A System for Typesetting Mathematics USD:23-1
A System for Typesetting Mathematics
Brian W. Kernighan and Lorinda L. Cherry
AT&T Bell Laboratories
Murray Hill, New Jersey 07974
ABSTRACT
This paper describes the design and implementation
of a system for typesetting mathematics. The language
has been designed to be easy to learn and to use by
people (for example, secretaries and mathematical typ-
ists) who know neither mathematics nor typesetting.
Experience indicates that the language can be learned
in an hour or so, for it has few rules and fewer excep-
tions. For typical expressions, the size and font
changes, positioning, line drawing, and the like neces-
sary to print according to mathematical conventions are
all done automatically. For example, the input
sum from i=0 to infinity x sub i = pi over 2
produces
oo _̅i_̅
≥̅ xi=2
The syntax of the=language is specified by a small
context-free grammar; a compiler-compiler is used to
make a compiler that translates this language into
typesetting commands. Output may be produced on either
a phototypesetter or on a terminal with forward and
reverse half-line motions. The system interfaces
directly with text formatting programs, so mixtures of
text and mathematics may be handled simply.
This paper is a revision of a paper originally
published in CACM, March, 1975.
1. Introduction
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``Mathematics is known in (``Requires'' is perhaps the
the trade as difficult, or wrong word, but mathematics
penalty, copy because it is has its own typographical con-
slower, more difficult, and ventions which are quite dif-
more expensive to set in type ferent from those of ordinary
than any other kind of copy text.) Typesetting such an
normally occurring in books expression by traditional
and journals.'' [1] methods is still an essen-
tially manual operation.
One difficulty with
mathematical text is the mul- A second difficulty is
tiplicity of characters, the two dimensional character
sizes, and fonts. An expres- of mathematics, which the
sion such as superscript and limits in the
preceding example showed in
lim (tan x)sin 2x = 1
x->i̅i̅/2
its simplest form. This is
requires an intimate mixture
carried further by
of roman, italic and greek ______1______
_____2____
letters, in three sizes, and a a0+
a1+ ___3___
a2+a3+...
special character or two.
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and still further by
phototypesetter is a device
|
| _ _ which exposes a piece of pho-
|_______ log _|_a__m__-__|_b_
|2m\|ab \|aemx+\|b
| _ tographic paper or film, plac-
_____x_____ |______ tanh-1(_|_a_emx)
aemx-be-mx |m\|ab \|b
| _ ing characters wherever they
|______ coth-1(_|_a_emx)
|m\|ab \|b
| are wanted. The Graphic Sys-
These examples also show line- tems phototypesetter[2] on the
drawing, built-up characters UNIX operating system[3] works
like braces and radicals, and by shining light through a
a spectrum of positioning character stencil. The charac-
problems. (Section 6 shows ter is made the right size by
what a user has to type to lenses, and the light beam
produce these on our system.) directed by fiber optics to
the desired place on a piece
2. Photocomposition of photographic paper. The
exposed paper is developed and
Photocomposition tech-
typically used in some form of
niques can be used to solve
photo-offset reproduction.
some of the problems of
typesetting mathematics. A
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On UNIX, the photo- ``assembly language,'' by
typesetter is driven by a for- designing a language for
matting program called TROFF describing mathematical
[4]. TROFF was designed for expressions, and compiling it
setting running text. It also into TROFF.
provides all of the facilities
that one needs for doing 3. Language Design
mathematics, such as arbitrary
The fundamental principle
horizontal and vertical
upon which we based our
motions, line-drawing, size
language design is that the
changing, but the syntax for
language should be easy to use
describing these special
by people (for example, secre-
operations is difficult to
taries) who know neither
learn, and difficult even for
mathematics nor typesetting.
experienced users to type
This principle implies
correctly.
several things. First, ``nor-
For this reason we
mal'' mathematical conventions
decided to use TROFF as an
about operator precedence,
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operators, and the like. This
parentheses, and the like can-
keeps the language easy to
not be used, for to give spe-
learn and remember. Further-
cial meaning to such charac-
more, there should be few
ters means that the user has
exceptions to the rules that
to understand what he or she
do exist: if something works
is typing. Thus the language
in one situation, it should
should not assume, for
work everywhere. If a variable
instance, that parentheses are
can have a subscript, then a
always balanced, for they are
subscript can have a sub-
not in the half-open interval
script, and so on without
(a,b]. Nor should it assume
___ limit.
that that \|a+b can be
replaced by (a+b)1/2, or that Third, ``standard''
1/(1-x) is better written as things should happen automati-
___ (or vice versa).
1-x cally. Someone who types
Second, there should be ``x=y+z+1'' should get
relatively few rules, key- ``x=y+z+1''. Subscripts and
words, special symbols and superscripts should automati-
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cally be printed in an typed on a computer terminal
appropriately smaller size, much like an ordinary type-
with no special intervention. writer. This implies an input
Fraction bars have to be made alphabet of perhaps 100 char-
the right length and posi- acters, none of them special.
tioned at the right height.
A secondary, but still
And so on. Indeed a mechanism
important, goal in our design
for overriding default actions
was that the system should be
has to exist, but its applica-
easy to implement, since nei-
tion is the exception, not the
ther of the authors had any
rule.
desire to make a long-term
We assume that the typist project of it. Since our
has a reasonable picture (a design was not firm, it was
two-dimensional representa- also necessary that the pro-
tion) of the desired final gram be easy to change at any
form, as might be handwritten time.
by the author of a paper. We
To make the program easy
also assume that the input is
to build and to change, and to
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guarantee regularity (``it significant examples required
should work everywhere''), the perhaps a person-month. Since
language is defined by a then, we have spent a modest
context-free grammar, amount of additional time over
described in Section 5. The several years tuning, adding
compiler for the language was facilities, and occasionally
built using a compiler- changing the language as users
compiler. make criticisms and sugges-
tions.
A priori, the
grammar/compiler-compiler We also decided quite
approach seemed the right early that we would let TROFF
thing to do. Our subsequent do our work for us whenever
experience leads us to believe possible. TROFF is quite a
that any other course would powerful program, with a macro
have been folly. The original facility, text and arithmetic
language was designed in a few variables, numerical computa-
days. Construction of a work- tion and testing, and condi-
ing system sufficient to try tional branching. Thus we have
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been able to avoid writing a Since our program is only use-
lot of mundane but tricky ful for typesetting mathemat-
software. For example, we ics, it is necessary that it
store no text strings, but interface cleanly with the
simply pass them on to TROFF. underlying typesetting
Thus we avoid having to write language for the benefit of
a storage management package. users who want to set inter-
Furthermore, we have been able mingled mathematics and text
to isolate ourselves from most (the usual case). The standard
details of the particular dev- mode of operation is that when
ice and character set a document is typed, mathemat-
currently in use. For example, ical expressions are input as
we let TROFF compute the part of the text, but marked
widths of all strings of char- by user settable delimiters.
acters; we need know nothing The program reads this input
about them. and treats as comments those
things which are not mathemat-
A third design goal is
ics, simply passing them
special to our environment.
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through untouched. At the same as they are handed to the
time it converts the mathemat- typesetting program
ical input into the necessary (hereinafter called ``EQN''),
TROFF commands. The resulting except that we won't show the
ioutput is passed directly to delimiters that the user types
TROFF where the comments and to mark the beginning and end
the mathematical parts both of the expression. The inter-
become text and/or TROFF com- face between EQN and TROFF is
mands. described at the end of this
section.
4. The Language
As we said, typing
We will not try to x=y+z+1 should produce
describe the language pre- x=y+z+1, and indeed it does.
cisely here; interested Variables are made italic,
readers may refer to the operators and digits become
appendix for more details. roman, and normal spacings
Throughout this section, we between letters and operators
will write expressions exactly are altered slightly to give a
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more pleasing appearance. may be typed as many short
Input is free-form.
lines.
Spaces and new lines in the
input are used by EQN to
separate pieces of the input; Extra white space can be
they are not used to create
space in the output. Thus forced into the output by
x = y
several characters of various
+ z + 1
also gives x=y+z+1. Free-form
sizes. A tilde ``~'' gives a
input is easier to type ini-
space equal to the normal word
tially; subsequent editing is
spacing in text; a circumflex
also easier, for an expression
gives half this much, and a
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tab charcter spaces to the Here spaces are necessary in
next tab stop. the input to indicate that
sin, pi, int, and omega are
Spaces (or tildes, etc.)
special, and potentially worth
also serve to delimit pieces
special treatment. EQN looks
of the input. For example, to
up each such string of charac-
get
ters in a table, and if
f(t)=2i̅i̅sin(wt)dt
we write
appropriate gives it a trans-
f(t) = 2 pi int sin ( omega t )dlation. In this case, pi and
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omega become their greek roman wherever found.
equivalents, int becomes the
Fractions are specified
integral sign (which must be
with the keyword over:
moved down and enlarged so it
a+b over c+d+e = 1
looks ``right''), and sin is
produces
made roman, following conven-
__±__=1
c+d+e
tional mathematical practice.
Similarly, subscripts and
Parentheses, digits and opera-
superscripts are introduced by
tors are automatically made
the keywords sub and sup:
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x2+y2=z2 is automatic. Multiple levels
is produced by
of subscripts or superscripts
x sup 2 + y sup 2 = z sup 2
are of course allowed:
The spaces after the 2's are
``xsupysupz'' is xyz. The con-
necessary to mark the end of
struct ``something sub some-
the superscripts; similarly
thing sup something'' is
the keyword sup has to be
recognized as a special case,
marked off by spaces or some 2
so ``x sub i sup 2'' is xi
equivalent delimiter. The instead of xi2.
return to the proper baseline
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More complicated expres- they indicate unambiguously
sions can now be formed with what goes over what on the
these primitives: left-hand side of the expres-
sion. The language defines the
`__f_=_2_+_2_
`x2 a2 b2
is produced by
precedence of sup to be higher
{partial sup 2 f} over {partial x sup 2} =
than that of over, so no
x sup 2 over a sup 2 + y sup 2 over b sup 2
braces are needed to get the
Braces {} are used to group correct association on the
objects together; in this case right side. Braces can always
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be used when in doubt about in braces can also occur in
precedence. that context.
The braces convention is There is a sqrt operator
an example of the power of for making square roots of the
using a recursive grammar to appropriate size: ``sqrt a+b''
___
define the language. It is produces \|a+b, and
part of the language that if a
x = {-b +- sqrt{b sup 2 -4ac}} over 2a
construct can appear in some
is
context, then any expression
______
_b__\__b__-__a__c
x= 2a
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Since large radicals look poor
sum from i=0 to inf x sub i -> 0
on our typesetter, sqrt is not
Centering and making the ≥̅ big
useful for tall expressions.
enough and the limits smaller
Limits on summations,
are all automatic. The from
integrals and similar con-
and to parts are both
structions are specified with
optional, and the central part
the keywords from and to. To
(e.g., the ≥̅) can in fact be
get
anything:
oo
≥̅ xi->0
i=0
we need only type
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of the right height, using the
lim from {x -> pi /2} ( tan~x) = inf
keywords left and right:
is
left [ x+y over 2a right ]~=~1
lim (tan x)=oo
x->i̅i̅/2
Again, the braces indicate
makes
just what goes into the from
|_±_|
|2a | = 1
part.
A left need not have a
There is a facility for corresponding right, as we
making braces, brackets, shall see in the next example.
parentheses, and vertical bars Any characters may follow left
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and right, but generally only
we can type
various parentheses and bars
sign (x) ~==~ left {
are meaningful.
rpile {1 above 0 above -1}
Big brackets, etc., are
~~lpile {if above if above if}
often used with another facil-
~~lpile {x>0 above x=0 above x<0}
ity, called piles, which make
The construction ``left {''
vertical piles of objects. For
makes a left brace big enough
example, to get
to enclose the ``rpile
|
| 1
sign(x) _ | 0 if
|-1 if x>0
| if x=0 {...}'', which is a right-
x<0
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justified pile of ``above ... Although EQN makes a
above ...''. ``lpile'' makes a valiant attempt to use the
left-justified pile. There are right sizes and fonts, there
also centered piles. Because are times when the default
of the recursive language assumptions are simply not
definition, a pile can contain what is wanted. For instance
any number of elements; any the italic sign in the previ-
element of a pile can of ous example would convention-
course contain piles. ally be in roman. Slides and
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transparencies often require is 1/72 inch; this paper is
larger characters than normal set in 9 point type.)
text. Thus we also provide
If necessary, an input
size and font changing com-
string can be quoted in "...",
mands: ``size 12 bold
which turns off grammatical
{A~x~=~y}'' will produce
significance, and any font or
A x = y. Size is followed by a
spacing changes that might
number representing a charac-
otherwise be done on it. Thus
ter size in points. (One point
we can say
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lim~ roman "sup" ~x sub n = 0 x dot under + x hat + y tilde
+ X hat + Y dotdot = z+Z bar
to ensure that the supremum
doesn't become a superscript:
There are also facilities
lim sup xn=0 for globally changing default
Diacritical marks, long a
sizes and fonts, for example
problem in traditional
for making viewgraphs or for
typesetting, are straightfor-
setting chemical equations.
ward:
The language allows for
. .. ___
_+^+y̅+^+Y =z+Z
matrices, and for lining up
is made by typing
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equations at the same horizon- expression will be expanded
tal position. into whatever was inside the
double quotes in its defini-
Finally, there is a
tion. This lets users tailor
definition facility, so a user
the language to their own
can say
specifications, for it is
define name "..."
quite possible to redefine
at any time in the document;
keywords like sup or over.
henceforth, any occurrence of
Section 6 shows an example of
the token ``name'' in an
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definitions. the sequence ``.EQ'' to mark
the beginning of an equation
The EQN preprocessor
and ``.EN'' to mark the end.
reads intermixed text and
The ``.EQ'' and ``.EN'' are
equations, and passes its out-
passed through to TROFF
put to TROFF. Since TROFF uses
untouched, so they can also be
lines beginning with a period
used by a knowledgeable user
as control words (e.g.,
to center equations, number
``.ce'' means ``center the
them automatically, etc. By
next output line''), EQN uses
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default, however, ``.EQ'' and input:
``.EN'' are simply ignored by
.ce
TROFF, so by default equations
.EQ
are printed in-line.
x sub i = y sub i ...
``.EQ'' and ``.EN'' can .EN
be supplemented by TROFF com-
Since it is tedious to
mands as desired; for example,
type ``.EQ'' and ``.EN''
a centered display equation
around very short expressions
can be produced with the
(single letters, for
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instance), the user can also
Let #x sub i#, #y# and #alpha# be positive
define two characters to serve
produces:
as the left and right delim-
Let xi, y and ( be positive
iters of expressions. These
Running a preprocessor is
characters are recognized any-
strikingly easy on UNIX. To
where in subsequent text. For
typeset text stored in file
example if the left and right
``f'', one issues the command:
delimiters have both been set
to ``#'', the input: eqn f | troff
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The vertical bar connects the various ways. For example,
output of one process (EQN) to something with a subscript is
the input of another (TROFF). just a box followed by another
box moved downward and shrunk
5. Language Theory
by an appropriate amount. A
The basic structure of
fraction is just a box cen-
the language is not a particu-
tered above another box, at
larly original one. Equations
the right altitude, with a
are pictured as a set of
line of correct length drawn
``boxes,'' pieced together in
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between them. guarantee that some keyword is
recognized early enough in the
The grammar for the
parsing process. Symbols in
language is shown below. For
capital letters are terminal
purposes of exposition, we
symbols; lower case symbols
have collapsed some produc-
are non-terminals, i.e., syn-
tions. In the original gram-
tactic categories. The verti-
mar, there are about 70 pro-
cal bar | indicates an alter-
ductions, but many of these
native; the brackets [ ] indi-
are simple ones used only to
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cate optional material. A TEXT
eqn: box | eqn box
is a string of non-blank char-
box: text
acters or any string inside
| { eqn }
double quotes; the other ter-
| box OVER box
minal symbols represent
| SQRT box
literal occurrences of the
| box SUB box | box SUP box
corresponding keyword.
| [ L | C | R ]PILE { list }
| LEFT text eqn [ RIGHT text ]
| box [ FROM box ] [ TO box ]
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| SIZE text box observation that something can
| [ROMAN | BOLD | ITALIC] box be replaced by a more compli-
| box [HAT | BAR | DOT | DOTDOTcatedLDsomething in braces is
| DEFINE text text implicit in the productions:
list: eqn | list ABOVE eqn
eqn : box | eqn box
text: TEXT
box : text | { eqn }
The grammar makes it Anywhere a single character
obvious why there are few could be used, any legal con-
exceptions. For example, the struction can be used.
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Clearly, our grammar is
a over {b over c} ?
highly ambiguous. What, for
To answer questions like
instance, do we do with the
this, the grammar is supple-
input
mented with a small set of
a over b over c ?
rules that describe the pre-
Is it
cedence and associativity of
{a over b} over c operators. In particular, we
specify (more or less arbi-
or is it
trarily) that over associates
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to the left, so the first resolve the ambiguity in a
alternative above is the one construction like
chosen. On the other hand, sub
a sup 2 over b
and sup bind to the right,
We define sup to have a higher
because this is closer to
precedence than over, so this
standard mathematical prac-
_2_
construction is parsed as b
tice. That is, we assume xab _2
b.
instead of a
is x(ab), not (xa)b.
Naturally, a user can
The precedence rules
always force a particular
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parsing by placing braces resolving ambiguity. Instead
around expressions. the supplemental information
about precedence and associa-
The ambiguous grammar
tivity (also small enough to
approach seems to be quite
be understood) provides the
useful. The grammar we use is
compiler-compiler with the
small enough to be easily
information it needs to make a
understood, for it contains
fast, deterministic parser for
none of the productions that
the specific language we want.
would be normally used for
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When the language is supple- mands are output. For example,
mented by the disambiguating when the lexical analyzer
rules, it is in fact LR(1) and reports that it has found a
thus easy to parse[5]. TEXT (i.e., a string of con-
tiguous characters), we have
The output code is gen-
recognized the production:
erated as the input is
scanned. Any time a production text : TEXT
of the grammar is recognized,
The translation of this is
(potentially) some TROFF com-
simple. We generate a local
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name for the string, then hand the production
the name and the string to
box : box OVER box
TROFF, and let TROFF perform
is:
the storage management. All we
save is the name of the
string, its height, and its
baseline.
As another example, the
translation associated with
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draw bottom box (i.e., copy string for bottom box);
Width of output box =
move up; move left enough to center top box;
slightly more than largest input width
draw top box (i.e., copy string for top box);
Height of output box =
move down and left; draw line full width;
slightly more than sum of input heights
return to proper base line.
Base of output box =
slightly more than height of botMostiofutheoother productions
String describing output box = have equally simple semantic
move down; actions. Picturing the output
move right enough to center bottas bax;set of properly placed
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boxes makes the right sequence one of our users suggested a
of positioning commands quite TENSOR operator, to make con-
obvious. The main difficulty structions like
is in finding the right kj
lT
mni
Grammatically, this is easy:
numbers to use for estheti-
it is sufficient to add a pro-
cally pleasing positioning.
duction like
With a grammar, it is
box : TENSOR { list }
usually clear how to extend
Semantically, we need only
the language. For instance,
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juggle the boxes to the right The first question is
places. easily addressed. This entire
paper has been set by the pro-
6. Experience
gram. Readers can judge for
There are really three
themselves whether it is good
aspects of interest-how well
enough for their purposes. One
EQN sets mathematics, how well
of our users commented that
it satisfies its goal of being
although the output is not as
``easy to use,'' and how easy
good as the best hand-set
it was to build.
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material, it is still better tages.
than average, and much better
Some of the deficiencies
than the worst. In any case,
in the output could be cleaned
who cares? Printed books can-
up with more work on our part.
not compete with the birds and
For example, we sometimes
flowers of illuminated
leave too much space between a
manuscripts on esthetic
roman letter and an italic
grounds, either, but they have
one. If we were willing to
some clear economic advan-
keep track of the fonts
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involved, we could do this As to ease of use, at the
better more of the time. time of writing, the system
has been used by two distinct
Some other weaknesses are
groups. One user population
inherent in our output device.
consists of mathematicians,
It is hard, for instance, to
chemists, physicists, and com-
draw a line of an arbitrary
puter scientists. Their typi-
length without getting a per-
cal reaction has been some-
ceptible overstrike at one
thing like:
end.
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(1) It's easy to write, The learning time is
although I make the fol- short. A few minutes gives the
lowing mistakes... general flavor, and typing a
page or two of a paper gen-
(2) How do I do...?
erally uncovers most of the
(3) It botches the following
misconceptions about how it
things.... Why don't you
works.
fix them?
The second user group is
(4) You really need the fol-
much larger, the secretaries
lowing features...
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and mathematical typists who critical of the esthetics of
were the original target of their output than users
the system. They tend to be trained in mathematics. After
enthusiastic converts. They a transition period, most find
find the language easy to using a computer more
learn (most are largely self- interesting than a regular
taught), and have little trou- typewriter.
ble producing the output they
The main difficulty that
want. They are of course less
users have seems to be
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remembering that a blank is a instead of
delimiter; even experienced
f(xi)
Since the EQN language knows
users use blanks where they
no mathematics, it cannot
shouldn't and omit them when
deduce that the right
they are needed. A common
parenthesis is not part of the
instance is typing
subscript.
f(x sub i)
The language is somewhat
which produces
prolix, but this doesn't seem
f(xi)
excessive considering how much
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is being done, and it is cer-
a sub 0 + b sub 1 over
tainly more compact than the
{a sub 1 + b sub 2 over
corresponding TROFF commands.
{a sub 2 + b sub 3 over
For example, here is the
{a sub 3 + ... }}}
source for the continued frac-
This is the input for the
tion expression in Section 1
large integral of Section 1;
of this paper:
notice the use of definitions:
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USD:23-44 A System for Typesetting Mathematics
1 over mab ~ tanh sup -1 ( sa over sb emx )
define emx "{e sup mx}"
above
define mab "{m sqrt ab}"
-1 over mab ~ coth sup -1 ( sa over sb emx )
define sa "{sqrt a}"
}
define sb "{sqrt b}"
int dx over {a emx - be sup -mx} ~=~
As to ease of construc-
left { lpile {
tion, we have already men-
1 over {2 mab} ~log~
tioned that there are really
{sa emx - sb} over {sa emx + sb}
only a few person-months
above
invested. Much of this time
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A System for Typesetting Mathematics USD:23-45
has gone into two things-fine- The program consists of a
tuning (what is the most number of small, essentially
esthetically pleasing space to unconnected modules for code
use between the numerator and generation, a simple lexical
denominator of a fraction?), analyzer, a canned parser
and changing things found which we did not have to
deficient by our users write, and some miscellany
(shouldn't a tilde be a delim- associated with input files
iter?). and the macro facility. The
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USD:23-46 A System for Typesetting Mathematics
program is now about 1600 commands can be changed to
lines of C [6], a high-level accommodate other formatting
language reminiscent of BCPL. languages and devices. For
About 20 percent of these example, in less than 24
lines are ``print'' state- hours, one of us changed the
ments, generating the output entire semantic package to
code. drive NROFF, a variant of
TROFF, for typesetting
The semantic routines
mathematics on teletypewriter
that generate the actual TROFF
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A System for Typesetting Mathematics USD:23-47
devices capable of reverse and sometimes even for ulti-
line motions. Since many mate use.
potential users do not have
7. Conclusions
access to a typesetter, but
We think we have shown
still have to type mathemat-
that it is possible to do
ics, this provides a way to
acceptably good typesetting of
get a typed version of the
mathematics on a photo-
final output which is close
typesetter, with an input
enough for debugging purposes,
language that is easy to learn
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and use and that satisfies like the only sensible way to
many users' demands. Such a do business. Our experience
package can be implemented in with the use of a grammar and
short order, given a compiler- a compiler-compiler has been
compiler and a decent typeset- uniformly favorable. If we had
ting program underneath. written everything into code
directly, we would have been
Defining a language, and
locked into our original
building a compiler for it
design. Furthermore, we would
with a compiler-compiler seems
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A System for Typesetting Mathematics USD:23-49
have never been sure where the Acknowledgements
exceptions and special cases
We are deeply indebted to
were. But because we have a
J. F. Ossanna, the author of
grammar, we can change our
TROFF, for his willingness to
minds readily and still be
modify TROFF to make our task
reasonably sure that if a con-
easier and for his continuous
struction works in one place
assistance during the develop-
it will work everywhere.
ment of our program. We are
also grateful to A. V. Aho for
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help with language theory, to [1] A Manual of Style, 12th
S. C. Johnson for aid with the Edition. University of
compiler-compiler, and to our Chicago Press, 1969. p
early users A. V. Aho, S. I. 295.
Feldman, S. C. Johnson, R. W.
[2] Model C/A/T Photo-
Hamming, and M. D. McIlroy for
typesetter. Graphic Sys-
their constructive criticisms.
tems, Inc., Hudson, N. H.
[3] Ritchie, D. M., and
References
Thompson, K. L., ``The
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A System for Typesetting Mathematics USD:23-51
UNIX time-sharing sys- S. C., ``LR Parsing.''
tem.'' Comm. ACM 17, 7 Comp. Surv. 6, 2 (June
(July 1974), 365-375. 1974), 99-124.
[4] Ossanna, J. F., TROFF [6] B. W. Kernighan and D. M.
User's Manual. Bell Ritchie, The C Program-
Laboratories Computing ming Language. Prentice-
Science Technical Report Hall, Inc., 1978.
54, 1977.
[5] Aho, A. V., and Johnson,
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