related to the ‘best fit’ algorithm we
discussed in the line breaking chapter.
Theoretically, page breaking is a more complicated problem than line breaking. We will
explore the issues in section 3.11, but first we will briefly go into the algorithms that TEX
actually uses.
3.10
TEX’s page breaking algorithm
The problem of page breaking has two components. One is that of stretching or shrinking
available glue (mostly around display math or section headings) to find typographically
desirable breakpoints. The other is that of placing ‘floating’ material, such as tables and
figures. These are typically placed at the top or the bottom of a page, on or after the first
page where they are referenced. These ‘inserts’, as they are called in TEX, considerably
complicate the page breaking algorithms, as well as the theory.
3.10.1
Typographical constraints
There are various typographical guidelines for what a page should look like, and TEX has
mechanisms that can encourage, if not always enforce, this behaviour.
1.
2.
3.
4.
The first line of every page should be at the same distance from the top. This
changes if the page starts with a section heading which is a larger type size.
The last line should also be at the same distance, this time from the bottom. This
is easy to satisfy if all pages only contain text, but it becomes harder if there are
figures, headings, and display math on the page. In that case, a ‘ragged bottom’
can be specified.
A page may absolutely not be broken between a section heading and the subsequent paragraph or subsection heading.
It is desirable that
(a) the top of the page does not have the last line of a paragraph started on the
preceding page
(b) the bottom of the page does not have the first line of a paragraph that continues on the next page.
TEX – LATEX – CS 594
3.10. TEX’S PAGE BREAKING ALGORITHM
3.10.2
119
Mechanisms
The basic goal of page breaking in TEX is to fill up a box of height \vsize. The is the
goal size of the material without header and footer lines. The box is constructed by adding
material to the vertical list until an optimal breakpoint is found. The material before that
breakpoint is then put in \box255, and the code in \output, the ‘output routine’ is
executed. The command to send a box to the output file is \shipout.
The typographical rules in the previous section can be realized fairly simply in TEX.
1
The vertical location of the first line on a page is controlled by \topskip. If the
baseline of the first line is closer to the top than this parameter, glue is inserted to
make up for the difference.
2
The vertical location of the last line of a page is controlled by \maxdepth. If
the last line of the page is deeper than this amount, the reference point of the box
is shifted down accordingly.
3
Preventing page breaks between vertical material of different kinds can be done
by the proper use of penalties and glue.
4a
A break after the first line of a paragraph is prevented by setting the \clubpenalty.
4b
A break before the last line of a paragraph is prevented by setting the \widowpenalty.
3.10.3
Breakpoints
TEX builds up a current page that is a vertical list of material. It regularly tests whether
there is enough material to fill a box of height \vsize while incurring a badness less
than 10, 000. The breakpoints are similar to those in the line breaking algorithm, mostly
occurring at a penalty, or at a glue that follows non-discardable material.
3.10.4
Some output routines
The very simplest output routine simply takes the vertical list and ships it to the output file:
\output={\shipout\box255}
Slighly more sophisticated, a header and footer are added:
\output={
\setbox255=\vbox{ <header>
\box255
<footer>
}
\shipout\box255
}
The following example makes the page one line longer if a widow (break before the last line
of a paragraph) occurs. First we save the original \vsize and we declare a recognizable
value for the \widowpenalty:
\newif\ifEnlargedPage \widowpenalty=147
\newdimen\oldvsize \oldvsize=\vsize
The output routine now works by testing for the widow penalty, and if it is found, increasing
the \vsize and returning the page material to the list by \unvbox255:
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CHAPTER 3. BREAKING THINGS INTO PIECES
\output={
\ifEnlargedPage <output the page>
\else \ifnum \outputpenalty=\widowpenalty
\global\EnlargedPagetrue
\global\advance\vsize\baselineskip
\unvbox255 \penalty\outputpenalty
\else \shipout\box255
\fi
\fi}
Here is the missing bit of code that outputs the enlarged page:
\ifEnlargedPage \shipout\box255
\global\LargePagefalse
\global\vsize=\oldvsize
3.10.5
Insertions
Floating material, such as tables and figures, are handled by a mechanism called ‘insertions’. Insertions fall in different classes, and insertion material is specified by
\insert<class number>{ <material> }
If the class number is n, then
•
•
•
When the output routine is active, \boxn contains the material of insertion class n.
\dimenn specifies the maximum amount of insertion material that is allowed
to be placed on one page. If there is more material, it is split with the remainder
carried over to the next page.
There are further fine points to this algorithm.
Insertions are thus added, for as far as possible, to the page being formed when the \insert
command is given. TEX has no way of moving insertions to an earlier page, although moving material to a later page – presumable where more space is available – is possible.
3.11
Theory of page breaking
At first glance, the page breaking problem is much like the line breaking problem, except
with larger basic blocks, and vertically instead of horizontally. In the line breaking problem,
a long list of words is broken into lines, adjusting the margins by stretching or shrinking
the interword space. In the page breaking problem, a vertical list of lines, display formulas,
and other vertical material, is broken into equal sized pages, using the various amounts of
vertical space for taking up the slack.
However, the page breaking problem becomes much harder if we include the possibility of
figures and other floating material. In that case, the computed badness (the measure we try
to minimize in the breaking process) will include reflect that we want a figure to be close
to pages that reference it, or satisfy other ordering rules involving references to it. Maybe
surprisingly, even rather simple cost functions make page breaking into an NP-complete
problem.
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3.11. THEORY OF PAGE BREAKING
121
To get an appreciation of the issues, consider this sequence of pages with figures:
References to the figures are here indicated with parenthesized numbers. We see that out
of 5 references, 3 are not to a figure on the same page. However, if we move one line from
page 1 to 2, and move figure 3 forward a page, we get:
where we see that only one reference is not to the same page. Unfortunately this is a backward reference.
In the remainder of this chapter we will investigate theoretical aspects of functions that try
to optimize placement of floating material in a global fashion. It should be noted that this is
considerably more sophisticated than what is implemented in TEX. The available algorithm
is closer to a ‘first fit’ strategy.
We will investigate two closely related page breaking algorithms. We show how a particular form of the page breaking problem (the ‘MQ’ problem) is equivalent to the 2satisfyability problem, which is known NP-complete. As is usual in the context of proving
NP-completeness, we transform our minimization problem ‘give the solution with minimum cost’ into a decision problem by asking ‘is there a solution with a cost ≤ B’ where
B is some predetermined bound.
The very similar ‘ML’ problem, which only differs in the form of the badness function,
does have a polynomial time solution.
Victor Eijkhout
122
3.11.1
CHAPTER 3. BREAKING THINGS INTO PIECES
The MQ problem is NP-complete
We consider the MQ page breaking problem: Multiply referenced figures, and Quadratic
badness measure. Under the simplifying assumptions that each figure takes a whole page,
we then have a set T = {t1 , . . . , tN } of text blocks and a set F = {f1 , . . . , fN } of figures
and a function W : T × F such that W (ti , fj ) describes how many times text block i
references figure j. We assume a bound W (ti , fj ) ≤ q(N ) (where q is a polynomial)
dependent on the size of the problem.
The MQ problem is now the question whether there is an page ordering of text blocks and
figures, that is, a mapping P : (T ∪ F ) → {1, 2, . . . , 2N } such that
P (ti ) < P (tj )
P (fi ) < P (fj )
∀1≤i<j≤N
and so that
S=
X
W (ti , fj )(P (ti ) − P (fj ))2 ≤ B
i,j
In order to show that this problem is NP-complete, we show that it can be transformed into
an instance of the maximum 2-satisfiability problem. This means that solving the one problem is equivalent to solving the other, and since the transformation is done in polynomial
time, the two problems have the same complexity.
The maximum 2-satisfiability (MAX 2- SAT problem can be formulated as follows. Let there
be given n binary variables x1 , . . . , xn and m clauses {u1 ∨ v1 , . . . , um ∨ vm }, where
ui = xj or ui = ¬xj for some j. Given a bound K, is there a way of setting the xi variables
such that at least K of the clauses are satisfied? This problem is known to be NP-complete.
We will now make a pagination problem that ‘translates’ an instance of the 2-satisfyability
problem. That is, given a configuration of binary variables and clauses and a bound K
to satisfy, we will make a pagination problem with bound B such that the one bound is
satisfied if and only if the other is. Then, since MAX 2- SAT is NP-complete, it follows that
this particular pagination problem is also NP-complete.
3.11.1.1 Construction of the translation
We make the translation between the two types of problems by constructing the page assignment function P , and the weight function W . There are three aspects we need to take
care of, so we let B = B1 + B2 + B3 , and we will determine the Bi bounds separately.
First of all we set W (ti , fi ) = b1 sufficiently large that only configuration with |P (ti ) −
P (fi )| = 1 will satisfy the bound. (Recall that the badness is a sum of W (ti , fj )(P (ti ) −
P (fj ))2 terms.) To allow for the pages to be ordered this way, we let B1 = N b1 . The b1
quantity will be defined in terms of B2 and B3 as
b1 = d(B2 + B3 )/3e + 1
Supposing that at least one ti , fi pair is not adjacent, then it follows from this bound that
the badness will be
S ≥ (N − 1)b1 + 22 b1 = (N + 3)b1 > B
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3.11. THEORY OF PAGE BREAKING
123
where the N − 1 corresponds to the pairs that are adjacent, and the 22 to at least one pair
at distance 2.
Since text blocks and are next to each other, the only remaining question is which comes
first. This limits the number of cases we need to consider in the remainder of this proof.
Now let parameters n for the number of variables and m for the number of clauses be
as described above, then our pagination problem will have N text blocks and figures,
where N = 2n + 2m. The first 4n pages encode the values of the variables xi , with
each consecutive 4 pages corresponding to one variable:
4i − 3 4i − 2 4i − 1 4i
t2i−1
f2i−1 f2i
t2i
f2i−1 t2i−1
t2i
f2i
if xi is true
if xi is false
To ensure that these are the only configuration that will satisfy the bound, set W (t2i−1 , f2i ) =
W (f2i−1 , t2i ) = b2 large enough. Either of the above patterns then contributes 2 · 22 b2 =
8b2 , while the other possibilities (t f t f and f t f t) would contribute (12 + 32 )b2 = 10b2 .
P
Correspondingly, we allow a bound of B2 = 4b2 (i − j)2 where i, j range over the pairs
that satisfy W (ti , fj ) = b2 . Defining
b2 = 8(m − k) + 5
is sufficient to make violation of this condition outweigh the gain from more clauses being
satisfied.
Next, the 4m remaining pages will encode the clauses in a similar manner:
4n + 4j − 3 4n + 4j − 2
t2n+2j−1
f2n+2j−1
f2n+2j−1
t2n+2j−1
4n + 4j − 1
4n + 4j
if uj is true
if uj is false
t2n+2j
f2n+2j
if vj is true
f2n+2j
t2n+2j
if vj is false
Further conditions on W ensure that the uj variables indeed correspond to the proper xi .
For instance
W (t2n+2j−1 , f2i ) = W (t2i = f2n+2j−1 ) = b2
if uj = xi
2
This term contributes 2d b2 to the badness, where d is twice the difference between the
subscripts, in this case d = (2n + 2j − 2i). With a mismatch, a t and f page assignment
are reversed, so the contribution becomes (d − 1)2 + (d + 1)2 = 2(d2 + 1)b2 .
Proper truth values of the clauses are enforced as follows. Observe that the combination
where uj and vj are both false is the only one that gives a false result. This corresponds to
the pagination
f2n+2j−1
t2n+2j−1
f2n+2j
t2n+2j
In this configuration f2n+2j1 and t2n+2j are spread the furthest apart, so we penalize that
with
W (t2n+2j , f2n+2j1 ) = 5,
W (t2n+2j−1 , f2n+2j ) = 3.
This gives a contribution of 32 for the three true cases, and 48 for the false result. Correspondingly, to allow for K clauses to be satisfied we allow B3 = 48(m − K) + 32K.
Victor Eijkhout
124
CHAPTER 3. BREAKING THINGS INTO PIECES
Finally, by defining the polynomial q(i) = 64(i + 1)4 , we have q(N ) > b1 ≥ b2 > 5 > 3,
so W is bounded as required.
3.11.1.2 NP-completeness
In order to establish NP-completeness of the problem MQ, we need to establish that something is a true instance of M AX 2- SAT iff the translated instance is true for MQ.
Given a truth assignment of the xi s that makes K clauses true, we can now construct a
pagination P with a satisfied bound of B.
Conversely, let a pagination with bound B be given, then the corresponding truth assignment makes K clauses true. To see this, inspect each of the details of the translation between the problems, and observe that any violation will cause the bound B to be exceeded.
3.11.2
The ML problem has a polynomial time solution
The ‘ML’ problem (Multiply referenced figures, Linear badness function) is identical to
MQ, except for the form of the badness function. Having a linear badness function makes
it possible to solve this problem by dynamic programming in linear time.
As in MQ, we have text blocks ti and figures fj that take up whole pages. We generalize
the problem slightly to having different numbers of text and figure blocks:
T = {t1 , . . . , tN },
F = {f1 , . . . , fM }
The function W : T × F is such that W (ti , fj ) ≥ 0 describes how many times text block i
references figure j.
The ML problem is now the question whether, given the above, and given a bound B,
there is an page ordering of text blocks and figures, that is, a mapping P : (T ∪ F ) →
{1, 2, . . . , M + N } such that
P (ti ) < P (tj )
∀1≤i≤N,1≤j≤M
P (fi ) < P (fj )
and so that
X
S=
W (ti , fj ) |P (ti ) − P (fj )| ≤ B
i,j
3.11.2.1 Dynamic programming solution
The key to a dynamic programming solution of ML is to identify subproblems. The subproblem we consider is
Given i text blocks and j figures, what is the least badness of placing
these on the first i + j pages. Call this partial badness Bij .
The problem here is the ‘dangling’ references (tr , fs ) with r > i, s ≤ j or r ≤ i, s > j.
The measure Ri,j is defined as the number of dangling references after these blocks and
figures have been placed.
A dangling reference is either
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3.11. THEORY OF PAGE BREAKING
125
A forward reference: A text block refering to a figure not yet placed. The number of
forward references
X from the i + j block is
Fij =
W (tr , fs )
1≤r≤i
j<s≤M
A backward reference: A figure that will be reference on a text block not yet placed.)
The number ofX
backward references from the i + j block is
Bij =
W (tr , fs )
i<r≤N
1≤s≤j
which makes Rij = Fij + Bij .
F
(i,j)
s
U
B
r
Figure 3.1: The Fij , Bij , and Uij areas in (r, s) space
For badness calculations involving dangling references, we count only the part to the
boundary of the i, j region. Formally:
(1)
(2)
Bij = Bij + Bij
where
(1)
Bij =
X
W (tr , fs ) |P (tr ) − P (fs )|
r≤i
s≤j
is the part of the badness due to references that are fully resolved within the pages already
placed; the part of the badness due to dangling references is
(1)
(2)
Figure 3.2: Resolved and dangling references of a block of pages
Victor Eijkhout
126
CHAPTER 3. BREAKING THINGS INTO PIECES
(2)
Bij =
X
W (tr , fs ) `(i, j; r, s) +
X
r>i
s≤j
W (tr , fs ) `(i, j; r, s)
r≤i
s>j
where
i + j − P (fs ) if r > i
i + j − P (tr ) if s > j
describes the part of the arc between tr and f2 that lies in the first i + j pages. These two
types of arcs are illustrated in figure 3.2.
`(i, j; r, s) =
Figure 3.3 illustrates how reference arcs change status when we go from i + j − 1 to i + j
pages, say by placing text block ti :
(2)
(1)
(3)
i+j-1
i+j
Figure 3.3: Change in status of resolved and dangling references upon extending a block of
pages
References that were unresolved with references originating in ti move their contribution from the B (2) term to the B (1) term. (Note that a reference to a page one
location outside the current block is already fully counted in the badness.)
(2)
DanglingP
references thatP
stay unresolved increase their contribution to the B (2)
term by ( r≤i−1,s>j + r>i−1,s≤j )W (tr , fs )
(3)
Arcs that used to fall completely outside the P
page block, but that are now dangling
in the new page block, add a contribution of r=i,s>j W (tr , fs ) to the B (2) term.
P
r>i,s≤j W (tr , fs ) In sum, Bij = Bi−1,j + Rij . The same story holds for extending
i + j − 1 pages by placing figure fj , so we have the recurrence
Bij = min(Bi−1,j , Bi,j−1 ) + Rij .
We still need to compute the Rij quantities, describing the number of dangling references
from placing i text blocks and j figures. Above we saw Rij = Fij + Bij . For efficient
calculation of these sums, it is convenient to make a table of
X
Uij =
W (tr , fs )
(1)
1≤r≤i
1≤s≤j
which takes time O(N M ), that is, quadratic in the total number of pages. Then
Rij = UiM + UN j − 2Uij ,
as is easily seen from figure 3.1.
3.11.3
Discussion
It is interesting to note that certain details of the NP-completeness proof of MQ rely on the
quadratic badness function, rather than on more ‘structural’ properties of the problem.
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3.11. THEORY OF PAGE BREAKING
127
Exercise 40. Find a place in the NP-completeness proof of MQ that
uses the quadratic badness function, and show that the underlying fact
does not hold for linear functions. Does it hold for other functions than
quadratic?
Similarly, ML only has a dynamic programming solution thanks to the linear badness function.
Exercise 41. Explain how the linearity of the badness function is essential for the dynamic programming solution of ML.
Exercise 42. The discussion of the ML problem only gave the cost
computation. How can the actual page assignment be derived from the
given construction? What is the time and space complexity of this part
of the algorithm?
Victor Eijkhout
128
CHAPTER 3. BREAKING THINGS INTO PIECES
Projects for this chapter.
Project 3.1. What is the paragraph breaking algorithm of OpenOffice? Replace by TEX’s
algorithm.
Project 3.2. Write a paragraph breaking algorithm that prevents ‘rivers’ in the text.
Project 3.3. TEX’s line breaking algorithm is not just for good looks. Many aesthetic decisions in typography actually influence readability of the document. Read ‘Digital
Typography’ by Rubinstein [19] and find an issue to investigate.
Project 3.4. Many page layout parameters (Rubinstein [19] and the references therein)
have an influence on legibility. Has typographic design deteriorated in that sense
now that authors publish their own works? Do a study, using various books in the
library.
Project 3.5. The following sentence
Only the fool would take trouble to verify that this sentence was composed of ten a’s, three b’s, four c’s, four d’s, forty-six e’s, sixteen f’s,
four g’s, thirteen h’s, fifteen i’s, two k’s, nine l’s, four m’s, twenty-five
n’s, twenty-four o’s. five p’s, sixteen r’s, forty-one s’s, thirty-seven t’s,
ten u’s, eight v’s, eight w’s, four x’s, eleven y’s, twenty-seven commas, twenty-three apostrophes, seven hyphens and, last but not least,
a single !
is called a pangram. (There are other pangrams. Google for the combination of
‘pangram’ and ‘Lee Sallows’ for this particular type.) Given a beginning of the
sentence (‘Only the fool. . . ’), solve the rest of the sentence by dynamic programming.
TEX – LATEX – CS 594
Chapter 4
Fonts
Knuth wrote a font program, Metafont, to go with TEX. The font descriptions involve some
interesting mathematics.
Handouts and further reading for this chapter
Bezier curves and raster graphics are standard topics in computer graphics. The book by
Foley and Van Dam (section 11.2 about Splines) has been placed on reserve, T385.C587 .
More theoretical information can be found de Boor’s Practical Guide to Splines [2], which
unfortunately limits itself to spline functions.
Digital typography is a very wide area, spanning from perception psychology and physiology to the electronics and mathematics of display equipment. The book by Rubinstein [19]
is a good introduction. This book has been placed on reserve, Z253.3.R8 .
The relation between Bezier curves and aesthetics is explicitly discussed in http://
www.webreference.com/dlab/9902/examples-ii.html.
129
130
CHAPTER 4. FONTS
Bezier curves.
4.1
Introduction to curve approximation
You may never have thought of it, but fonts (actually, typefaces) usually have a mathematical definition somehow. If a font is given as a bitmap, this is typically the result from a
more compact description. Imagine the situation that you have bitmaps at 300dpi, and you
buy a 600dpi printer. It wouldn’t look pretty.
There is then a need for a mathematical way of describing arbitrary shapes. These shapes
can also be three-dimensional; in fact, a lot of the mathematics in this chapter was developed by a car manufacturer for modeling car body shapes. But let us for now only look in
two dimensions, which means that the curves are lines, rather than planes.
A mathematical formalism for curves should have these properties:
•
•
•
•
•
The description should be clear and unambiguous.
It should be easy to tinker with the shapes. This is important for the design phase.
Constructing and evaluating the curves should be computationally cheap.
The curves should be well behaved: small changes in the definition should not
lead to unexpected bulges or spikes.
Curves should be easily composable: if two curves describe adjacent pieces of the
shape, they should connect smoothly.
We actually have two problems that we want to solve:
1.
2.
The exact curve is known, and we want to approximate it, for instance by something that is cheaper to compute, or
Only certain points are given, and we want to draw a smooth curve through them.
We will tackle the second problem first.
4.1.1
Interpolation
The interpolation problem is that of, given points (x1 , f1 ) . . . (xn , fn ), drawing a curve
through them, for instance for purposes of computing intermediate values. Suppose that
we have decided on a polynomial for the interpolating curve. With n points we need an
n − 1st degree polynomial p(x) = pn−1 xn−1 + · · · + p1 x + p0 , which takes n coefficients.
We can draw up the set of equations p(xi ) = fi and solve that. The system
pn−1 xn−1
+ · · · + p1 x1 + p0
1
pn−1 xn−1
+ · · · + p1 xn + p0
n
can be written as X p¯ = f¯, where
p1
.
j
X = xi , p¯ = .. ,
pn−1
=
...
=
f1
fn
f1 − p0
.
..
f¯ =
fn − p0
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4.1. INTRODUCTION TO CURVE APPROXIMATION
131
Figure 4.1: A family of functions that converges pointwise but not uniformly.
Solving this system is not overly expensive, but its numerical stability is questionable. A
better way of computing the same polynomial p is to define auxiliary polynomials p(k) :
p(k) (x) = ck (x − x1 ) · · · (x − xk−1 ) (x − xk+1 ) · · · (x − xn )
where ck is chosen so that p(k) (xk ) = 1. From the fact that p(i) (xj ) = δij , it follows that
X
Y x − xj
p(x) =
fi p(i) (x),
p(i) (x) =
(4.1)
xi − xj
i
j6=i
interpolates the points as intended. It is easy enough to prove that polynomials are uniquely
defined by these interpolation points, so we have now computed the same polynomial in a
more stable way. A polynomial that is based on exact interpolation of values in a number
of points is called a ‘Lagrange interpolation’ polynomial.
Another type of interpolation is ‘Hermite interpolation’, where the derivatives are dictated,
rather than function values. Analogous to the above construction, we can define polynomials
q (k) = ck (x − x1 )2 · · · (x − xk−1 )2 · (x − xk ) · (x − xk+1 )2 · · · (x − xn )2
0
where ck is chosen so that q (k) (xk ) = 1.
4.1.2
Approximation
The above notion of interpolation is sometimes applied to known curves. For instance, by
finding an interpolating polynomial we may aim to find a cheaper way of computing values
on the curve. This raises the question how well the interpolation curve approximates the
original curve.
In classical approximation theory there is usually a family of functions {fn }, typically
polynomials of increasing degree, such that kfn − f k → 0, typically on a closed interval I.
The Weierstrass approximation theorem tells us that every continuous function on a closed
bounded interval can be approximated by polynomials.
Note that this is uniform convergence:
∀ ∃N ∀x∈I,n≥N : |fn (x) − f (x)| ≤ .
Victor Eijkhout
132
CHAPTER 4. FONTS
Figure 4.2: The Runge effect of interpolating with a high degree polynomial
This is a stronger statement than pointwise convergence:
∀x∈I, ∃N ∀n≥N : |fn (x) − f (x)| ≤ .
It is easy to find families of functions fn that convergence in the pointwise sense, but not
uniformly; see figure 4.1.
The spline curves that we will study later in this chapter are a special case of Bernstein
polymials: the n-th Bernstein polynomials for a function f is
Bn (f )(t) =
n
X
n p
f
(1 − t)n−p tp .
p
n
p=0
If f is continuous on [0, 1], this sequence converges uniformly to f . It is worth remarking
that these polynomials do not require computation of derivatives.
While the ideas behind Lagrange and Hermite interpolation will find applicability later in
this story, the idea of interpolating with a single, high degree, polynomial may not be a
good one from a point of uniform convergence. The error can be unacceptably large, as
can be seen in figure 4.2, where the dashed line is an interpolation on equispaced points. In
this case there is in fact not even pointwise convergence. There are a few ways out of that,
such as better choice of interpolation points or of basis functions. In figure 4.2 the dotted
line uses Tchebyshev interpolation points which is seen to remedy the problem to a large
extent.
However, the approach we will use here is that of piecewise approximations with relatively
low degree polynomials. This simplifies certain aspects, but we need to take care to piece
together such curves smoothly. For instance, with Lagrange interpolation the direction of
the curve at the end points can not be specified.
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4.1. INTRODUCTION TO CURVE APPROXIMATION
4.1.3
133
Computation with interpolation curves
While we will mostly investigate theoretical properties of interpolation curves, the practical
matter of how efficient it is to work with them, also deserves attention. In equation (4.1)
there are n terms involving n multiplications and additions each, making for an O(n2 ) cost.
This can be considerably reduced by rewriting the formula as
Y
Y
X yi
,
yi = fi / (xi − xj ),
p(x) =
(x − ti ) ·
x − ti
i
i
j6=i
which takes n additions, multiplications, and additions if the yi quantities are precomputed.
We will now see a formulation that dispenses with the divisions, and that will also be
simpler if derivatives need to be computed.
The k-th ‘divided difference’ of a function g in the points τ1 . . . τk+1 , notation [τ1 , . . . , τk+1 ]g,
is the leading coefficient of the k-th order1 polynomial pk+1 that agrees with g in the
points τ1 . . . τk+1 .
The simplest examples of divided differences are:
•
•
The zeroeth divided difference of a function is the leading coefficient of a zeroth
order polynomial p(x) = c that agrees with that function in one point: g(τ1 ) = g1 .
Clearly [τ1 ]g = g1 .
The first divided difference is the leading coefficient of a linear function that
agrees in two points:
g(τ2 ) − g(τ1 )
[τ2 ]g − [τ1 ]g
[τ1 , τ2 ]g =
=
τ2 − τ1
τ2 − τ1
This equation may suggest to the reader a relation for higher divided differences.
We shall see in lemma 4 that this indeed holds.
We now prove some facts about divided differences.
Lemma 3 Let pk+1 ∈
τ1 . . . τk , then
Q
<k+1
agree with g in τ1 . . . τk+1 , and pk ∈
pk+1 (x) − pk (x) = [τ1 , . . . , τk+1 ]g
k
Y
(x − τi ).
Q
<k
with g in
(4.2)
i=1
Proof. Since pk is of a lower order, we immediately have
pk+1 − pk = [τ1 , . . . , τk+1 ]gxk + cxk−1 + · · · .
Observing that pk+1 − pk is zero in ti for i ≤ k, it follows that
pk+1 − pk = C
k
Y
(x − τi ).
i=1
From this we get that C = [τ1 , . . . , τk+1 ]g.
•
1.
It is convenient to talk about polymomials of aQcertain order rather than degree: a polynomial of order k has
a degree no more than k. We denote this set with <k+1 . One advantage of this set is that it is closed under
summing.
Victor Eijkhout
134
CHAPTER 4. FONTS
If we repeat this lemma we find that
pk+1 (x) =
k+1
X
[τ1 , . . . , τm ]g
m=1
m−1
Y
(x − τi ),
(4.3)
i=1
which we can evaluate as
pk+1 (x)
Qk
Qk−1
= [τ1 , . . . , τk+1 ]g
(x − τi ) + [τ1 , . . . , τk ]g
(x − τi )
= [τ1 , . . . , τk+1 ]g(x − τk ) ck + [τ1 , . . . , τk ]g(x − τk−1 ) ck−1 + · · ·
where ck = [τ1 , . . . , τk ]g/[τ1 , . . . , τk+1 ]g. This is a very efficient evaluation by Horner’s
rule.
The remaining question is how to construct the divided differences. We approach this recursively.
Lemma 4 Divided differences can be constructed by, eh, dividing differences
[τ1 , . . . , τn+1 ]g = ([τ1 , . . . , τn ]g − [τ2 , . . . , τn+1 ]g) /(τ1 − τn+1 ).
(4.4)
Proof. Let three polynomials be given:
Q
(1)
•
pn ∈ <n agrees with g on τ1 . . . τn ;
Q
(2)
•
pn ∈ <n agrees with g on τ2 . . . τn+1 ;
Q
•
pn−1 ∈ <n−1 agrees with g on τ2 . . . τn .
Then by lemma 3
(1)
pn − pn−1
(2)
pn − pn−1
Qn
= [τ1 , . . . , τn ]g j=2 (x − τj )
Qn
= [τ2 , . . . , τn+1 ]g j=2 (x − τj )
Now let pn+1 be the polynomial that agrees with g on τ1 . . . τn+1 , then
Qn
pn+1 − p(1) = [τ1 , . . . , τn+1 ]g j=1 (x − τj )
Qn+1
pn+1 − p(2) = [τ1 , . . . , τn+1 ]g j=2 (x − τj )
(1)
(2)
Subtracting both pairs of equations, we find two expressions for pn − pn :
n
n+1
n
Y
Y Y
(x − τj )
([τ1 , . . . , τn ]g − [τ2 , . . . , τn+1 ]g)
(x − τj ) = [τ1 , . . . , τn+1 ]g
−
j=2
j=2
j=1
Filling in τ2 . . . τn in this equation, we find zero for both sides. Using x = τ1 we find
([τ1 , . . . , τn ]g − [τ2 , . . . , τn+1 ]g)
n
Y
j=2
from which the lemma follows.
(τ1 − τj ) = [τ1 , . . . , τn+1 ]g
n+1
Y
(τ1 − τj )
j=2
•
From this lemma, we see easily that [τ1 , . . . , τn ]g can be computed in approximately n2 /2
additions and divisions.
TEX – LATEX – CS 594
4.2. PARAMETRIC CURVES
4.2
135
Parametric curves
So far we have been looking at approximation by a function of a single value. That is,
we have y as a function of x. This rules out many curves, such as circles. We could try
expressing x as a function of y, or more generally rotating them, but that is a big hassle,
and it would still rule out some curves.
Another approach is to let the curve be defined implicitly by f (x, y, z) = 0. This suffers
from several problems. We could get too many solutions, for instance as in x2 +y 2 −1 = 0,
requiring constraints. Tracking a path is possible in theory, but is not trivial in practice.
Finally, many shapes are defined piecewise, and joining shapes in a smooth way is also
tricky.
Rather than considering a curve in the plane as a function, it is more productive to describe
the shape of the curve, as some imaginary point moves over the curve. That is, we have a
description of the points on the curve as
x(t)
P = P (t) =
.
y(t)
(Feel free to think of t as time.)
A simple example of how this works is to consider two points P1 and P2 , and the curve
P = tP2 + (1 − t)P1 . Then for t = 0, P (0) = P1 , for t = 1, P (1) = P2 , and for
intermediate values of t we get some point between P1 and P2 .
That this solves our problems with multiple solutions that were present in both function
and implicit approaches is clear if we look at the example P (t) = (cos 2πt, sin 2πt), which
traces out a circle as t goes from 0 to 1.
While a description in terms of piecewise linear functions would be very simple, it is not
smooth. Using the various ideas sketched above, we will now concentrate on curves that
are piecewise parametric cubic splines. Cubics have the following property that they are the
lowest degree that allows specification of location and direction in the end points. Higher
degree functions would allow for instance the specification of higher derivatives, but unless
great care is taken, they would introduce unwanted ‘wiggles’ in the curves.
Using piecewise cubic parametric curves then is a good mean between ease of use and
power of approximation.
4.2.1
Parametrized basis functions
To begin with we will concentrate on a single curve Q(t) where t = 0 . . . 1. We often write
this as Q(t) = C · T where the coefficient matrix
1
c11 c12 c13 c14
t
C = c21 c22 c23 c24 ,
T = 2
t
c31 c32 c33 c34
t3
Victor Eijkhout
136
CHAPTER 4. FONTS
The direction of the curve is then given by
0
dQ(t)
dT
1
=C·
=C ·
2t
dt
dt
2
3t
We see that the C matrix can be derived if we know a total of four locations or directions
of the curve. For instance, if P1 = Q(0), R1 = Q0 (0), P2 = Q(1), and R2 = Q0 (1) are
given, then
1 0 1 0
0 1 1 1
C ·
= [P1 , R1 , P2 , R2 ],
0 0 1 2
0 0 1 3
from which C follows.
Now, often we have a set of basis polynomials given, and we want to take combinations of
them to satisfy the constraints. That can be done by splitting C = GM , where M describes
the basis polynomials, and G is the ‘geometry matrix’. We get the equation
m11 . . . m14
g11 g12 g13 g14
.
..
Q(t) = G · M · T = g21 g22 g23 g24 · ..
·T
(4.5)
.
g31 g32 g33 g34
m41 . . . m44
If we introduce new basis polynomials πi (t) = Mi∗ · T , then we see that Qx = G11 π1 +
G12 π2 + G13 π3 + G14 π4 , Qy = G21 π1 + · · ·, et cetera.
4.2.2
Hermite basis functions
In equation (4.5) the matrix M describes the basis functions, so it is fixed for a certain
class of curves: we will have one set of basis functions for Lagrange type curves, one for
Hermite curves, et cetera. However, we have not yet seen a way to compute the matrix M .
The geometry matrix G is used to derive a specific curve in the class described by M : each
choice of G corresponds to one curve. The columns of G will be points or direction vectors
that somehow describe the intended curve.
Let us now consider Hermite curves. Here we want G to be the matrix of geometric constraints, [P1 , R1 , P2 , R2 ] in the above example. Recall that these constraints, using the locations of the end points and the derivatives of the curve there, give us indeed an example
of Hermite interpolation.
We write out the equations. From Q = G · M · T we get
Q(t) = GH · MH · T (t),
Q0 (t) = GH · MH · T 0 (t).
Applying both these formulas to t = 0 and t = 1, we get
QH ≡ [Q(0), Q0 (0), Q(1), Q0 (1)] = GH · MH · TH
where
1 0 1 0
0 1 1 1
TH = [T (0), T 0 (0), T (1), T 0 (1)] =
0 0 1 2
0 0 1 3
TEX – LATEX – CS 594
4.2. PARAMETRIC CURVES
137
1.2
P1(x)
P2(x)
P3(x)
1
0.8
0.6
0.4
0.2
0
-0.2
0
0.2
0.4
0.6
0.8
1
Figure 4.3: Hermite polynomials
But this QH is the matrix that we had stipulated as the matrix of geometry constraints, in
other words: GH = QH . It now follows that
1 0 −3 2
0 1 −2 1
−1
MH = T H
=
.
0 0 3 −2
0 0 −1 1
Writing this out, we find the cubic Hermite polynomials
P1 (t) = 2t3 − 3t2 + 1,
P2 (t) = t3 − 2t2 + t,
P3 (t) = −2t3 + 3t2 ,
P1 (t) = t3 − t2
illustrated in figure 4.3, and we get Q(t) = G · BH where BH = M · T is the matrix of
‘Hermite blending functions’.
With this formula, we can take a set of geometric constraints, in this case the two endpoints
and the directions there, and derive the formula for the Hermite curve that satisfies these
constraints. As an example, figure 4.5 is the curve .3P1 − 2P2 + P3 − 2P 4, that is, the
curve through (0, .3) and (1, 1), with slope −2 in both x = 0, 1.
We make the transition to parametric curves by expressing both components as Hermite
curves. For instance, figure 4.7 shows the curve
Qx (t) = .1 ∗ P1 (t) + P2 (t) + .9 ∗ P3 (t),
#
# 4
#
set
set
set
cubic Hermite polynomials
terminal pdf
xrange [0:1]
yrange [-.2:1.2]
Qy (t) = .2 ∗ P1 (t) + .3 ∗ P3 (t) − P4 (t)
P1(x) = 2*x**3-3*x**2+1
P2(x) = x**3-2*x**2+x
P3(x) = -2*x**3+3*x**2
P4(x) = x**3-x**2
plot P1(x), P2(x), P3(x), P4(x) title ""
Figure 4.4: The gnuplot source for figure 4.3
Victor Eijkhout
138
CHAPTER 4. FONTS
1.2
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
Figure 4.5: An example of Hermite interpolation
that is
.1
1
.9
0
Q=
P1 +
P2 +
P3 +
P4 .
.2
0
.3
−1
There is one aspect of these Hermite
curves worth
remarking on. In figure 4.9 we have
replaced the direction vector 10 in (0, 0) by x0 , where x = 1, 1.5, 2, 2.5, which all
have the same direction, but a different magnitude. Clearly there is a visual effect.
4.2.3
Splines
We will now take a close look at Bernshtein polynomials of degree 3:
z(t) = (1 − t)3 z1 + 3(1 − t)2 tz2 + 3(1 − t)t2 z3 + t3 z4 ,
(4.6)
also known as Bezier curves or Bezier cubics after Pierre Bezier, an engineer at Renault2 .
2.
Pierre B´ezier was born September 1, 1910 and died November 25, 1999 at the age of 89. In 1985 he
was recognized by ACM SIGGRAPH with a ‘Steven A. Coons’ award for his lifetime contribution to computer
graphics and interactive techniques.
#
# Hermite interpolation
#
set terminal pdf
set multiplot
set xrange [0:1]
set yrange [0:1.3]
P1(x) = 2*x**3-3*x**2+1
P2(x) = x**3-2*x**2+x
P3(x) = -2*x**3+3*x**2
P4(x) = x**3-x**2
p1y = .3
p1slope = -2
p2y = 1
p2slope = -2
plot p1y*P1(x) + p1slope*P2(x) \
+ P3(x) + p2slope*P4(x) title ""
set parametric
set style function lines
plot [t=0:.1] t, p1y+t*p1slope \
title "" with lines 2
plot [t=0:.1] 1-t,p2y-t*p2slope \
title "" with lines 2
Figure 4.6: The source for figure 4.5
TEX – LATEX – CS 594
4.2. PARAMETRIC CURVES
139
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
Figure 4.7: A Hermite interpolation curve
There are a couple of ways of looking at these polynomials. Consider the function z(t) to
be the sum of four basis functions, (1 − t)3 , (1 − t)2 t, (1 − t)t2 , and t3 , each multiplied by
a factor deriving from a control point. From the formulas, and from a picture (figure 4.11)
we see that the first term p1 (t) = (1 − t)3 is the only one with p(0) 6= 0. Likewise, p4 is the
only one with p(1) 6= 0. That means z(0) = z1 and z(1) = z4 . Furthermore, the second
term is (after the first term) the only remaining one with p0 (0) 6= 0, so by choosing z2 we
can change z 0 (0) without changing z(0) or z(1). Likewise z3 for z 0 (1).
Bezier curves can be derived from cubic Hermite splines by replacing the direction vectors R1 , R2 by control points P10 , P20 , so that R1 = 3(P10 − P 1) and R2 = 3(P2 − P20 ). For
the Bezier geometry vector we then have
GB = [P1 , P10 , P20 , P2 ]
and the relation with the Hermite geometry vector
GH = [P1 , R1 , P2 , R2 ] = [P1 , P10 , P20 , P2 ]MBH = GB · MBH
#
# Parametric Hermite curve
#
set terminal pdf
set parametric
set multiplot
set xrange [0:1]
set yrange [0:.7]
P1(t) = 2*t**3-3*t**2+1
P2(t) = t**3-2*t**2+t
P3(t) = -2*t**3+3*t**2
P4(t) = t**3-t**2
p1x = .1 ; p1y = .2
p1dx = 1 ; p1dy = 0
p2x = .9 ; p2y = .3
p2dx = 0 ; p2dy = -1
plot [t=0:1] \
p1x*P1(t)+p1dx*P2(t)+p2x*P3(t)+p2dx*P4(t), \
p1y*P1(t)+p1dy*P2(t)+p2y*P3(t)+p2dy*P4(t) \
title ""
plot [t=0:.2] p1x+t*p1dx,p1y+t*p1dy \
title "" with lines 2
plot [t=0:.2] p2x-t*p2dx,p2y-t*p2dy \
title "" with lines 2
Figure 4.8: The source of figure 4.7
Victor Eijkhout
140
CHAPTER 4. FONTS
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
Figure 4.9: The same curve, tinkering with the direction vector
where
1 −3
0
3
=
0 0
0 0
MBH
0 0
0 0
0 −3
1 3
(4.7)
# direction 2:
set terminal pdf
p1dx = 1.5 ; p1dy = 0
set parametric
plot [t=0:1] \
set multiplot
p1x*P1(t)+p1dx*P2(t)+p2x*P3(t)+p2dx*P4(t), \
set dummy t
p1y*P1(t)+p1dy*P2(t)+p2y*P3(t)+p2dy*P4(t) \
set xrange [0:1]
set yrange [0:.7]
title ""
# direction 3:
P1(t) = 2*t**3-3*t**2+1
p1dx = 2 ; p1dy = 0
P2(t) = t**3-2*t**2+t
plot [t=0:1] \
P3(t) = -2*t**3+3*t**2
p1x*P1(t)+p1dx*P2(t)+p2x*P3(t)+p2dx*P4(t), \
P4(t) = t**3-t**2
p1y*P1(t)+p1dy*P2(t)+p2y*P3(t)+p2dy*P4(t) \
p1x = .1 ; p1y = .2
title ""
p2x = .9 ; p2y = .3
p2dx = 0 ; p2dy = -1
# direction 4:
p1dx = 2.5 ; p1dy = 0
# direction 1:
plot [t=0:1] \
p1dx = 1 ; p1dy = 0
plot [t=0:1] \
p1x*P1(t)+p1dx*P2(t)+p2x*P3(t)+p2dx*P4(t), \
p1y*P1(t)+p1dy*P2(t)+p2y*P3(t)+p2dy*P4(t)
\
p1x*P1(t)+p1dx*P2(t)+p2x*P3(t)+p2dx*P4(t),
\
title
p1y*P1(t)+p1dy*P2(t)+p2y*P3(t)+p2dy*P4(t)
\ ""
title ""
Figure 4.10: The source for figure 4.9
TEX – LATEX – CS 594
4.2. PARAMETRIC CURVES
141
1
x**3
3*x**2*(1-x)
3*x*(1-x)**2
(1-x)**3
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
Figure 4.11: Bernshtein polynomials
Defining
1 −3 3 −1
0 3 −6 3
=
0 0
3 −3
0 0
0
1
MB = MBH · MH
(4.8)
we now get for Bezier curves
Q(t) = GH · MH · T (t) = GB · MBH · MH · T (t) = GB · MB · T (t)
We can also write that as Qx (t) = g11 B1 (t) + g12 B2 (t) + · · · where
B1 (t) = 1 − 3t + 3t2 − t3
B2 (t) = 3t − 6t2 + 3t3
B3 (t) = 3t2 − 3t3
B4 (t) =
= (1 − t)3
= 3t(1 − t)2
= 3t2 (1 − t)
= t3
which are the Bernstein polynomials we started this section with.
The sum of these polynomials (this is equivalent to setting the zi coefficients to one in
equation (4.6)) is z(t) = (t + (1 − t)))3 ≡ 1. Also, the polynomials are positive on [0, 1],
so the components Qx , Qy , Qz are weighted averages of the polynomials. This means that
the curve, consisting of weighted sums of the control points, is contained in the convex hull
of the control points.
Exercise 43. One could also define quadratic Bezier curves. These
have only a single control point, and the curve is such that in both the
endpoints it is aimed at the control point.
Victor Eijkhout
142
CHAPTER 4. FONTS
2
Quadratic Bezier curve
1.5
1
0.5
0
0
0.2
0.4
0.6
0.8
Derive the basis functions and geometry matrix for this case. Make a
gnuplot figure of a single quadratic Bezier curve, and of two curves
joined smoothly.
Hint: you can follow the construction of the cubic splines in the lecture
notes. The only problem is defining the control point. First draw up the
Hermite geometry matrix based on end points q0 and q1 , and the derivative q00 in the first end point. Derive from them the derivative q10 in the
other end point. The control point then lies on the intersection of two
lines. Solving this looks like a single equation in two unknowns, but it
can be solved: write it as a matrix-vector equation that is satisfied no
matter the choice of the geometry matrix.
4.2.4
Calculation of Bezier curves
Suppose we have a Bezier curve defined by four control points, and we want to draw points
on the curve, meaning that we have to evaluate the function Q(t) for a number of values
of t. The relation Q(t) = G · M · T (t) allows us to do this calculation in
2 multiplications to form the terms t2 and t3 in T ;
16 multiplications and 12 additions forming M · T ;
12 multiplications and 9 additions forming G · (M · T ).
˜ = G · M , which brings the cost down to two
An obvious improvement is to store M
multiplications for T and
˜ · T.
•
12 multiplications and 9 additions for forming M
•
•
•
A similar count is arrived at by looking at divided differences. Additionally, this way of
computing is more stable.
TEX – LATEX – CS 594
1
4.3. PRACTICAL USE
143
From the formula Q(t) = G · M · T (t) we get for each component
X
X
qi (t) =
Gij (M T )j =
Gij Mjk tk−1 .
j
j,k
Looking at only one component for simplicity, we find, for instance
X
X
x(t) =
ck tk−1
ck =
G1j Mjk .
j
k
We recall equation (4.8):
1 −3 3 −1
0 3 −6 3
MB =
0 0
3 −3
0 0
0
1
and writing gj ≡ G1j we find
c1 = g1 ,
c2 = 3(g2 − g1 ),
c3 = 3(g3 − 2g2 + g1 ),
c4 = g4 − 3g3 + 3g2 − g1 .
In this we recognize divided differences of g:
[2, 1]g
= g2 − g1 ,
[3, 2, 1]g = [3, 2]g − [2, 1]g = (g3 − g2 ) − (g2 − g1 )
= g3 − 2g2 + g1
[4, 3, 2, 1] = [4, 3, 2]g − [3, 2, 1]g = (g4 − 2g3 + g2 ) − (g3 − 2g2 + g1 )
= g4 − 3g3 + 3g2 − g1
using lemma 4.
4.3
Practical use
4.3.1
Piecewise curves
As we indicated earlier, complicated shapes can be approximated by piecewise cubics. Figure 4.12 shows two Hermite curves joined together. The curve is continuous and the directions at the join are proportional. This is called ‘geometric continuity’, and is denoted G1 .
So-called B-splines (‘basis splines’) are constructed by piecing together Bezier curves,
not only continuously, but differentiably. For this, if P1 . . . P4 and P4 . . . P7 are the control
points of the two curves, then we require
P4 = (P3 + P5 )/2.
This is shown in figure 4.14.
4.3.2
Drawing splines
Even if we can evaluate a Bezier curve efficiently (section 4.2.4) in a given point, we
can improve on this considerably in the context of line drawing. Consider the problem of
drawing a cubic curve on the interval [0, 1] by drawing consecutive points Q(nδ) for n =
0, 1, . . ..
We will discuss one line-drawing technique in the chapter on raster graphics. A technique
that is attractive for splines, and which is used in M ETAFONT, is recursive subdivision.
Victor Eijkhout
144
CHAPTER 4. FONTS
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
Figure 4.12: Two Hermite curves, joined together smoothly in (.5, .3)
Here, a curve is subdivided until a segment is a straight line within the pixel resolution, so
that a line drawing algorithm can be used. The test for a straight line can be implemented
efficiently through using the spline control points.
set terminal pdf
set parametric
set multiplot
set dummy t
set xrange [0:1]
set yrange [0:.8]
P1(t) = 2*t**3-3*t**2+1
P2(t) = t**3-2*t**2+t
P3(t) = -2*t**3+3*t**2
P4(t) = t**3-t**2
p1x = .1 ; p1y = .2
p2x = 1 ; p2y = 0
p3x = .5 ; p3y = .3
p4x = 0 ; p4y = -1
plot [t=0:1] \
p1x*P1(t)+p2x*P2(t)+p3x*P3(t)+p4x*P4(t), \
p1y*P1(t)+p2y*P2(t)+p3y*P3(t)+p4y*P4(t) \
title ""
p5x = .9 ; p5y = .6
p6x = 0 ; p6y = -1
plot [t=0:1] \
p3x*P1(t)+.5*p4x*P2(t)+p5x*P3(t)+p6x*P4(t), \
p3y*P1(t)+.5*p4y*P2(t)+p5y*P3(t)+p6y*P4(t) \
title "" with lines 2
plot [t=0:1] \
p3x*P1(t)+2*p4x*P2(t)+p5x*P3(t)+p6x*P4(t), \
p3y*P1(t)+2*p4y*P2(t)+p5y*P3(t)+p6y*P4(t) \
title "" with lines 2
Figure 4.13: The source for figure 4.12
TEX – LATEX – CS 594
4.4. INTRODUCTION
145
3
2.5
2
1.5
1
0.5
0
0
1
2
3
4
5
6
Figure 4.14: Two Bezier curves, joined together smoothly
Curve plotting with gnuplot.
The gnuplot utility can be used for
plotting sets of points. However, here we will only discuss drawing curves.
4.4
Introduction
The two modes for running gnuplot are interactive and from file. In interactive mode,
you call gnuplot from the command line, type commands, and watch output appear (see
next paragraph); in the second case you call gnuplot <your file>.
The output of gnuplot can be a picture on your screen, or drawing instructions in a file.
set terminal pdf
set xrange [0:6]
set yrange [0:3]
set parametric
B1(x) = x**3
B2(x) = 3*x**2*(1-x)
B3(x) = 3*x*(1-x)**2
B4(x) = (1-x)**3
P1x = .5 ; P1y = .2
P2x = 1.2 ; P2y = .4
P3x = 2.2 ; P3y = 1.3
P4x =
3 ; P4y = 1.2
P5x = 2*P4x-P3x
P5y = 2*P4y-P3y
P6x = 4.5 ; P6y = .2
P7x = 5
; P7y = 2.5
set multiplot
plot [t=0:1] \
P1x*B1(t)+P2x*B2(t)+P3x*B3(t)+P4x*B4(t), \
P1y*B1(t)+P2y*B2(t)+P3y*B3(t)+P4y*B4(t) \
title ""
plot [t=0:1] \
P4x*B1(t)+P5x*B2(t)+P6x*B3(t)+P7x*B4(t), \
P4y*B1(t)+P5y*B2(t)+P6y*B3(t)+P7y*B4(t) \
title ""
plot [t=-1:1] \
P4x+t*(P5x-P4x),P4y+t*(P5y-P4y) \
title "" with lines 2
plot [t=0:1] \
P1x+t*(P2x-P1x),P1y+t*(P2y-P1y) \
title "" with lines 3
plot [t=0:1] \
P7x+t*(P6x-P7x),P7y+t*(P6y-P7y) \
title "" with lines 3
Figure 4.15: The source for figure 4.14
Victor Eijkhout
146
CHAPTER 4. FONTS
Where the output goes depends on the setting of the terminal. By default, gnuplot will
try to draw a picture. This is equivalent to declaring
set terminal x11
or aqua, windows, or any choice of graphics hardware.
For output to file, declare
set terminal pdf
or fig, latex, pbm, et cetera.
4.5
Plotting
The basic plot command is plot. By specifying
plot x**2
you get a plot of f (x) = x2 ; gnuplot will decide on the range for x. With
set xrange [0:1]
plot 1-x title "down", x**2 title "up"
you get two graphs in one plot, with the x range limited to [0, 1], and the appropriate legends
for the graphs. The variable x is the default for plotting functions.
Plotting one function against another – or equivalently, plotting a parametric curve – goes
like this:
set parametric
plot [t=0:1.57] cos(t),sin(t)
which gives a quarter circle.
To get more than one graph in a plot, use the command set multiplot.
4.5.1
Styles
You can change the default drawing style with
set style function dots
(lines, dots, points, et cetera), or change on a single plot with
plot f(x) with points
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147
Figure 4.16: Line with slope ≤ 1 and one pixel per column on
Raster graphics.
4.6
Vector graphics and raster graphics
We may use fancy techniques such as Bezier curves for describing shapes, but at some
point the graphic needs to be rendered on an actual device with pixels or ink dots. Thus we
need algorithms for deciding which pixels to turn on or off, and, in case the device has a
larger bitdepth, with what intensity.
The technical terms here are ‘vector graphics’ for a description of the lines and curves, and
‘raster graphics’ for the pixel-by-pixel description. A description of a raster is also called
‘bitmap’. Common bitmap-based formats are JPEG, GIF, TIFF, PNG, PICT, and BMP.
Vector graphics make for a much more compact file, but they rely on the presence of a final
rendering stage. For instance, Macromedia’s Flash format (now an open Internet standard)
is a vector graphics format, that relies on a browser plugin. However, presence of a Flash
renderer is pretty much standard in browsers these days. Adobe Postscript is also a vector
format. The first popular printer to incorporate it, the Apple Laserwriter, had a Motorola
68000 processor, exactly as powerful as the Macintosh computer it connected to.
Two vector standards are being proposed to the W3C: the Precision Graphics Markup
Language and the Vector Markup Language. PGML is backed by Adobe Systems, IBM,
Netscape, and Sun Microsystems. VML is supported by Microsoft, Hewlett-Packard, Autodesk, Macromedia, and Visio. Both standards are based on Extensible Markup Language
(XML).
4.7
Basic raster graphics
4.7.1
Line drawing
We limit the discussion to lines with slope less than 1. For those, on a grid of pixels, one
pixel per column will be switched on, and the question to be addressed is which pixel to
turn on in each column.
Probably the simplest way to draw a line is by using an ‘incremental’ drawing algorithm.
Let a line y = mx + B be given, and we write the slope as m = δy/δx. In the case of pixel
graphics, we set δx ≡ 1, so δy = m and we can recursively state
yi+1 = yi + δy.
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Figure 4.17: The midpoint algorithm for line drawing
The simplest implementation of this is
let x0 , y0 and m be given, then
for i = 0 . . . n − 1
WritePixel(xi , Round(yi ))
xi+1 = xi + 1
yi+1 = yi + m
Since (xi , yi ) is never related to the actual formula for the line, there will be an accumulation of round-off error. However, this will be negligible. More seriously, the rounding
operation is relatively expensive. In the next algorithm we will eliminate it. If possible, we
want to operate completely in integer quantities.
The ‘midpoint algorithm’ proceeds fully by integer addition. First we write the equation
for the line in two different ways as
dy
y=
x + B,
F (x, y) = ax + by + c = 0.
dx
Clearly, a = dy, b = −dx, c = B, and we can derive dx, dy from the end pixels of the line.
The function F is zero on the line, positive in the half plane under it, and negative above it.
Now we consider how to progress given that we have switched on a pixel at (xp , yp ). With
the initial assumption that the line has a slope between 0 and 1, the next pixel will be either
(xP + 1, yP + 1), or (xp + 1, yp ), depending on which is closer to the line.
Instead of measuring the distance of the candidate next pixels to the line, we decide whether
their midpoint M is above or under the line. For this, we use the function F (·, ·) introduced
above, and evaluate the ‘decision value’ of the midpoint:
d = F (xp + 1, yp + 1/2).
The two cases to consider then are
d < 0: M lies over the line, so we take yp+1 = yp ;
d ≥ 0: M lies under the line, so we take yp+1 = yp + 1.
Similarly we update the mid point: if d ≥ 0, the midpoint moves up. Note that the new
midpoint is at xp+1 + 1.
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149
Now, we do not actually use the midpoint, only the value of d. The algorithm is then
complete once we find a way to update d cheaply. For this we look at its next value
d0 = F (xp+1 + 1, yp+1 + 1/2).
Corresponding to the above two cases:
d0 = a(xp+1 + 1) + b(yp+1 + 1/2) + c =
d < 0 : = a(xp + 2) + b(yp + 1/2)
= d + a = d + dy
d ≥ 0 : = a(xp + 2) + b(yp + 3/2) + c
= d + a + b = d + dy − dx
In other words, we update d with dy or dy − dx depending on whether it’s negative or
non-negative.
To start off the algorithm, dx and dy are computed from the endpoints, and the initial value
of d follows from
d0 = F (x0 + 1, y0 + 1/2) = F (x0 , y0 ) + a + b/2 = 0 + dy − dx/2.
To get rid of the division by 2, which would cause real rather than integer values to be
used throughout the algorithm, we can redefine F˜ (x, y) = 2F (x, y); correspondingly we
update d with 2dy and 2(dy − dx) in the two cases.
Exercise 44. Can you modify the DDA line drawing algorithm so that
it works (as well as possible) when the line is given between points that
are not on pixels?
These algorithms are sometimes referred to as ‘Digital Differential Analyzers’, since they
trace out a curve by proceeding with small differences. The line algorithm was first derived
by Bressenham.
4.7.2
Circle drawing
Circles can be drawn with a similar algorithm. We observe that, because of 8-fold symmetry, we can limit the algorithm to the part of a circle from x = 0 to x = y. The midpoint
argument is now slightly more complicated. The function for the circle is
F (x, y) = x2 + y 2 − R2 ,
and the decision value in the midpoint M is
d = F (xp + 1, yp + 1/2) = x2 + 2x + y 2 + y + 5/4.
The two cases to consider are similar to before:
d < 0: M lies in the circle, so we take yp+1 = yp ;
d ≥ 0: M lies outside the circle, so we take yp+1 = yp + 1.
To update the decision value we get the two cases
d0 = F (xp+1 + 1, yp+1 + 1/2) =
d < 0 : = x2 + 4x + y 2 + y + 4 1/4
= d + 2x + 3
d ≥ 0 : = x2 + 4x + y 2 + 3y + 6 1/4 = d + 2(x + y) + 5
Exercise 45. Why is there no need to consider bigger increments of yp
in the circle drawing algorithm? After all, a circle has curvature so the
slope increases.
The circle algorithm can be further improved by observing that the quantities 2x and 2y
can themselves easily by constructed by updating. This removes the need for any multiplication.
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Figure 4.18: The midpoint algorithm for circle drawing
4.7.3
Cubics
Suppose we have a cubic function f (t) = at3 + bt2 + ct + d. Instead of evaluating this
polynomial directly, using Horner’s rule, we compute the value f (t + δ) by updating:
f (t + δ) = f (t) + ∆f (t).
We find
∆f (t) = f (t + δ) − f (t)
= a(3t2 δ + 3tδ 2 + δ 3 ) + b(2tδ + δ 2 ) + cδ
= 3aδt2 + (3aδ 2 + 2bδ)t + aδ 3 + bδ 2 + cδ
This still leaves us with a quadratic function to evaluate, so we define
∆2 f (t) = ∆f (t + δ) − ∆f (t)
= 3aδ(2tδ + δ 2 ) + (3aδ 2 + 3bδ)δ
= 6aδ 2 t + 6aδ 3 + 2bδ 2
Finally, we derive ∆3 f (t) = ∆2 f (t + δ) − ∆2 f (t) = 6aδ 2 . Taken all together, we can
now compute fn+1 ≡ f ((n + 1)δ) by initializing
∆3 f0 = 6aδ 2 , ∆2 f0 = 6aδ 3 + 2bδ 2 , ∆f0 = aδ 3 + bδ 2 + cδ
and computing by update
fn+1 = fn + ∆fn , ∆fn+1 = ∆fn + ∆2 fn , ∆2 fn+1 = ∆2 fn + ∆3 f0
The advantage of this algorithm is its low operation count, and the fact that it works fully
by integer operations.
4.8
Rasterizing type
Typefaces can be described by curves, but several aspects to them make it necessary to do
more than just rendering these curves, when rasterizing them. Part of the problem is that
characters in a font are relatively small, and satisfy all sorts of constraints that both may be
hard to satisfy (especially at low resolution), and are immediately noticed when rendered
incorrectly.
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151
Figure 4.19: Problems in rasterizing type, and resulting illegible output
Such problems result from using too simple algorithms for converting the character outlines
to rasters. For instance, an obvious algorithm is
Figure 4.20: A bad and a good way of rendering a Times Roman ‘e’ at low resolution
A pixel is turned on if its center is within the curve.
Now consider the case where a curve with large radius exceeds location y = n + 1/2 for
only one x. This results in the ‘pimple’ on top of the ‘e’ in figure 4.20. On the other hand,
if such a curve stays just under such a halfpoint, we get a long plateau, as in the left side
curve of the ‘e’.
4.8.1
Scaled fonts
These rasterizing problems are largely due to the facts that
•
Characters are scalable, so the relations between top/bottom or left/right are not
always mapped the same way to a pixel grid;
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CHAPTER 4. FONTS
•
Even if characters are used at the same size, they need to be displayed on various
kinds of rasters (printer, screen);
Characters can be placed in any horizontal or vertical location, so relations to
pixel boundaries are also flexible.
•
The conversion process goes more or less as follows3 :
Figure 4.21: Scaled and rasterized character outline
•
Character outlines are based on coordinates on some grid, often expressed as fixed
point, or other scheme with a finite mesh size.
The character is scaled to the raster on which it will be rendered.
The result is rounded to the raster, in figure 4.21 this puts the left and right sides on
pixel boundaries; note that other vertical parts of the character are not necessarily
pixel-aligned.
The scaled and rounded outline is then rasterized by turning a set of pixels on.
•
•
•
We see that in both final steps, rounding to the raster, and switching on pixels, we can have
unwanted effects.
4.8.2
Pixelation
Above we said that pixels are switched on if their center falls within the curve. There are
two problems with this:
•
•
Sometimes not enough pixels are turned on to render the shape accurately, and
Deciding whether a pixel is within the shape is not trivial to begin with. For instance, letters such as ‘o’ or ‘p’ have an inner region that should not be filled. In
another example, sometimes it is convenient to describe a shape as non-disjoint
union of other shapes; see figure 4.22
The second problem can be solved a number of ways. We could for instance look at a scan
line, and switch to on/off mode every time we cross a curve boundary. This approach does
not work for intersecting curves.
3.
Much of this discussion is based on the definition of TrueType fonts.
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153
Figure 4.22: A shape consisting of overlapping contours
Figure 4.23: The effects of different winding number rules
Better solutions are based on looking at the so-called ‘winding number’. This number
counts, for a given point in the plane, how often a curve winds around the point. If this
is zero, the point is outside the curve, otherwise it is inside it.
That implementing winding number rules is not trivial can be seen from two screen shots
of Acrobat Reader version 4; figure 4.24.
4.8.3
Font hinting / instructing
To prevent some of the problems indicated above, characters in a font file consist of more
than just the outline description. Additionally, each character can have a short program in a
language defined by the font file format. Such a program can enforce that certain distances
in the font as exact multiples of pixel distances.
For instance, the letter ‘O’ in figure 4.25 has the following constraints
1.
A certain amount of white space buffering the character; distance 3 is the ‘transport’;
2.
The width of the band
5,6
Visual under and overshoot.
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CHAPTER 4. FONTS
Figure 4.24: Acrobat 4 rendering of a complicated figure at different magnifications
7
The height of the band
Distances 5 and 6 are over and undershoot: a letter with curved top/bottom like ‘O’ would
seem too small if it stayed between the baseline and the cap height. To compensate for that,
the letter is made slightly higher and deeper. However, for small sizes and low resolutions,
this compensation needs to be switched off, since it would look too drastic.
4.8.4
Dropouts
In certain cases, pixels can not be switched on that are necessary for continuity of the figure
drawn. Figure 4.26 shows a case where a connecting bit is too thin to hit the centers of any
pixels. To cover such cases, the renderer usually has an algorithm that detects when a scan
line enters and leaves a contour without setting any pixels. It will then set, for instance, the
left pixel.
4.9
Anti-aliasing
In the algorithms so far, the decision was between switching a pixel on or off. There are a
few problems with this. For instance, for certain slopes the rendered curve can have a very
‘jagged’ look. Also, a curve at slope 1 will
√ have the same number of pixels on as a curve at
slope 0, but on a path that is longer by 2. Thus, it may look lighter or thinner.
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155
Figure 4.25: Constraints on the letter ‘O’
If the display supports a range of values (‘grayscale’), we can try to use this and find a
better visual rendering. First we will look at an example, then briefly consider the general
theory of this phenomenon.
4.9.1
Raster graphics with larger bitdepths
In the algorithms above, pixels close to a line or other curve were switched on. If the display
supports it, we can compute a measurement of proximity of the pixel to the line, and set a
brightness based on that. There are various ways to implement this. For instance, one could
consider the line to have an area, and to compute the intersection area of the line and the
pixel box. Here we will use a ‘filter function’. The support of this function will be larger
than the pixel box.
We will modify the midpoint method for line drawing so that in each column three pixels
will be nonzero, with an intensity based on the Euclidean distance to the line. Let v be the
(signed) vertical
p distance from the midpoint to the line, and D the euclidean distance, then
D = vdx/ dx2 + dy 2 . The denominator we can compute once and for all at the start of
the algorithm, but a simple way of computing vdx is needed.
Consider the case where d < 0, in which case we choose yp+1 = yp . Now we have
0 = F (xp + 1, yp + v) = F (xp + 1, yp ) + 2bv ⇒ 2vdx = F (xp + 1, yp ),
and
d = F (M ) = F (xp + 1, yp + 1/2) = F (xp + 1, yp ) + b
so 2vdx = d + dx. Likewise, if d ≥ 0, 2vdx = d − dx. Since we know how to update d
cheaply, we can now iteratively compute D.
√
√
For the top and bottom point, D = 2(1−v)dx/ . . . and D = 2(1 +v)/ . . ., respectively.
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Figure 4.26: A case of ‘dropout’: missing pixels give a disconnected curve
4.9.2
The general idea
Smoothing out a picture with grayscales is usually called ‘anti-aliasing’. To see why, consider how the parts of a graphics system fit together. After we derive a curve or other
two-dimensional object (this could be a projected surface) to display, the exact values are
sampled according to the pixel resolution. By computing a Fourier transform of these samples, we get an impression of the visual ‘frequencies’ in the picture.
If, instead of simply switching on pixels based on the sampled curve, we compute pixel
values so that sampling them reproduces the frequency spectrum, we get a picture that
looks closer to the intended one.
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157
Projects for this chapter.
Project 4.1. Bezier curves can also be used for graphics problems such as enlarging a
bitmat or projecting text on a waving flag. Consult http://www.tinaja.
com/cubic01.asp and report on the techniques used.
Project 4.2. (very mathematical) Explain elliptical integrals and how they are used to compute the length of a Bezier curve. Explain approximations. Same resource as the
previous project.
Project 4.3. (very mathematical) Study the theory of NURBS (Non-Uniform Rational BSplines); what are their capabilities and limitations? Start at http://devworld.
apple.com/dev/techsupport/develop/issue25/schneider.html
for an introduction.
Project 4.4. Investigate perception issues in font design or display technology. Start by
browsing through Rubinstein’s book.
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TEX – LATEX – CS 594
Chapter 5
TEX’s macro language – unfinished chapter
The programming language of TEX is rather idiosyncratic. One notable feature is the difference between expanded and executed commands. The expansion mechanism is very
powerful: it is in fact possible to implement lambda calculus in it.
Handouts and further reading for this chapter
The inspiration for this chapter was the article about lists by Alan Jeffrey [8].
159
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CHAPTER 5. TEX’S MACRO LANGUAGE – UNFINISHED CHAPTER
Lambda calculus in TEX.
5.1
Logic with TEX
5.1.1
Truth values, operators
We start by defining a couple of simple tools.
\def\Ignore#1{}
\def\Identity#1{#1}
\def\First#1#2{#1}
\def\Second#1#2{#2}
For example:
Taking first argument:
input : \First {first}{second}
output : first
Taking second argument:
input : \Second {first}{second}
output : second
We define truth values:
\let\True=\First
\let\False=\Second
and logical operators:
\def\And#1#2{#1{#2}\False}
\def\Or#1#2{#1\True{#2}}
\def\Twiddle#1#2#3{#1{#3}{#2}}
\let\Not=\Twiddle
Explanation: And x y is y if x is true, false is x is false. Since True and False are defined as taking the first and second component, that gives the definition of And as above.
Likewise Or.
To test logical expressions, we attach TF to them before evaluting; that was \True TF
will print T, and \False TF will print F.
Let us test the truth values and operators:
True takes first of TF:
Not true is false:
input : \True
input : \Not \True
output : T
output : F
False takes second of TF:
And truth table TrueTrue:
input : \False
input : \And \True \True
output : F
output : T
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5.1. LOGIC WITH TEX
And truth table TrueFalse:
input : \And \True \False
output : F
And truth table FalseTrue:
input : \And \False \True
output : F
And truth table FalseFalse:
input : \And \False \False
output : F
Or truth table TrueTrue:
input : \Or \True \True
output : T
5.1.2
161
Or truth table TrueFalse:
input : \Or \True \False
output : T
Or truth table FalseTrue:
input : \Or \False \True
output : T
Or truth table FalseFalse:
input : \Or \False \False
output : F
Conditionals
Some more setup. We introduce conditionals
\def\gobblefalse\else\gobbletrue\fi#1#2{\fi#1}
\def\gobbletrue\fi#1#2{\fi#2}
\def\TeXIf#1#2{#1#2 \gobblefalse\else\gobbletrue\fi}
\def\IfIsPositive{\TeXIf{\ifnum0<}}
with the syntax
\TeXIf <test> <arg>
We test this:
Numerical test:
input : \IfIsPositive {3}
output : T
Numerical test:
input : \IfIsPositive {-2}
output : F
5.1.3
Lists
A list is defined as a construct with a head, which is an element, and a tail, which is another
list. We will denote the empty list by Nil.
\let\Nil=\False
We implement a list as an operator with two arguments:
•
If the list is not empty, the first argument is applied to the head, and the tail is
evaluated;
•
If the list is empty, the second argument is evaluated.
In other words
a2 if L = ()
L a1 a2 =
a1 (x) Y if L = (x, Y )
In the explanation so far, we only know the empty list Nil. Other lists are formed by taking
an element as head, and another list as tail. This operator is called Cons, and its result is a
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list. Since a list is a two argument operator, we have to make Cons itself a four argument
operator:
% \Cons <head> <tail> <arg1> <arg2>
\def\Cons#1#2#3#4{#3{#1}{#2}}
Since Cons#1#2 is a list, applied to #3#4 it should expand to the second clause of the
list definition, meaning it applies the first argument (#3) to the head (#1), and evaluates
the tail (#2).
The following definitions are typical for list operations: since a list is an operator, applying
an operation to a list means applying the list to some other objects.
\def\Error{{ERROR}}
\def\Head#1{#1\First\Error}
\def\Tail#1{#1\Second\Error}
Let us take some heads and tails of lists. As a convenient shorthand, a singleton is a list
with an empty tail:
\def\Singleton#1{\Cons{#1}\Nil}
Head of a singleton:
input : \Head {\Singleton \True }
output : T
Head of a tail of a 2-elt list:
input : \Head {\Tail {\Cons \True {\Singleton \False
}}}
output : F
We can also do arithmetic tests on list elements:
Test list content:
input : \IfIsPositive {\Head {\Singleton {3}}}
output : T
Test list content:
input : \IfIsPositive {\Head {\Tail {\Cons
{3}{\Singleton {-4}}}}}
output : F
Exercise 46.
Write a function \IsNil and test with
\test{Detect NIL}{\IsNil\Nil}
\test{Detect non-NIL}{\IsNil{\Singleton\Nil}}
5.1.3.1
A list visualization tool
If we are going to be working with lists, it will be a good idea to have a way to visualize
them. The following macros print a ‘1’ for each list element.
\def\Transcribe#1{#1\TranscribeHT\gobbletwo}
\def\TranscribeHT#1#2{1\Transcribe{#2}}
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5.1.3.2
163
List operations
Here are some functions for manipulating lists. We want a mechanism that takes a function f , an initial argument e, and a list X, so that
Apply f e X ⇒ f x1 (f x2 (. . . (f xn e) . . .))
% #1=function #2=initial arg #3=list
\def\ListApply#1#2#3{#3{\ListApplyp{#1}{#2}}{#2}}
\def\ListApplyp#1#2#3#4{#1{#3}{\ListApply{#1}{#2}{#4}}}
This can for instance be used to append two lists:
\def\Cat#1#2{\ListApply\Cons{#2}{#1}}
For example:
Cat two lists:
input : \Transcribe {\Cat {\Singleton \Nil }{\Cons
\Nil {\Singleton \Nil }}}
output : 111
From now on the \Transcribe macro will be implicitly assumed; it is no longer displayed in the examples.
5.1.4
Numbers
We can define integers in terms of lists: zero is the empty list, and to add one to a number
is to Cons it with an empty list as head element. In other words,
n + 1 ≡ (0, n).
This defines the ‘successor’ function on the integers.
\let\Zero\Nil
\def\AddOne#1{\Cons\Nil{#1}}
Examples:
Transcribe zero:
input : \Zero
output :
Transcribe one:
input : \AddOne \Zero
output : 1
Transcribe three:
input : \AddOne {\AddOne {\AddOne \Zero }}
output : 111
Writing this many \AddOnes get tiring after a while, so here is a useful macro:
\newtoks\dtoks\newcount\nn
\def\ndef#1#2{\nn=#2 \dtoks={\Zero}\nndef#1}
\def\nndef#1{
\ifnum\nn=0 \edef\tmp{\def\noexpand#1{\the\dtoks}}\tmp
\else \edef\tmp{\dtoks={\noexpand\AddOne{\the\dtoks}}}\tmp
\advance\nn by -1 \nndef#1
\fi}
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which allows us to write
\ndef\One1 \ndef\Two2 \ndef\Three3 \ndef\Four4 \ndef\Five5
\ndef\Seven7\ndef\Six6
et cetera.
It is somewhat surprising that, even though the only thing we can do is compose lists, the
predecessor function is just as computable as the successor:
\def\SubOne#1{#1\Second\Error}
Predecessor of two:
input : \SubOne {\AddOne {\AddOne \Zero }}
output : 1
(If we had used \Ignore instead of \Second a subtle TEXnicality would come into play:
the list tail would be inserted as {#2}, rather than #2, and you would see an Unexpected }
error message.)
Some simple arithmetic: we test if a number is odd or even.
\def\IsEven#1{#1\IsOddp\True}
\def\IsOddp#1#2{\IsOdd{#2}}
\def\IsOdd#1{#1\IsEvenp\False}
\def\IsEvenp#1#2{\IsEven{#2}}
Zero even?:
input : \IsEven \Zero
output : T
Zero odd?:
input : \IsOdd \Zero
output : F
Test even:
input : \IsEven {\AddOne
{\AddOne {\AddOne \Zero
}}}
output : F
Test odd:
input : \IsOdd {\AddOne
{\AddOne {\AddOne \Zero
}}}
output : T
Test even:
input : \IsEven {\AddOne
{\AddOne {\AddOne {\AddOne
{\Zero }}}}}
output : T
Test odd:
input : \IsOdd {\AddOne
{\AddOne {\AddOne {\AddOne
{\Zero }}}}}
output : F
Exercise 47. Write a test \IsOne that tests if a number is one.
Zero:
input : \IsOne \Zero
output : F
One:
input : \IsOne \One
output : T
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165
Two:
input : \IsOne \Two
output : F
5.1.4.1
Arithmetic: add, multiply
Above, we introduced list concatenation with \Cat. This is enough to do addition. To save
typing we will make macros \Three and such that stand for the usual string of \AddOne
compositions:
\let\Add=\Cat
Adding numbers:
input : \Add {\Three }{\Five }
output : 11111111
Instead of adding two numbers we can add a whole bunch
\def\AddTogether{\ListApply\Add\Zero}
For example:
Adding a list of numbers:
input : \AddTogether {\Cons \Two {\Singleton \Three
}}
output : 11111
Adding a list of numbers:
input : \AddTogether {\Cons \Two {\Cons \Three
{\Singleton \Three }}}
output : 11111111
This is one way to do multiplication: to evaluate 3 × 5 we make a list of 3 copies of the
number 5.
\def\Copies#1#2{#1{\ConsCopy{#2}}\Nil}
\def\ConsCopy#1#2#3{\Cons{#1}{\Copies{#3}{#1}}}
\def\Mult#1#2{\AddTogether{\Copies{#1}{#2}}}
Explanation:
•
If #1 of \Copies is empty, then Nil.
•
Else, \ConsCopy of #2 and the head and tail of #1.
•
The tail is one less than the original number, so \ConsCopy makes that many
copies, and conses the list to it.
For example:
Multiplication:
input : \Mult {\Three }{\Five }
output : 111111111111111
However, it is more elegant to define multiplication recursively.
\def\MultiplyBy#1#2{%
\IsOne{#1}{#2}{\Add{#2}{\MultiplyBy{\SubOne{#1}}{#2}}}}
Multiply by one:
input : \MultiplyBy \One \Five
output : 11111
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CHAPTER 5. TEX’S MACRO LANGUAGE – UNFINISHED CHAPTER
Multiply bigger:
input : \MultiplyBy \Three \Five
output : 111111111111111
5.1.4.2
More arithmetic: subtract, divide
The recursive definition
of subtraction is
m
if n = 0
m−n=
(m − 1) − (n − 1) otherwise
Exercise 48. Implement a function \Sub that can subtract two numbers. Example:
Subtraction:
input : \Sub \Three \Five
output : 11
5.1.4.3
Continuing the recursion
The same mechanism we used for defining multiplication from addition can be used to
define taking powers:
\def\ToThePower#1#2{%
\IsOne{#1}{#2}{%
\MultiplyBy{#2}{\ToThePower{\SubOne{#1}}{#2}}}}
Power taking:
input : \ToThePower {\Two }{\Three }
output : 111111111
5.1.4.4
Testing
Some arithmetic tests. Greater than: if
X = (x, X 0 ), Y = (y, Y 0 )
then Y > X is false if Y ≡ 0:
\def\GreaterThan#1#2{#2{\GreaterEqualp{#1}}\False}
Otherwise, compare X with Y 0 = Y − 1: Y > X ⇔ Y 0 ≥ X; this is true if X ≡ 0:
\def\GreaterEqualp#1#2#3{\GreaterEqual{#1}{#3}}
\def\GreaterEqual#1#2{#1{\LessThanp{#2}}\True}
Otherwise, compare X 0 = X − 1 with Y 0 = Y − 1:
\def\LessThanp#1#2#3{\GreaterThan{#3}{#1}}
Greater (true result):
input : \GreaterThan \Two
\Five
output : T
Greater (false result):
input : \GreaterThan \Three
\Two
output : F
Greater (equal case):
input : \GreaterThan \Two
\Two
TEX – LATEX – CS 594
5.1. LOGIC WITH TEX
output : F
Greater than zero:
input : \GreaterThan \Two
\Zero
output : F
167
Greater than zero:
input : \GreaterThan \Zero
\Two
output : T
Instead of just printing ‘true’ or ‘false’, we can use the test to select a number or action:
Use true result:
input : \GreaterThan \Two \Five \Three \One
output : 111
Use false result:
input : \GreaterThan \Three \Two \Three \One
output : 1
Let’s check if the predicate can be used with arithmetic.
3 < (5 − 1):
input : \GreaterThan \Three {\Sub \One \Five }
output : T
3 < (5 − 4):
input : \GreaterThan \Three {\Sub \Four \Five }
output : F
Equality:
\def\Equal#1#2{#2{\Equalp{#1}}{\IsZero{#1}}}
\def\Equalp#1#2#3{#1{\Equalx{#3}}{\IsOne{#2}}}
\def\Equalx#1#2#3{\Equal{#1}{#3}}
Equality, true:
input : \Equal
output : T
Equality, true:
input : \Equal
output : T
Equality, false:
input : \Equal
output : F
Equality, false:
input : \Equal
output : F
\Five \Five
\Four \Four
\Five \Four
(1 + 3) ≡ 5: false:
input : \Equal {\Add \One
\Three }\Five
output : F
(2 + 3) ≡ (7 − 2): true:
input : \Equal {\Add \Two
\Three }{\Sub \Two \Seven
}
output : T
\Four \Five
Fun application:
\def\Mod#1#2{%
\Equal{#1}{#2}\Zero
{\GreaterThan{#1}{#2}%
{\Mod{#1}{\Sub{#1}{#2}}}%
{#2}%
}}
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CHAPTER 5. TEX’S MACRO LANGUAGE – UNFINISHED CHAPTER
Mod(27, 4) = 3:
input : \Mod \Four \TwentySeven
output : 111
Mod(6, 3) = 0:
input : \Mod \Three \Six
output :
With the modulo operation we can compute greatest common divisors:
\def\GCD#1#2{%
\Equal{#1}{#2}%
{#1}%
{\GreaterThan{#1}{#2}%
{\IsOne{#1}\One
{\GCD{\Sub{#1}{#2}}{#1}}}%
{\IsOne{#2}\One
{\GCD{\Sub{#2}{#1}}{#2}}}}}
GCD(27,4)=1:
input : \GCD \TwentySeven \Four
output : 1
GCD(27,3)=3:
input : \GCD \TwentySeven \Three
output : 111
% #2>#1
% then take GCD(#2-#1,#1)
% else GCD(#1-#2,#2)
and we can search for multiples:
\def\DividesBy#1#2{\IsZero{\Mod{#1}{#2}}}
\def\NotDividesBy#1#2{\GreaterThan\Zero{\Mod{#1}{#2}}}
\def\FirstDividesByStarting#1#2{%
\DividesBy{#1}{#2}{#2}{\FirstDividesByFrom{#1}{#2}}}
\def\FirstDividesByFrom#1#2{\FirstDividesByStarting{#1}{\AddOne{#2}}}
5|25:
input : \DividesBy \Five \TwentyFive
output : T
5 6 |27:
input : \DividesBy \Five \TwentySeven
output : F
5 6 |27:
input : \NotDividesBy \Five \TwentySeven
output : T
10 = min{i : i ≥ 7 ∧ 5|i}:
input : \FirstDividesByFrom \Five \Seven
output : 1111111111
5.1.5
Infinite lists
So far, we have dealt with lists that are finite, built up from an empty list. However, we can
use infinite lists too.
TEX – LATEX – CS 594
5.1. LOGIC WITH TEX
169
\def\Stream#1{\Cons{#1}{\Stream{#1}}}
Infinite objects:
input : \Head {\Tail {\Stream 3}}
output : 3
Infinite objects:
input : \Head {\Tail {\Tail {\Tail {\Tail {\Tail
{\Stream 3}}}}}}
output : 3
Even though the list is infinite, we can easily handle it in finite time, because it is never
constructed further than we ask for it. This is called ‘lazy evaluation’.
We can get more interesting infinite lists by applying successive powers of an operator to
the list elements. Here is the definition of the integers by applying the AddOne operator a
number of times to zero:
% \StreamOp <operator> <initial value>
\def\StreamOp#1#2{\Cons{#2}{\StreamOp{#1}{#1{#2}}}}
\def\Integers{\StreamOp\AddOne\Zero}
\def\PositiveIntegers{\Tail\Integers}
\def\TwoOrMore{\Tail\PositiveIntegers}
Again, the Integers object is only formed as far as we need it:
Integers:
input : \Head {\Tail {\Integers }}
output : 1
Integers:
input : \Head {\Tail {\Tail {\Tail {\Tail {\Tail
{\Integers }}}}}}
output : 11111
Let us see if we can do interesting things with lists. We want to make a list out of everything
that satisfies some condition.
\def\ConsIf#1#2#3{#1{#2}{\Cons{#2}{#3}}{#3}}
\def\Doubles{\ListApply{\ConsIf{\DividesBy\Two}}\Nil\PositiveIntegers}
\def\AllSatisfy#1{\ListApply{\ConsIf{#1}}\Nil\PositiveIntegers}
\def\FirstSatisfy#1{\Head{\AllSatisfy{#1}}}
third multiple of two:
input : \Head {\Tail {\Tail \Doubles }}
output : 111111
old enough to drink:
input : \FirstSatisfy {\GreaterThan \TwentyOne }
output : 1111111111111111111111
We add the list in which we test as a parameter:
\def\AllSatisfyIn#1#2{\ListApply{\ConsIf{#1}}\Nil{#2}}
\def\FirstSatisfyIn#1#2{\Head{\AllSatisfyIn{#1}{#2}}}
:
input : \FirstSatisfyIn {\NotDividesBy
{\FirstSatisfyIn {\NotDividesBy \Two }\TwoOrMore
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CHAPTER 5. TEX’S MACRO LANGUAGE – UNFINISHED CHAPTER
}} {\AllSatisfyIn {\NotDividesBy \Two }\TwoOrMore
}
output : 11111
And now we can repeat this:
\def\FilteredList#1{\AllSatisfyIn{\NotDividesBy{\Head{#1}}}{\Tail{#1}}}
\def\NthPrime#1{\Head{\PrimesFromNth{#1}}}
\def\PrimesFromNth#1{\IsOne{#1}\TwoOrMore
{\FilteredList{\PrimesFromNth{\SubOne{#1}}}}}
Third prime; spelled out:
input : \Head {\FilteredList {\FilteredList
\TwoOrMore }}
output : 11111
Fifth prime:
input : \NthPrime \Four
output : 1111111
However, this code is horrendously inefficient. To get the 7th prime you can go make a cup
of coffee, one or two more and you can go pick the beans yourself.
%\def\FilteredList#1{\AllSatisfyIn{\NotDividesBy{\Head{#1}}}{\Tail{#1}}}
\def\xFilteredList#1#2{\AllSatisfyIn{\NotDividesBy{#1}}{#2}}
\def\FilteredList#1{\xFilteredList{\Head{#1}}{\Tail{#1}}}
Fifth prime:
input : \NthPrime \Five
output : 11111111111
TEX – LATEX – CS 594
Chapter 6
Character encoding
This chapter is about how to interpret the characters in an input file – no there ain’t such a
thing as a plain text file – and how the printed characters are encoded in a font.
Handouts and further reading for this chapter
There is very little printed material on this topic. A good introduction is http://www.
joelonsoftware.com/articles/Unicode.html; after that, http://www.cs.
tut.fi/˜jkorpela/chars.html is a good tutorial for general issues, and http:
//en.wikipedia.org/wiki/Unicode for Unicode.
For the technical details on Unicode consult http://www.unicode.org/. An introduction to ISO 8859: http://www.wordiq.com/definition/ISO_8859.
171
172
CHAPTER 6. CHARACTER ENCODING
Input file encoding.
6.1
History and context
6.1.1
One-byte character sets; Ascii
Somewhere in the depths of prehistory, people got to agree on a standard for character codes
under 127, A SCII. Unlike another encoding scheme, E BCDIC, it has a few nice properties.
•
•
•
•
All letters are consecutive, making a test ‘is this a letter’ easy to perform.
Uppercase and lowercase letters are at a distance of 32.
The first 31 codes, everything below the space character, as well as position 127,
are ‘unprintable’, and can be used for such purposes as terminal cursor control.
Unprintable codes are accessible through the control modifier (for this reason
they are also called ‘control codes’), which zeros bits 2 and 3: hit Ctrl-[ to get
Esc1 .
The ISO 646 standard codified 7-bit A SCII, but it left certain character positions (or ‘code
points’) open for national variation. For instance, British usage put a pound sign (£) in the
position of the dollar. The A SCII character set was originally accepted as ANSI X3.4 in
1968.
6.1.2
Code pages
This left the codes with the high bit set (‘extended A SCII’) undefined, and different manufacturers of computer equipment came up with their own way of filling them in. These
standards were called ‘code pages’, and IBM gave a standard numbering to them. For
instance, code page 437 is the MS-DOS code page with accented characters for most European languages, 862 is DOS in Israel, 737 is DOS for Greeks.
1.
The way key presses generate characters is typically controlled in software. This mapping from keyboard
scan codes to 7 or 8-bit characters is called a ‘keyboard’, and can be changed dynamically in most operating
systems.
TEX – LATEX – CS 594
6.1. HISTORY AND CONTEXT
173
Here is cp473:
MacRoman:
and Microsoft cp1252:
More code pages are displayed on http://aspell.net/charsets/codepages.
html. These diagrams can be generated from Unicode mapping tables, which look like
=20
=21
=22
U+0020
U+0021
U+0022
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SPACE
EXCLAMATION MARK
QUOTATION MARK
174
CHAPTER 6. CHARACTER ENCODING
...
=A3
U+00A3 POUND SIGN
=A4
U+20AC EURO SIGN
=A5
U+00A5 YEN SIGN
...
The international variants were standardized as ISO 646-DE (German), 646-DK (Danish),
et cetera. Originally, the dollar sign could still be replaced by the currency symbol, but after
a 1991 revision the dollar is now the only possibility.
6.1.3
ISO 8859
The different code pages were ultimately standardized as ISO 8859, with such popular code
pages as 8859-1 (‘Latin 1’) for western European,
8859-2 for eastern, and 8859-5 for Cyrillic.
These ISO standards explicitly left the first 32 extended positions undefined. Microsoft
code page 1252 uses ISO 8859-1.
More useful information about A SCII: http://jimprice.com/jim-asc.htm. History of A SCII out of telegraph codes: http://www.wps.com/projects/codes/
index.html. A history, paying attention to multilingual use: http://tronweb.super-nova.
co.jp/characcodehist.html History as written by the father of A SCII: Bob Bemer
http://www.bobbemer.com/HISTORY.HTM.
A good inventory of ISO 8859, Latin-1: http://www.cs.tut.fi/˜jkorpela/latin1/
index.html, with a discussion by logical grouping: http://www.cs.tut.fi/˜jkorpela/
latin1/4.html.
6.1.4
DBCS
Since certain Asian alphabets do not fit in 256 positions, a system called the ‘Double Byte
Character Set’ was invented where some characters were stored in one, others in two bytes.
TEX – LATEX – CS 594
6.2. UNICODE
175
This is very messy, since you can not simply write s++ or s-- to traverse a string. Instead
you have to use functions from some library that understands these encodings. This system
is now only of historical interest.
6.2
Unicode
The systems above functioned quite well as long as you stuck to one language or writing
system. Poor dictionary makers. More or less simultaneously two efforts started that aimed
to incorporate all the world’s character sets in one standard: Unicode standard (originally
2-byte), and ISO 10646 (oringally 4-byte). Unicode then was extended, so that it has all
numbers up to 10FFFFF, which is slightly over a million.
6.2.1
ISO 10646 and Unicode
Two international standards organizations, the Unicode Consortium and ISO/IEC JTC1/SC2,
started designing a universal standard that was to be a superset of all existing character sets.
These standards are now synchronized. Unicode has elements that are not in 10646, but
they are compatible where it concerns straight character encoding.
ISO 10646 defines UCS, the ‘Universal Character Set’. This is in essence a table of official
names and code numbers for characters. Unicode adds to this rules for hyphenation, bidirectional writing, and more.
The full Unicode list of code points can be found, broken down by blocks, online at http:
//www.fileformat.info/info/unicode/index.htm, or downloadable at http:
//www.unicode.org/charts/.
6.2.2
BMP and earlier standards
Characters in Unicode are mostly denoted hexadecimally as U+wxyz, for instance U+0041
is ‘Latin Capital Letter A’. The range U+0000–U+007F (0–127) is identical to US-ASCII
(ISO 646 IRV), and U+0000–U+00FF (0–255) is identical to Latin 1 (ISO 8859-1).
The original 2-byte subset is now called ‘BMP’ for Basic Multilingual Plane.
From http://www.hyperdictionary.com/:
BMP (Basic Multilingual Plane) The first plane defined in Unicode/ISO 10646, designed
to include all scripts in active modern use. The BMP currently includes the Latin,
Greek, Cyrillic, Devangari, hiragana, katakana, and Cherokee scripts, among others, and a large body of mathematical, APL-related, and other miscellaneous characters. Most of the Han ideographs in current use are present in the BMP, but due
to the large number of ideographs, many were placed in the Supplementary Ideographic Plane.
SIP (Supplementary Ideographic Plane) The third plane (plane 2) defined in Unicode/ISO
10646, designed to hold all the ideographs descended from Chinese writing (mainly
found in Vietnamese, Korean, Japanese and Chinese) that aren’t found in the Basic Multilingual Plane. The BMP was supposed to hold all ideographs in modern
Victor Eijkhout
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CHAPTER 6. CHARACTER ENCODING
use; unfortunately, many Chinese dialects (like Cantonese and Hong Kong Chinese) were overlooked; to write these, characters from the SIP are necessary. This
is one reason even non-academic software must support characters outside the
BMP.
6.2.3
Unicode encodings
Unicode is basically a numbered list of characters. When they are used in a file, their
numbers can be encoded in a number of ways. To name the obvious example: if only the
first 128 positions are used, the long Unicode code point can be truncated to just one byte.
Here are a few encodings:
UCS-2 Obsolete: this was the original ‘native’ two-byte encoding before Unicode was
extended.
UTF-32 Little used: this is a four-byte encoding. (UTF stands for ‘UCS Transformation
Format’.)
UTF-16 This is the BMP.
UTF-8 A one-byte scheme; details below.
UTF-7 Another one-byte scheme, but now the high bit is always off. Certain byte values
act as ‘escape’, so that higher values can be encoded. Like UTF-1 and SCSU, this
encoding is only of historical interest.
There is an important practical reason for UTF-8. Encodings such as UCS-2 are wasteful
of space, if only traditional A SCII is needed. Furthermore, they would break software that
is expecting to walk through a file with s++ and such. Also, they would introduce many
zero bytes in a file, which would play havoc with Unix software that uses null-termination
for strings. Then there would be the problem of whether two bytes are stored in low-endian
or high-endian order. For this reason it was suggested to store FE FF or FF FE at the
beginning of each file as the ‘Unicode Byte Order Mark’. Of course this plays havoc with
files such as shell scripts which expect to find #! at the beginning of the file.
6.2.4
UTF-8
UTF-8 is an encoding where the positions up to 127 are encoded ‘as such’; higher numbers
are encoded in groups of 2 to 6 bytes. (UTF-8 is standardized as RFC 3629.) In a multibyte group, the first byte is in the range 0xC0–0xFD (192–252). The next up to 5 bytes are
in the range 0x80–0xBF (128–191, bit pattern starting with 10). Note that 8 = 1000 and
B = 1011, so the highest two bits are always 10, leaving six bits for encoding).
U-00000000
U-00000080
U-00000800
U-00010000
U-00200000
U-04000000
-
U-0000007F
U-000007FF
U-0000FFFF
U-001FFFFF
U-03FFFFFF
U-7FFFFFFF
7 bits
11 = 5 + 6
16 = 4 + 2 × 6
21 = 3 + 3 × 6
26 = 2 + 4 × 6
31 = 1 + 5 × 6
0xxxxxxx
110xxxxx
1110xxxx
11110xxx
111110xx
1111110x
10xxxxxx
10xxxxxx 10xxxxxx
10xxxxxx (3 times)
10xxxxxx (4 times)
10xxxxxx (5 times)
All bites in a multi-byte sequence have their high bit set.
Exercise 49. Show that a UTF-8 parser will not miss more than two
characters if a byte becomes damaged (any number of bits arbitrarily
changed).
TEX – LATEX – CS 594
6.2. UNICODE
177
IETF documents such as RFC 2277 require support for this encoding in internet software.
Here is a good introduction to UTF-8 use in Unix: http://www.cl.cam.ac.uk/
˜mgk25/unicode.html. The history of it: http://www.cl.cam.ac.uk/˜mgk25/
ucs/utf-8-history.txt.
6.2.5
Unicode tidbits
6.2.5.1
Line breaking
See http://www.cs.tut.fi/˜jkorpela/unicode/linebr.html and http:
//www.unicode.org/reports/tr14/
6.2.5.2
Bi-directional writing
Most scripts are left-to-right, but Arabic and Hebrew run right-to-left. Characters in a file
are stored in ‘logical order’, and usually it is clear in which direction to render them, even
if they are used mixed. Letters have a ‘strong’ directionality: unless overridden, they will
be displayed in their natural direction. The first letter of a paragraph with strong direction
determines the main direction of that paragraph.
However, when differently directional texts are embedded, some explicit help is needed.
The problem arises with letters that have only weak directionality. The following is a sketch
of a problematic case.
Memory: he said ”I NEED WATER!”, and expired.
Display: he said ”RETAW DEEN I!”, and expired.
If the exclamation mark is to be part of the Arabic quotation, then the user can select the
text ‘I NEED WATER!’ and explicitly mark it as embedded Arabic (<RLE> is Right-Left
Embedding; <PDF> Pop Directional Format), which produces the following result:
Memory: he said ”<RLE>I NEED WATER!<PDF>”, and expired.
Display: he said ”!RETAW DEEN I”, and expired.
A simpler method of doing this is to place a Right Directional Mark <RLM> after the
exclamation mark. Since the exclamation mark is now not on a directional boundary, this
produces the correct result.
Memory: he said ”I NEED WATER!<RLM>”, and expired.
Display: he said ”!RETAW DEEN I”, and expired.
For the full definition, see http://www.unicode.org/reports/tr9/.
Victor Eijkhout
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6.2.6
CHAPTER 6. CHARACTER ENCODING
Unicode and oriental languages
‘Han unification’ is the Unicode strategy of saving space in the oriental languages (traditional Chinese, simplified Chinese, Japanese, Korean: ‘CJK’) by recognizing common
characters. This idea is not uncontroversial; see http://en.wikipedia.org/wiki/
Han_unification.
6.3
More about character sets and encodings
6.3.1
Character sets
Informally, the term ‘character set’ (also ‘character code’ or ‘code’) used to mean something like ‘a table of bytes, each with a character shape’. With only the English alphabet to
deal with that is a good enough definition. These days, much more general cases are handled, mapping one octet into several characters, or several octets into one character. The
definition has changed accordingly:
A ‘charset is a method of converting a sequence of octets into a sequence
of characters. This conversion may also optionally produce additional
control information such as directionality indicators.
(From RFC 2978) A conversion the other way may not exist, since different octet combinations may map to the same character. Another complicating factor is the possibility of
switching between character sets; for instance, ISO 2022-JP is the standard A SCII character
set, but the escape sequence ESC $ @ switches to JIS X 0208-1978.
6.3.2
From character to encoding in four easy steps
To disentangle the concepts behind encoding, we need to introduce a couple of levels:
ACR Abstract Character Repertoire: the set of characters to be encoded; for example,
some alphabet or symbol set. This is an unordered set of characters, which can be
fixed (the contents of ISO 8859-1), or open (the contents of Unicode).
CCS Coded Character Set: a mapping from an abstract character repertoire to a set of nonnegative integers. This is what is meant by ‘encoding’, ‘character set definition’,
or ‘code page’; the integer assigned to a character is its ‘code point’.
There used to be a drive towards unambiguous abstract character names across
repertoires and encodings, but Unicode ended this, as it provides (or aims to provide) more or less a complete list of every character on earth.
CEF Character Encoding Form: a mapping from a set of nonnegative integers that are
elements of a CCS to a set of sequences of particular code units. A ‘code unit’ is
an integer of a specific binary width, for instance 8 or 16 bits. A CEF then maps
the code points of a coded character set into sequences of code point, and these
sequences can be of different lengths inside one code page. For instance
•
A SCII uses a single 7-bit unit
•
UCS-2 uses a single 16-bit unit
•
DBCS uses two 8-bit units
•
UTF-8 uses one to four 8-bit units.
TEX – LATEX – CS 594
6.3. MORE ABOUT CHARACTER SETS AND ENCODINGS
179
•
UTF-16 uses a mix of one and two 16-bit code units.
CES Character Encoding Scheme: a reversible transformation from a set of sequences of
code units (from one or more CEFs to a serialized sequence of bytes. In cases
such as A SCII and UTF-8 this mapping is trivial. With UCS-2 there is a single
‘byte order mark’, after which the code units are trivially mapped to bytes. On the
other hand, ISO 2022, which uses escape sequences to switch between different
encodings, is a complicated CES.
Additionally, there are the concepts of
CM Character Map: a mapping from sequences of members of an abstract character repertoire to serialized sequences of bytes bridging all four levels in a single operation.
These maps are what gets assigned MIBenum values by IANA; see section 6.3.3.
TES Transfer Encoding Syntax: a reversible transform of encoded data. This data may
or may not contain textual data. Examples of a TES are base64, uuencode, and
quoted-printable, which all transform a byte stream to avoid certain values.
6.3.3
A bootstrapping problem
In order to know how to interpret a file, you need to know what character set it uses. This
problem also occurs in MIME mail encoding (section 6.3.5), which can use many character sets. Names and numbers for character sets are standardized by IANA: the Internet
Assigned Names Authority (http://www.iana.org/). However, in what character
set do you write this name down?
Fortunately, everyone agrees on (7-bit) A SCII, so that is what is used. A name can be up to
40 characters from us-ascii.
As an example, here is the iana definition of A SCII:
name
ANSI_X3.4-1968
reference RFC1345,KXS2
MIBenum 3
source ECMA registry
aliases iso-ir-6, ANSI_X3.4-1986, ISO_646.irv:1991, ASCII, ISO646-US,
US-ASCII (preferred MIME name), us, IBM367, cp367, csASCII
The MIBenum (Management Information Base) is a number assigned by IANA2 . The full
list of character sets is at http://www.iana.org/assignments/character-sets,
and RFC 3808 is a memo that describes the IANA Charset MIB.
6.3.4
Character codes in HTML
HTML can access unusual characters in several ways:
•
With a decimal numerical code:   is a space token. (HTML 4 supports
hexadecimal codes.)
•
With a vaguely symbolic name: © is the copyright symbol. See http:
//www.cs.tut.fi/˜jkorpela/HTML3.2/latin1.html for a list of
symbolic names in Latin-1.
2.
Apparently these numbers derive from the Printer MIB, RFC 1759.
Victor Eijkhout
180
•
CHAPTER 6. CHARACTER ENCODING
The more interesting way is to use an encoding such as UTF-8 (section 6.2.3) for
the file. For this it would be nice if the server could state that the file is
Content-type: text/html;charset=utf-8
but it is also all right if the file starts with
<META HTTP-EQUIV="Content-Type" CONTENT="text/html;charset=utf-8">
It is requirement that user agents can at least parse the charset parameter, which means
they have to understand us-ascii.
Open this link in your browser, and additionally view the source: http://www.unicode.
org/unicode/iuc10/x-utf8.html. How well does your software deal with it?
See also section 6.7.1.
6.3.5
Characters in email
6.3.6
FTP
FTP is a very old ARPA protocol. It knows ‘binary’ and ‘text’ mode, but the text mode is
not well defined. Some ftp programs adjust line ends; others, such as Fetch on the Mac,
actually do code page translation.
6.3.7
Character encodings in editors and programming languages
Software must be rewritten to use character encodings. Windows NT/2000/XP, including
Visual Basic, uses UCS-2 as native string type. Strings are declared of type wchar_t
instead of char, and the programmer uses wcslen instead of strlen, et cetera. A literal
string is created as L"Hello world".
6.4
Character issues in TEX / LATEX
6.4.1
Diacritics
Original TEX is not very good at dealing with diacritics. They are implemented as things to
put on top of characters, even when, as with the cedilla, they are under the letter. Furthermore, TEX can not hypenate a word with accents, since the accent introduces a space in the
word (technically: an explicit kern). Both problems were remedied to a large extent with
the ‘Cork font encoding’, which contains most accented letters as single characters. This
means that accents are correctly placed by design, and also that the word can be hyphenated, since the kern has disappeared.
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6.4. CHARACTER ISSUES IN TEX / LATEX
181
These fonts with accented characters became possible when TEX version 3 came out around
1990. This introduced full 8-bit compatibility, both in the input side and in the font addressing.
6.4.2
LATEX input file access to fonts
If an input file for LATEX is allowed to contain all 8-bit octets, we get all the problems of
compatibility that plagued regular text files. This is solved by the package inputenc:
\usepackage[code]{inputenc}
where codes is applemac, ansinew, or various other code pages.
This package makes all unprintable A SCII characters, plus the codes over 127, into active
characters. The definitions are then dynamically set depending on the code page that is
loaded.
6.4.3
LATEX output encoding
The inputenc package does not solve the whole problem of producing a certain font
character from certain keyboard input. It only mapped a byte value to the TEX command
for producing a character. To map such commands to actual code point in a font file, the
TEX and LATEX formats contain lines such as
\chardef\i="10
declaring that the dotless-i is at position 16. However, this position is a convention, and
other people – type manufacturers – may put it somewhere else.
This is handled by the ‘font encoding’ mechanism. The various people working on the
LATEX font schemes have devised a number of standard font encodings. For instance, the
OT1 encoding corresponds to the original 128-character set. The T1 encoding is a 256character extension thereof, which includes most accented characters for Latin alphabet
languages.
A font encoding is selected with
\usepackage[T1]{fontenc}
A font encoding definition contains lines such as
\DeclareTextSymbol{\AE}{OT1}{29}
\DeclareTextSymbol{\OE}{OT1}{30}
\DeclareTextSymbol{\O}{OT1}{31}
\DeclareTextSymbol{\ae}{OT1}{26}
\DeclareTextSymbol{\i}{OT1}{16}
6.4.4
Virtual fonts
Exercise 50. What does an ALT key do?
Exercise 51. What is E BCDIC? What is the basic idea?
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Exercise 52. Find the Unicode definition. Can you find an example of
a character that has two functions, but is not defined as two characters?
Find two characters that are defined seperately for compatibility, but that
are defined equivalent.
Exercise 53. ISO 8859 has the ‘non-breaking space’ at position A0.
How does TEX handle the nbsp? How do TEX, HTML, Latin-1, MS
Word, et cetera handle multiple spaces? Discuss the pros and cons.
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6.5. BASIC TERMINOLOGY
183
Figure 6.1: Different shapes of ‘lowercase roman a’
Font encoding.
6.5
Basic terminology
Terminology of fonts and typefaces is quite confused these days. Traditionally, a typeface
was a design, realized in number of fonts, that could be sorted in families, such as roman,
and italic. A font would then be a style (medium weight bold italic) in a particular size.
Somewhat surprisingly, once you start throwing computers at this problem, even talking
about characters becomes very subtle.
In Unicode, there are abstract characters and characters. They don’t differ by much: an
abstract character is a concept such as ‘Latin lowercase a with accent grave’, and a character
is that concept plus a position in the Unicode table. The actually visible representation of
a character is called a ‘glyph’. According to ISO 9541, a glyph is ‘A recognizable abstract
graphic symbol which is independent of any specific design’.
6.5.1
The difference between glyphs and characters
Often, the mapping between character and glyph is clear: we all know what we mean by
‘Uppercase Roman A’. However, there may be different glyph shapes that correspond to
the same character.
An abstract character is defined as
abstract character: a unit of information used for the organization, control, or representation of textual data.
This definition has some interesting consequences. Sometimes one glyph can correspond
to more than one character, and the other way around.
For example, in Danish, the ligature ‘æ’ is an actual character. On the other hand, the
ligature ‘fl’, which appears in English texts, is merely a typographical device to make the
combination ‘fl’ look better, so one glyph corresponds to two characters.
The opposite case is rarer. In Tamil, a certain character is split, because it is positioned
around other characters. It can then even happen that one of the split parts forms a ligature
with adjacent characters.
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Figure 6.2: The f-i ligature
Figure 6.3: A split character in Tamil
A tricker question is how to handle accented letters: is ‘´e’ one character or a combination
of two? In math, is the relation in a 6= b one symbol, or an overstrike of one over another?
Another problem with ligatures is that a single glyph needs to be displayed, but two glyphs
need to be stored to make searching for the string possible.
6.5.2
The identity of a character
Another problem in defining a character is whether two glyphs that look the same, or sometimes ever are the same, should be the same character. For example, uppercase Latin a,
uppercase Greek α, and uppercase Cyrillic a, are all rendered ‘A’. Still, in Unicode they are
three distinct characters.
Similarly, in A SCII, there are no separate glyphs for minus, hyphen, and dash. In Unicode,
these are three characters. Is the character ‘superscript 2 ’ a separate glyph, or a typographical variant of the character ‘digit 2’? The latter should be the logical solution, but for
compatibility reasons with other standards it is defined as a separate glyph. There are quite
a few of these ‘compatibility characters.
Yet another example is the Greek letter Ω, which can be that letter, or the sign for electrical resistance in physics texts. Unicode defines them as two characters, but with identical
glyphs.
A capital ‘A’ in Times Roman and in Helvetica are the same character, but what about
italic?
All these matters are hard to settle objectively: everything is a matter of definition, or a
judgement call. The official Unicode white paper on characters versus glyphs is http:
//www.unicode.org/reports/tr17/.
Here are some of the guidelines of the Unicode project:
•
The Unicode Standard encodes characters, not glyphs.
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6.6. ÆSTHETICS
185
Figure 6.4: Different interpretations of an accented character glyph
•
•
•
Characters have well-defined semantics.
The Unicode Standard encodes plain text.
And:
The Unicode Standard avoids duplicate encoding of characters by unifying them within scripts across languages; characters that are equivalent in form are given a single code. Common letters, punctuation
marks, symbols, and diacritics are given one code each, regardless of
language, [. . . ]
6.5.3
Diacritics
Unicode, a bit like TEX, has two ways of dealing with diacritics. It has precomposed accented characters, but it can also compose accented characters by listing accents (yes, plural: transliterated Vietnamese regularly has two accents over a letter one relating to vowel
quality, and one to tone) after the base symbol. This mechanism can also deal with languages such as Hangul (Korean) which have composite characters.
6.6
Æsthetics
6.6.1
Scaling versus Design sizes
Lots of attention is devoted to font scaling, with the implicit assumption that that is the
way to get a font to display at different sizes. This is only true to an extent: a small version
of a typeface was traditionally of a different design than the same typeface at larger sizes.
With metal type, independent designs for the different sizes were of course the only way
one could proceed at all, but with photo typesetters and computers the need went away,
and with it the realization that independent designs are visually actually a Good Thing.
Figure 6.5 shows the difference between a typeface set at its ‘design size’, and a scaled up
Figure 6.5: A typeface and a smaller version scaled up
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Figure 6.6: Adobe’s optical masters for a typeface
smaller version of it.
Adobe incorporated this idea in their Multiple Masters typefaces, which could interpolate
between different designs. This technology seems to have been abandoned, but Adobe’s
Originals now have so-called ‘Optical masters: four different designs of the same typeface,
to be set at different sizes. Adobe labels their purposes as ‘display’, ‘subhead’, ‘text’, and
‘caption’ in decreasing design size; see figure 6.6.
Apple developed their own version of multiple design sizes in TrueType GX, released in
1994. The ideas in TrueType GX are incorporated in Apple Advanced Typography (AAT)
in OS X, but there few AAT typefaces, and certainly very few non-Apple ones.
6.7
Font technologies
6.7.1
Unicode in fonts
It is unrealistic to expect any single font to support even a decent fraction of the Unicode
character repertoire. However, TrueType and OpenType do support Unicode.
The few fonts that support (almost) the whole of Unicode are called ‘pan-Unicode’. There
are only a few of those. However, software these days is pretty sophisticated in gathering together symbols from disparate fonts. Some browsers do this, prompting the user for
‘install on demand’ of fonts if necessary.
6.7.2
Type 1 and TrueType
Type 1 (‘Postscript fonts’) was the outline font format developed by Adobe that was adopted
by Apple in the mid 1980s. Since it was proprietary (Adobe had release the specifications
for Type 3 fonts, but not Type 1), Apple and Microsoft later developed TrueType.
With Type 1 fonts, information is stored in two files, one for shape data and one for hinting
and such. With TrueType, all information is in the one font file.
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6.7. FONT TECHNOLOGIES
6.7.2.1
187
Type1
Adobe Type 1 fonts are stored in two common formats, .pfa (PostScript Font ASCII) and
.pfb (PostScript Font Binary). These contain descriptions of the character shapes, with each
character being generated by a small program that calls on other small programs to compute
common parts of the characters in the font. In both cases, the character descriptions are
encrypted.
Before such a font can be used, it must be rendered into dots in a bitmap, either by the
PostScript interpreter, or by a specialized rendering engine, such as Adobe Type Manager,
which is used to generate low-resolution screen fonts on Apple Macintosh and on Microsoft
Windows systems.
The Type 1 outline files do not contain sufficient information for typesetting with the font,
because they have only limited metric data, and nothing about kerning (position adjustments of particular adjacent characters) or ligatures (replacement of adjacent characters by
a single character glyph, those for fi, ffi, fl, and ffl being most common in English typography).
This missing information is supplied in additional files, called .afm (Adobe Font Metric)
files. These are ASCII files with a well-defined easy-to-parse structure. Some font vendors,
such as Adobe, allow them to be freely distributed; others, such as Bitstream, consider them
to be restricted by a font license which must be purchased.
6.7.2.2
TrueType ⇔ Type1 conversion
Beware! There is no such thing as a one-to-one reversible conversion. There are several
problems:
The outlines are stored in different ways in both formats. In truetype, second-order Bezier
curves are used, and in type 1, third-order Bezier curves are employed. One second order
Bezier can be transformed into a third-order Bezier, but a third-order Bezier cannot be
transformed into one, two or seventeen second-order Beziers–approximations are in order
for that conversion. So, type 1 to truetype is problematic, right from the start. For truetype
to type 1, there is a snake in the grass, in the form of integer grid rounding (see below).
Both formats require all control points to be integers (whole numbers), falling in a grid.
Truetype uses a 2048x2048 grid, type 1 typically a 1000x1000 grid. For the truetype to type
1 direction, one could divide all grid values by two, but then what? Should 183.5 become
183 or 184? The type 1 to truetype direction is easier, at least from this point of view, as
we could multiply each grid coordinate by two, so no rounding loss would be involved.
However, in the truetype to type 1 direction, the rounding causes additional problems for
the new control points needed for the perfect third-order Bezier outlines mentioned above.
Placing ink on paper: the formats have different rules for placing ink on paper in case of
outlines that are nested or intersecting. These differences are not caught by many conversion programs. In most cases, the user should not worry about this—only rarely do we have
overlapping outlines (I was forced once to have them, for other reasons).
Complexity of the outlines: truetype permits more complex outlines, with more control
points. For example, I am sure you have all seen fonts made from scans of pictures of faces
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CHAPTER 6. CHARACTER ENCODING
of people. Typically, these outlines are beyond the type 1 limit, so this restriction makes
the truetype to type 1 conversion impossible for ultra complex fonts.
Encoding: truetype can work with a huge number of glyphs. There are truetype fonts for
Chinese and Japanese, for example. In type 1, the number of active glyphs is limited to
256. Again, for most Latin fonts, this is a non-issue.
The remarks about grid rounding also apply to all metrics, the bounding boxes, the character widths, the character spacing, the kerning, and so forth.
Finally, there is the hinting. This is handled very differently in both formats, with truetype
being more sophisticated this time. So, in truetype to type 1 conversions of professionally
(hand-hinted) fonts, a loss will occur. Luckily, 99% of the truetype fonts do not make use
of the fancy hinting possibilities of truetype, and so, one is often safe.
All this to tell people to steer away like the plague from format conversions. And a plea
to the font software community to develop one final format. My recommendation: get rid
of truetype, tinker with the type 1 format (well, tinker a lot). More about that ideal format
elsewhere.
6.7.2.3
Downsampling bitmaps
In principle, given adequate resolution, the screen preview quality of documents set in
bitmap fonts, and set in outline fonts, should be comparable, since the outline fonts have to
be rasterized dynamically anyway for use on a printer or a display screen.
Sadly, this is not the case with versions of Adobe Acrobat Reader, acroread, and Exchange,
acroexch (version 5.x or earlier); they do a poor job of downsampling high-resolution
bitmap fonts to low-resolution screen fonts. This is particularly inexcusable, inasmuch as
the co-founder, and CEO, of Adobe Systems, is the author of one of the earliest publications on the use of gray levels for font display: [ John E. Warnock, The display of characters
using gray level sample arrays, Computer Graphics, 14 (3), 302–307, July, 1980. ]
6.7.3
FreeType
FreeType is an Open Source implementation of TrueType. Unfortunately this runs into
patent problems, since Apple has patented some of the hinting mechanism. Recently FreeType
has acquired an automatic hinting engine.
6.7.4
OpenType
OpenType is a standard developed by Adobe and Microsoft. It combines
bitmap, outline, and metric information in a single cross-platform file. It has Unicode support, and can use ‘Optical Masters’ (section 6.6.1) multiple designs. It knows about the
distinction between code points and glyphs, so applications can render a character differently based on context.
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6.8. FONT HANDLING IN TEX AND LATEX
6.8
189
Font handling in TEX and LATEX
TEX has fairly sophisticated font handling, in the sense that it knows a lot about the characters in a font. However, its handling of typefaces and relations between fonts is primitive.
LATEX has a good mechanism for that.
6.8.1
TEX font handling
Font outlines can be stored in any number of ways; TEX is only concerned with the ‘font
metrics’, which are stored in a ‘tfm file’. These files contain
•
•
•
Global information about the font: the \fontdimen parameters, which describe
the spacing of the font, but also the x-height, and the slant-per-point, which describes the angle of italic and slanted fonts.
Dimensions and italic corrections of the characters.
Ligature and kerning programs.
We will look at these in slightly more detail.
6.8.1.1
Font dimensions
The tfm file specifies the natural amount of space, with stretch and shrink for a font, but
also a few properties related to the size and shape of letters. For instance, it contains the
x-height, which is the height of characters without ascenders and descenders. This is, for
instance, used for accents: TEX assumes that accents are at the right height for characters
as high as an ‘x’: for any others the accent is raised or lowered.
The ‘slant per point’ parameters is also for use in accents: it determines the horizontal offset
of a character.
6.8.1.2
Character dimensions
The height, width, and depth of a character is used to determine the size of the enclosing
boxes of words. A non-trivial character dimension is the ‘italic correction’. A tall italic
character will protrude from its bounding box (which apparently does not always bound).
The italic correction can be added to a subsequent space.
‘TEX has’ versus ‘TEX has’
6.8.1.3
Ligatures and kerning
The tfm file contains information that certain sequences of characters can be replaced by
another character. The intended use of this is to replace sequences such as fi or fl by ‘fi’
or ‘fl’.
Kerning is the horizontal spacing that can bring characters closer in certain combinations.
Compare
‘Von’ versus ‘Von’
Kerning programs are in the tfm file, not accessible to the user.
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6.8.2
CHAPTER 6. CHARACTER ENCODING
Font selection in LATEX
Font selection in LATEX (and TEX) was rather crude in the early versions. Commands such
as \bf and \it switched to boldface and italic respectively, but could not be combined to
give bold italic. The New Font Selection Scheme improved that situation considerably.
With NFSS, it becomes possible to make orthogonal combinations of the font family (roman, sans serif), series (medium, bold), and shape (upright, italic, small caps). A quick
switch back to the main document font is \textnormal or \normalfont.
6.8.2.1
Font families
It is not necessary for a typeface to have both serifed and serifless (sans serif) shapes.
Often, therefore, these shapes are taken from different, but visually compatible typefaces,
for instance combining Times New Roman with Helvetica. This is the combination that
results from
\usepackage{times}
Loading the package lucidabr instead, gives Lucida Bright and Lucida Sans.
The available font families are
roman using the command \textrm and the declaration \rmfamily.
sans serif using the command \textsf and the declaration \sffamily.
typewriter type using the command \texttt and the declaration \ttfamily. Typewriter type is usually a monospaced font – all characters of the same width – and
is useful for writing about LATEX or for giving code samples.
6.8.2.2
Font series: width and weight
The difference between normal and medium width, or normal and bold weight, can be
indicated with font series commands:
medium width/weight using the command \textmd and the declaration \mdseries.
bold using the command \textbf and the declaration \bfseries.
6.8.2.3
Font shape
The final parameter with which to classify fonts is their shape.
upright This is the default shape, explicitly available through \textup or \upshape.
italic and slanted These are often the same; they are available through \textit, \textsl,
and \itshape, \slshape.
small caps Here text is set through large and small capital letters; this shape is available
through \textsc and \scshape.
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6.9. THE FONTENC PACKAGE
191
Input and output encoding in LATEX.
6.9
The fontenc package
Traditionally, in TEX accented characters were handled with control characters, such as
in \’e. However, many keyboards – and this should be understood in a software sense –
are able to generate accented characters, and other non-latin characters, directly. Typically,
this uses octets with the high bit set.
As we have seen, the interpretation of these octets is not clear. In the absense of some
Unicode encoding, the best we can say is that it depends on the code page that was used.
This dependency could be solved by having the TEX software know, on installation, what
code page the system is using. While this may be feasible for one system, if the input files
are moved to a different system, they are no longer interpreted correctly. For this purpose
the inputenc package was developed.
An input encoding can be stated at the load of the package:
\usepackage[cp1252]{inputenc}
or input encodings can be set and switched later:
\inputencoding{latin1}
With this, a (part of a) file can be written on one machine, using some code page, and still
be formatted correctly on another machine, which natively has a different code page.
These code pages are all conventions for the interpretation of singly octets. The inputenc
package also has limited support for UTF-8, which is a variable length (up to four octets)
encoding of Unicode.
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Projects for this chapter.
Project 6.1. What is the problem with ‘Han unification’? (section 6.2.6) Discuss history,
philology, politics, and whatever may be appropriate.
Project 6.2. How do characters get into a file in the first place? Discuss keyboard scan
codes and such. How do modifier keys work? How can an OS switch between
different keyboard layouts? What do APIs for different input methods look like?
Project 6.3. Dig into history (find the archives of alt.folklore.computers!) and
write a history of character encoding, focusing on the pre-ascii years. Describe
design decisions made in various prehistoric computer architectures. Discuss.
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Chapter 7
Software engineering
In the course of writing TEX and Metafont, Knuth developed some interesting ideas about
software engineering. We will put those in the context of other writings on the subject. One
of the by-products of TEX is the Web system for ‘literate programming’. We will take a
look at that, as well as at the general idea of markup.
Handouts and further reading for this chapter
Knuth wrote a history of the TEX project in [10], reprinted in ‘Literate Programming’,
which is on reserve in the library, QA76.6.K644 1992 .
One of the classics of software engineering is Frederick Brooks’ ‘The Mythical ManMonth’ [6].
For software engineering research, consult the following journals:
•
Software practice and experience
•
Journal of systems and software
•
ACM Transactions on Software Engineering and Methodology
•
IEEE Transactions on Reliability
Some of these the library has available online.
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CHAPTER 7. SOFTWARE ENGINEERING
Literate programming.
7.1
The Web system
7.2
Knuth’s philosophy of program development
7.2.1
The Trip and Trap tests
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7.3. EXTREMELY BRIEF HISTORY OF TEX
Software engineering.
7.3
195
(Quotes by Knuth in this chapter taken from [10].)
Extremely brief history of TEX
Knuth wrote a first report on TEX in early 1977, and left it to graduate students Frank Liang
and Michael Plass to implement it over the summer. Starting from their prototype, he then
spent the rest of 1977 and early 1978 implementing TEX and producing fonts with the first
version of M ETAFONT.
TEX was used by Knuth himself in mid 1978 to typeset volume 2 of The Art of Computer
Programming; it was in general use by August of 1978. By early 1979, Knuth had written
a system called Doc that was the precursor of W EB, and that produced both documentation
and a portable Pascal version of the source; the original program was written in Sail.
In 1980 Knuth decided to rewrite TEX and M ETAFONT. He started on this in 1981, and
finished, including producing the five volumes of Computer and Typesetting, in 1985.
7.4
TEX’s development
7.4.1
Knuth’s ideas
Inspecting the work of his students, Knuth found that they had had to make many design
decisions, despite his earlier conviction to have produced ‘a reasonably complete specification of a language for typesetting’.
The designer of a new kind of system must participate fully in the implementation.
Debugging happened in about 18 days in March 1978. Knuth contrasts that with 41 days
for writing the program, making debugging about 30% of the total time, as opposed to
70% in his earlier experience. The whole code at that time was under 5000 statements. The
rewritten TEX82 runs to about 14 000 statements, in 1400 modules of W EB.
He considered this his first non-trivial program written using the structured programming
methodology of Dijkstra, Hoare, Dahl, and others. Because of the confidence this gave him
in the correctness of the program, he did not test TEX until both the whole program and the
fonts were in place. ‘I did not have to prepare dummy versions of non-existent modules
while testing modules that were already written’.
By mid 1979, Knuth was using TEX, and was improving TEX ‘at a regular rate of about one
enhancement for every six pages typed’.
Thus, the initial testing of a program should be done by the designer/implementor.
Triggered by a challenge of John McCarthy, Knuth wrote a manual, which forced him to
think about TEX as a whole, and which led to further improvements in the system.
The designer should also write the first user manual.
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‘If I had not participated fully in all these activities, literally hundreds of improvements
would never have been made, because I would never have thought of them or perceived
why they were important.’
Knuth remarks that testing a compiler by using it on a large, real, input typically leaves
many statements and cases unexecuted. He therefore proposes the ‘torture test’ approach.
This consists of writing input that is as far-fetched as possible, so that it will explore many
subtle interactions between parts of the compiler. He claims to have spent 200 hours writing
and maintaining the ‘trip test’; there is a similar ‘trap test’ for M ETAFONT.
7.4.2
Context
Software engineering is not an exact science. Therefore, some of Knuth’s ideas can be fit
in accepted practices, others are counter. In this section we will mention some schools of
thought in software engineering, and see how Knuth’s development of TEX fits in it.
7.4.2.1
Team work
The upshot of the development of TEX seems to be that software development is a one-man
affair. However, in industry, programming teams exist. Is the distinction between TEX and
commercial products then that between academic and real-world?
Knuth’s ideas are actually not that unusual. Programming productivity is not simply expressible as the product of people and time, witness the book ‘The Mythical man-Month’.
However, some software projects are too big, even for one really clever designer / programmer / tester / manual writer.
Dividing programming work is tricky, because of the interdependence of the parts. The
further you divide, the harder coordination becomes, and the greater the danger of conflicts.
Harlan Mills proposed that software should be written by groups, where each group works
like a surgical team: one chief surgeon who does all the real work, with a team to assist in
the more mundane tasks. Specifically:
•
•
•
•
•
•
•
•
The Chief Programmer designs the software, codes it, tests, it, and writes the
documentation.
The Co-Pilot is the Chief Programmer’s alter ego. He knows all the code but
writes none of it. He thinks about the design and discusses it with the Chief Programmer, and is therefore insurance against disaster.
The Administrator takes care of the mundane aspects of a programming project.
This can be a part-time position, shared between teams.
The Editor oversees production of documentation.
Two Secretaries, one each for the Administrator and Editor.
The Program Clerk is responsible for keeping records of all code and the test runs
with their inputs. This post is also necessary because all coding and testing will
be matter of public record.
The Toolsmith maintains the utilities used by the other team members.
The Tester writes the test cases.
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7.4. TEX’S DEVELOPMENT
197
•
The Language Lawyer investigates different constructs that can realize the Chief
Programmer’s algorithms.
With such teams of 10 people, coordination problems are divided by 10. For the overall
design there will be a system architect, or a small number of such people.
Recently, a development methodology name ‘Extreme Programming’ has become popular.
One aspect of this is pair programming: two programmers share one screen, one keyboard.
The advantage of this is that all code is immediately reviewed and discussed.
7.4.2.2
Top-down and bottom-up
Knuth clearly favours the top-down approach that was proposed by Nicklaus Wirth in ‘Program Development by Stepwise Refinement’ [20], and by Harlan Mills, who pioneered it
at IBM. The advantage of top-down programming is that the design of the system is set
from the beginning. The disadvantage is that it is hard to change the design, and testing
that shows inadequacies can only start relatively late in the development process.
Bottom-up programming starts from implementing the basic blocks of a code. The advantage is that they can immediately be tested; the disadvantage is that the design is in danger
of becoming more ad hoc.
An interesting form of bottom-up programming is ‘test-driven development’. Here, first a
test is written for a unit (a ‘unit test’), then the code. At all times, all tests need to be passed.
Rewriting code, usually to simplify it, with preservation of functionality as defined by the
tests, is known as ‘refactoring’.
7.4.2.3
Program correctness
The Trip test is an example of ‘regression testing’: after every change to the code, a batch
of tests is run to make sure that previous bugs do not reappear. This idea dates back to
Brooks; it is an essential part of Extreme Programming.
However, the Trip test only does regression testing of the whole code. TDD uses both
Unit tests and Integration tests. A unit is a specific piece of code that can easily be tested
since it has a clear interface. In testing a unit, the code structure can be used to design
the tests. Integration testing is usually done as Black Box testing: only the functionality
of an assemblage of units is known and tested, rather than the internal structure. One way
of doing integration testing is by ‘equivalence partitioning’: the input space is divided into
classes such that within each classes the input are equivalent in their behaviour. Generating
these classes, however, is heuristic, and it is possible to overlook cases.
On the opposite side of the testing spectrum is program proving. However, as Knuth wrote
in a memo to Peter van Emde Boas: ‘Beware of bugs in the above code; I have only proved
it correct, not tried it.’
Victor Eijkhout
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CHAPTER 7. SOFTWARE ENGINEERING
Markup.
7.5
History
The idea of markup has been invented several times. The ideas can be traced by to William
Tunnicliffe, chairman of the Graphic Communications Association (GCA) Composition
Committee, who presented a talk on the separation of information content of documents
from their format in 1967. He called this ‘generic coding’. Simultaneously, a New York
book designer named Stanley Rice was publishing articles about ”Standardized Editorial
Structures”, parameterized style macros based on the structural elements of publications.
7.5.1
Development of markup systems
Probably the first software incorporating these ideas comes out of IBM. Charles Goldfarb
recounts (http://www.sgmlsource.com/history/roots.htm) how in 1969 he
invented GML with Ed Mosher and Ray Lorie. They were tackling the problem of having a
document storage system, an editor, and a document printing system talk to each other, and
found that each was using different ‘procedural markup’ for its own purposes. Gradually
the idea grew to use markup for a logical description of the document. The project was
called ‘Integrated Text Processing’, and the first prototype ‘Integrated Textual Information
Management Experiment’: InTIME.
GML was officially published in 1973, and by 1980 an extension, SGML, was under development. This was published in 1986 as ISO 8879. Actually, SGML is a standard for
defining markup languages, rather than a language itself. Markup languages are defined in
SGML through ‘Document Type Definitions’ (DTDs).
The most famous application of SGML is HTML. However, HTML quickly began violating the separation of function and presentation that was the core idea of markup languages.
A renewed attempt was made to introduce a system for markup languages that truly defined content, not form, and this led to the ‘eXtensible Markup Language’. XHTML is a
realization of HTML as an XML ‘schema’. While SGML made some attempts at readability (and saving keystrokes for poor overworked typists), XML is aimed primarily at being
generated and understood by software, not by humans.
Another well-known application of SGML is ‘DocBook’. However, this has also been defined as an XML DTD1 , and this seems to be the current definition. DocBook is a good
illustration of the separation of content and presentation: there are XSL style sheets that
render DocBook files as Pdf, Rtf, HTML, or man pages.
1.
XML has both DTDs, which are SGML-like, and Schemas, which themselves XML. Schemas are the more
powerful mechanism, but also newer, so there are established DTDs that may not be redefined as Schemas. Furthermore, Schemas are more complicated to transmit and parse.
TEX – LATEX – CS 594
7.5. HISTORY
7.5.2
199
Typesetting with markup
In the early 1970s, nroff/troff was written at Bell Labs, at first in PDP assembler and targetting a specific photo typesetter for producing Unix documentation. Later it was recoded
in C, with device independent output. Various tasks such as tables and equations were hard
in nroff/troff, so preprocessors existed: eqn for formulas, tbl for tables, and refer for bibliographies.
Brian Reid’s thesis of 1980 descibed a markup system called Scribe. Scribe source files can
be compiled to several target languages. For instance, recent versions Scribe can compile
to LATEX, HTML, or man pages.
Victor Eijkhout
200
CHAPTER 7. SOFTWARE ENGINEERING
Projects for this chapter.
Project 7.1. Do a literature study of code/documentation development. Here are some
places to start:
POD Plain Old Documentation; used for Perl. http://www.perl.com/pub/
a/tchrist/litprog.html
JavaDoc http://java.sun.com/j2se/javadoc/
Doxygen http://www.stack.nl/˜dimitri/doxygen/
Fitnesse http://fitnesse.org/
Leo http://webpages.charter.net/edreamleo/front.html
What schools of thought are there about developing medium size codes such as
TEX? How does Knuth’s philosophy relate to the others?
Project 7.2. Compare the TeX ”way” to MS Word, PageMaker, FrameMaker, Lout, Griff,
previewLaTeX.
Project 7.3. TEX has been criticized for its arcane programming language. Would a more
traditional programming language work for the purpose of producing text output? Compare TEX to other systems, in particular lout, http://www.pytex.
org/, ant http://www-mgi.informatik.rwth-aachen.de/˜blume/
Download.html and write an essay on the possible approaches. Design a system of your own.
Project 7.4. TEX and HTML were designed primarily with output in mind. Later systems
(XML, DocBook) were designed so that output would be possible, but to formalize the structure of a document better. However, XML is impossible to write
by hand. What would be a way out? Give your thoughts about a better markup
system, conversion between one tool and another, et cetera.
Project 7.5. Several improvements on TEX and LATEX have been developed or are under
development. Investigate NTS, LaTeX3, Context, Lollipop TEX describe their
methodologies, and evaluate relative merits.
Project 7.6. Knuth has pretty liberal ideas about publishing software; somewhat against
the spirit of the times. Report on software patents, the difference between patents
and copyright, the state of affairs in the world. Read http://swpat.ffii.
org/gasnu/knuth/index.en.html
Project 7.7. Knuth devised the ‘torture test’ approach to program correctness. Report on
various schools of thought on this topic. Where does Knuth’s approach stand?
TEX – LATEX – CS 594
Bibliography
[1] Richard E. Bellman and Stuart E. Dreyfus. Applied Dynamic Programming. Princeton University Press, 1962.
[2] Carl de Boor. A Practical Guide to Splines, Revised Edition. Springer-Verlag, New
York, Berlin, Heidelberg, 2001.
[3] V. Eijkhout. An indentation scheme. TUGboat, 11:613–616.
[4] V. Eijkhout. A paragraph skip scheme. TUGboat, 11:616–619.
[5] V. Eijkhout and A. Lenstra. The document style designer as a separate entity. TUGboat, 12:31–34, 1991.
[6] jr Frederick P. Brooks. The Mythical Man-Month, essays on software engineering.
Addison-Wesley, 1995. Aniversary edition; originally published in 1975.
[7] Michael R. Garey and David S. Johnson. Computers and Intractibility, a guide to the
theory of NP-completeness. W.H. Freeman and company, San Francisco, 1979.
[8] A. Jeffrey. Lists in TEX’s mouth. TUGboat, 11:237–245, 1990.
[9] D.E. Knuth. Digital Typography.
[10] D.E. Knuth. The errors of TEX. Software Practice and Experience, 19:607–681.
[11] D.E. Knuth. TEX: the Program. Addison-Wesley, 1986.
[12] D.E. Knuth. The TEX book. Addison-Wesley, reprinted with corrections 1989.
[13] D.E. Knuth and M.F. Plass. Breaking paragraphs into lines. Software practice and
experience, 11:1119–1184, 1981.
[14] Donald E. Knuth. The Art of Computer Programming, Volume 3, Sorting and Searching. Addison Wesley Longman, 1998. Second edition.
[15] Helmut Kopka and Patrick W. Daly. A Guide to LATEX. Addison-Wesley, first published 1992.
[16] L. Lamport. LATEX, a Document Preparation System. Addison-Wesley, 1986.
[17] Frank Mittelbach, Michel Goossens, Johannes Braams, David Carlisle, and Chris
Rowley. The LATEX Companion, 2nd edition. Addison-Wesley, 2004.
[18] Michael F. Plass. Optimal Pagination Techniques for Automatic Typesetting Systems.
PhD thesis, 1981. also Xerox technical report ISL-81-1.
[19] Richard Rubinstein. Digital Typography. Addison-Wesley, 1988.
[20] Niklaus Wirth. Program development by stepwise refinement. Comm. ACM, 14:221–
227, 1971.
201
Index
&, 12
\˜, 15
˜, 15
accents, 187
active character, 6
\addcontentsline, 20
\addtocontents, 20
alphabet, 33
ambiguity, 50
analysis
lexical, 32
syntactical, 32
anti-aliasing, 154
Approximation, of curves, 129
ascii, 170
7-bit, 170
8-bit, 170
control codes, 170
extended, 170
unprintable, 170
associate storage, 77
automaton, 34
deterministic, 36
B-splines, 141
\backmatter, 8
baddness
of line breaks, 97
\bfseries, 14, 188
\Big, 18
\big, 18
bitmap, 145
BMP, 173
bold, 14
bottom-up parsing, 45
byte order mark, 177
202
charset, 176
code page, 170, 179
code point, 170, 176
code unit, 176
compatibility characters (Unicode),
182
control space, 6
control symbol, 6
CTAN, 25
cursive, 14
dangling else, 51
DBCS, 172
DDA, 147
demerit
of line breaking, 101
derivation, 41
design size, 183
divided difference, 131
DocBook, 196
DTD, 196
DVI file format, 8
ebcdic, 170
\emph, 14, 16
environment
abstract, 9
center, 9
flushleft, 9
flushright, 9
quotation, 9
quote, 9
tabular, 11
verbatim, 9, 10
verbatim*, 10
Equivalence partitioning, 195
Extreme programming, 195
INDEX
\fboxrule, 16
filter function, 153
Finite State Automaton, 34
Finite State Automaton, 32
Follow, 49
Font
Postscript, 184
TrueType, 184
Type 1, 184
Type 3, 184
font, 181
font encoding, 179
font metrics, 187
\footnote, 13
\footnotemark, 13
\footnotetext, 13
FreeType, 186
\frontmatter, 8
FSA, 32
FSA, see Finite State Automaton,
34
geometric continuity, 141
geometry matrix, 134
glyph, 181
GML, 196
Goldfarb, Charles, 196
grammar, 33
normal form, 42
handle, 46
Hash function, 78
Hermite interpolation, 129
\hline, 12
horizontal mode, 16
HTML, 196
IANA, 177
\include, 9
\includeonly, 9
incremental drawing, 145
\index, 20
\input, 9
inputenc package, 179
inserts, 116
Interpolation, 128
Victor Eijkhout
203
Hermite, 129
Lagrange, 129
ISO
10646, Unicode, 173
646, 7-bit ascii, 170
8859, Latin alphabets, 172
9541, 181
ISO 2022, 176
italic, 14
italic correction, 187
Item (in LR parser), 48
\itshape, 14, 188
label
after \item, 10
\label, 19
Lagrange interpolation, 129
language
context-free, 33
context-sensitive, 33
recursive, 33
regular, 33
language, formal, 33
lazy evaluation, 167
\leftmark, 22
leftmost derivation, 45
lexical analysis, 32
Linear Bounded automaton, 34
linear rehashing, 84
\linebreak, 15
list
description, 10
enumerate, 10
itemize, 10
\listoffigures, 17
\listoftables, 17
look ahead, 44
in FSAs, 40
LR parser, 47
macro, 4
\mainmatter, 8
\markboth, 22
\markright, 22
Markup
procedural, 196
204
INDEX
roman, 14
\mbox, 13, 15, 16
rule, 33
\mdseries, 14, 188
type of, 33
Multiple Master font technology,
184
\rule, 16
\newenvironment, 23
\newline, 15
\newpage, 15
\newtheorem, 23
NFA, see FSA, non-deterministic,
36
NFSS, 188
\nolinebreak, 15
non-deterministic FSA, 36
nonlinear rehashing, 84
\nonumber, 18
\nopagebreak, 15
\normalfont, 188
NP-hard, 112
scan code, 170
schema, 196
\scshape, 14, 188
\section*, 8
separate chaining, 80
\sffamily, 14, 188
SGML, 196
shift-reduce parsing, 45
shift/reduce conflict, 51
slanted, 14
\slshape, 14, 188
structured programming, 193
surgical team, 194
syntactical analysis, 32
open hashing, 80
OpenType, 186
operator grammar, 46
Optical masters, 184
OT1, 179
output routine, 117
T1, 179
\tableofcontents, 20
Test-driven development, 195
\texrm, 14
text
bold, 14
italic, 14
roman, 14
slanted, 14
\textbf, 14, 188
\textit, 14, 188
\textmd, 188
\textnormal, 188
\textrm, 188
\textsc, 188
\textsf, 188
\textsl, 14, 188
\texttt, 188
\textup, 188
tfm file, 187
\thanks, 13
\thispagestyle, 21
token, 32
tokenizer, 32
Torture test, 194
Trip test, 194
\pagebreak, 15
\pagestyle, 21
\par, 6, 16
parser, 32
parsing, 41
preamble, 7
\printindex, 20
production, 33
Pushdown automaton, 34
raster graphics, 145
Reduce/reduce conflict, 52
Refactoring, 195
Regression testing, 195
regular expressions, 33
regular expression, 35
\rightmark, 22
rightmost derivation, 45
\rmfamily, 14, 188
TEX – LATEX – CS 594
INDEX
\ttfamily, 14, 188
Turing machine, 34
typeface, 181
UCS, Universal Character Set,
173
underlining, 16
Unit test, 195
\upshape, 14, 188
vector graphics, 145
\verb, 10
\verb*, 10
\verbatiminput, 10
vertical mode, 16
XHTML, 196
XML, 196
yyerror, 69
Victor Eijkhout
205
