A System for Typesetting Mathematics USD:26-1A System for Typesetting MathematicsBrian W. Kernighan and Lorinda L. Cherry AT&T Bell Laboratories Murray Hill, New Jersey 07974ABSTRACTThis 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. IntroductionOctober 25, 2015 USD:26-2 A System for Typesetting Mathematics ``Mathematics is known in (``Requires'' is perhaps the the trade asdifficult, or wrong word, but mathematicspenalty,copybecause 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. October 25, 2015 A System for Typesetting Mathematics USD:26-3 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 tems phototypesetter[2] on the line-drawing, built-up charac- UNIX operating system[3] works ters like braces and radicals, by shining light through a and 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 piece2. Photocompositionof 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 October 25, 2015 USD:26-4 A System for Typesetting Mathematics 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 doing3. Language Designmathematics, 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, October 25, 2015 A System for Typesetting Mathematics USD:26-5 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- October 25, 2015 USD:26-6 A System for Typesetting Mathematics 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 October 25, 2015 A System for Typesetting Mathematics USD:26-7 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 October 25, 2015 USD:26-8 A System for Typesetting Mathematics 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. October 25, 2015 A System for Typesetting Mathematics USD:26-9 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 LanguageAs 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 October 25, 2015 USD:26-10 A System for Typesetting Mathematics more pleasing appearance. several characters of various sizes. A tilde ``~'' gives a Input is free-form. space equal to the normal word Spaces and new lines in the spacing in text; a circumflex input are used by EQN to gives half this much, and a separate pieces of the input; tab charcter spaces to the they are not used to create next tab stop. space in the output. Thus Spaces (or tildes, etc.) x = y also serve to delimit pieces + z + 1 of the input. For example, to also gives x=y+z+1. Free-form get input is easier to type ini- f(t)=2i̅i̅sin(wt)dt tially; subsequent editing is we write also easier, for an expression may be typed as many short f(t) = 2 pi int sin ( omega t )dt lines. Extra white space can be Here spaces arenecessaryin forced into the output by the input to indicate that October 25, 2015 A System for Typesetting Mathematics USD:26-11sin, pi, int, andomegaare Fractions are specified special, and potentially worth with the keywordover: special treatment. EQN looks a+b over c+d+e = 1 up each such string of charac- ters in a table, and if produces appropriate gives it a trans-__±__=1 lation. In this case,piand c+d+eomegabecome their greek Similarly, subscripts and equivalents,intbecomes the superscripts are introduced by integral sign (which must be the keywordssubandsup: moved down and enlarged so it x2+y2=z2 looks ``right''), andsinis is produced by made roman, following conven- x sup 2 + y sup 2 = z sup 2 tional mathematical practice. Parentheses, digits and opera- tors are automatically made The spaces after the 2's are roman wherever found. necessary to mark the end of the superscripts; similarly October 25, 2015 USD:26-12 A System for Typesetting Mathematics the keywordsuphas to be {partial sup 2 f} over {partial x sup 2} = marked off by spaces or some equivalent delimiter. The x sup 2 over a sup 2 + y sup 2 over b sup 2 return to the proper baseline is automatic. Multiple levels Braces {} are used to group of subscripts or superscripts objects together; in this case are of course allowed: they indicate unambiguously ``xsupysupz'' is xyz. The con- what goes over what on the struct ``somethingsubsome- left-hand side of the expres- thingsupsomething'' is sion. The language defines the recognized as a special case, precedence ofsupto be higher 2 so ``x sub i sup 2'' is xi than that ofover, so no instead of xi2. braces are needed to get the More complicated expres- correct association on the sions can now be formed with right side. Braces can always these primitives: be used when in doubt about`__f_=_2_+_2_precedence.`x2 a2 b2 is produced by October 25, 2015 A System for Typesetting Mathematics USD:26-13 Since large radicals look poor The braces convention is on our typesetter,sqrtis not an example of the power of useful for tall expressions. using a recursive grammar to define the language. It is Limits on summations, part of the language that if a integrals and similar con- construct can appear in some structions are specified with context, thenany expressionthe keywordsfromandto. To in braces can also occur in get that context. oo ≥̅ xi->0 i=0 There is asqrtoperator we need only type for making square roots of the sum from i=0 to inf x sub i -> 0 appropriate size: ``sqrt a+b'' ___ produces \|a+b, and Centering and making the ≥̅ big x = {-b +- sqrt{b sup 2 -4ac}} over 2a enough and the limits smaller are all automatic. Thefromis andtoparts are both ______ optional, and the central part_b__\__b__-__a__c x= 2a October 25, 2015 USD:26-14 A System for Typesetting Mathematics (e.g., the ≥̅) can in fact be makes anything: |_±_||2a | = 1 lim from {x -> pi /2} ( tan~x) = inf Aleftneed not have a correspondingright, as we is shall see in the next example. Any characters may followleftlim (tan x)=oo x->i̅i̅/2 andright, but generally only Again, the braces indicate various parentheses and bars just what goes into thefromare meaningful. part. Big brackets, etc., are There is a facility for often used with another facil- making braces, brackets, ity, calledpiles, which make parentheses, and vertical bars vertical piles of objects. For of the right height, using the example, to get keywordsleftandright: | | 1 left [ x+y over 2a right ]~=~1 sign(x)_| 0 if |-1 if x>0 | if x=0 x<0 we can type October 25, 2015 A System for Typesetting Mathematics USD:26-15 any number of elements; any sign (x) ~==~ left { element of a pile can of course contain piles. rpile {1 above 0 above -1} Although EQN makes a ~~lpile {if above if above ifvaliant attempt to use the right sizes and fonts, there ~~lpile {x>0 above x=0 aboveare} times when the default assumptions are simply not what is wanted. For instance The construction ``left {'' the italicsignin the previ- makes a left brace big enough ous example would convention- to enclose the ``rpile ally be in roman. Slides and {...}'', which is a right- transparencies often require justified pile of ``above ... larger characters than normal above ...''. ``lpile'' makes a text. Thus we also provide left-justified pile. There are size and font changing com- also centered piles. Because mands: ``size 12 bold of the recursive language {A~x~=~y}'' will produce definition, a pile can contain October 25, 2015 USD:26-16 A System for Typesetting Mathematics A x = y.Sizeis followed by a Diacritical marks, long a number representing a charac- problem in traditional ter size in points. (One point typesetting, are straightfor- is 1/72 inch; this paper is ward: . .. ___ set in 9 point type.)_+^+y̅+^+Y =z+Z If necessary, an input is made by typing string can be quoted in "...", x dot under + x hat + y tilde which turns off grammatical significance, and any font or + X hat + Y dotdot = z+Z bar spacing changes that might otherwise be done on it. Thus we can say There are also facilities for globally changing default lim~ roman "sup" ~x sub n = 0 sizes and fonts, for example for making viewgraphs or for to ensure that the supremum setting chemical equations. doesn't become a superscript: The language allows for lim sup xn=0 matrices, and for lining up October 25, 2015 A System for Typesetting Mathematics USD:26-17 equations at the same horizon- keywords likesuporover. tal position. Section 6 shows an example of definitions. Finally, there is a definition facility, so a user The EQN preprocessor can say reads intermixed text and equations, and passes its out- define name "..." put to TROFF. Since TROFF uses lines beginning with a period at any time in the document; as control words (e.g., henceforth, any occurrence of ``.ce'' means ``center the the token ``name'' in an next output line''), EQN uses expression will be expanded the sequence ``.EQ'' to mark into whatever was inside the the beginning of an equation double quotes in its defini- and ``.EN'' to mark the end. tion. This lets users tailor The ``.EQ'' and ``.EN'' are the language to their own passed through to TROFF specifications, for it is untouched, so they can also be quite possible to redefine used by a knowledgeable user October 25, 2015 USD:26-18 A System for Typesetting Mathematics to center equations, number .ce them automatically, etc. By default, however, ``.EQ'' and .EQ ``.EN'' are simply ignored by TROFF, so by default equations x sub i = y sub i ... are printed in-line. ``.EQ'' and ``.EN'' can .EN be supplemented by TROFF com- mands as desired; for example, Since it is tedious to a centered display equation type ``.EQ'' and ``.EN'' can be produced with the around very short expressions input: (single letters, for instance), the user can also define two characters to serve as the left and right delim- iters of expressions. These characters are recognized any- October 25, 2015 A System for Typesetting Mathematics USD:26-19 where in subsequent text. For output of one process (EQN) to example if the left and right the input of another (TROFF). delimiters have both been set to ``#'', the input:5. Language TheoryLet #x sub i#, #y# and #alpha# be posThevebasic structure of the language is not a particu- larly original one. Equations produces: are pictured as a set of Let xi, y and(be positive ``boxes,'' pieced together in various ways. For example, something with a subscript is Running a preprocessor is just a box followed by another strikingly easy on UNIX. To box moved downward and shrunk typeset text stored in file by an appropriate amount. A ``f'', one issues the command: fraction is just a box cen- eqn f | troff tered above another box, at the right altitude, with a The vertical bar connects the line of correct length drawn October 25, 2015 USD:26-20 A System for Typesetting Mathematics between them. cate optional material. A TEXT is a string of non-blank char- The grammar for the acters or any string inside language is shown below. For double quotes; the other ter- purposes of exposition, we minal symbols represent have collapsed some produc- literal occurrences of the tions. In the original gram- corresponding keyword. mar, there are about 70 pro- ductions, but many of these are simple ones used only to guarantee that some keyword is recognized early enough in the parsing process. Symbols in capital letters are terminal symbols; lower case symbols are non-terminals, i.e., syn- tactic categories. The verti- cal bar | indicates an alter- native; the brackets [ ] indi- October 25, 2015 A System for Typesetting Mathematics USD:26-21 | SIZE text box eqn: box | eqn box | [ROMAN | BOLD | ITALIC] box box: text | box [HAT | BAR | DOT | DOTDOT | TILDE] | { eqn } | DEFINE text text | box OVER box list: eqn | list ABOVE eqn | SQRT box text: TEXT | box SUB box | box SUP box | [ L | C | R ]PILE { list } The grammar makes it obvious why there are few | LEFT text eqn [ RIGHT text ] exceptions. For example, the observation that something can | box [ FROM box ] [ TO box ] be replaced by a more compli- cated something in braces is October 25, 2015 USD:26-22 A System for Typesetting Mathematics implicit in the productions: {a over b} over c eqn : box | eqn box or is it box : text | { eqn } a over {b over c} ? Anywhere a single character could be used,anylegal con- To answer questions like struction can be used. this, the grammar is supple- mented with a small set of Clearly, our grammar is rules that describe the pre- highly ambiguous. What, for cedence and associativity of instance, do we do with the operators. In particular, we input specify (more or less arbi- a over b over c ? trarily) thatoverassociates to the left, so the first Is it alternative above is the one chosen. On the other hand,subandsupbind to the right, October 25, 2015 A System for Typesetting Mathematics USD:26-23 The ambiguous grammar because this is closer to approach seems to be quite standard mathematical prac- useful. The grammar we use is tice. That is, we assume xab small enough to be easily is x(ab), not (xa)b. understood, for it contains The precedence rules none of the productions that resolve the ambiguity in a would be normally used for construction like resolving ambiguity. Instead the supplemental information a sup 2 over b about precedence and associa- tivity (also small enough to We definesupto have a higher be understood) provides the precedence thanover, so this compiler-compiler with the_2_ construction is parsed as b_information it needs to make a b. instead of a fast, deterministic parser for Naturally, a user can the specific language we want. always force a particular When the language is supple- parsing by placing braces mented by the disambiguating around expressions. October 25, 2015 USD:26-24 A System for Typesetting Mathematics rules, it is in fact LR(1) and name for the string, then hand thus easy to parse[5]. the name and the string to TROFF, and let TROFF perform The output code is gen- the storage management. All we erated as the input is save is the name of the scanned. Any time a production string, its height, and its of the grammar is recognized, baseline. (potentially) some TROFF com- mands are output. For example, As another example, the when the lexical analyzer translation associated with reports that it has found a the production TEXT (i.e., a string of con- box : box OVER box tiguous characters), we have recognized the production: is: text : TEXT The translation of this is simple. We generate a local October 25, 2015 A System for Typesetting Mathematics USD:26-25 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 have equally simple semantic String describing output box = actions. Picturing the output as a set of properly placed move down; boxes makes the right sequence of positioning commands quite move right enough to center bottobvious. The main difficulty is in finding the right October 25, 2015 USD:26-26 A System for Typesetting Mathematics6. Experiencenumbers to use for estheti- cally pleasing positioning. There are really three aspects of interest-how well With a grammar, it is EQN sets mathematics, how well usually clear how to extend it satisfies its goal of being the language. For instance, ``easy to use,'' and how easy one of our users suggested a it was to build. TENSOR operator, to make con- structions like The first question is kj easily addressed. This entire lT mni paper has been set by the pro- Grammatically, this is easy: gram. Readers can judge for it is sufficient to add a pro- themselves whether it is good duction like enough for their purposes. One box : TENSOR { list } of our users commented that although the output is not as Semantically, we need only good as the best hand-set juggle the boxes to the right material, it is still better places. than average, and much better October 25, 2015 A System for Typesetting Mathematics USD:26-27 than the worst. In any case, Some other weaknesses are who cares? Printed books can- inherent in our output device. not compete with the birds and It is hard, for instance, to flowers of illuminated draw a line of an arbitrary manuscripts on esthetic length without getting a per- grounds, either, but they have ceptible overstrike at one some clear economic advan- end. tages. As to ease of use, at the Some of the deficiencies time of writing, the system in the output could be cleaned has been used by two distinct up with more work on our part. groups. One user population For example, we sometimes consists of mathematicians, leave too much space between a chemists, physicists, and com- roman letter and an italic puter scientists. Their typi- one. If we were willing to cal reaction has been some- keep track of the fonts thing like: involved, we could do this (1) It's easy to write, better more of the time. although I make the fol- October 25, 2015 USD:26-28 A System for Typesetting Mathematics lowing mistakes... were the original target of the system. They tend to be (2) How do I do...? enthusiastic converts. They (3) It botches the following find the language easy to things.... Why don't you learn (most are largely self- fix them? taught), and have little trou- (4) You really need the fol- ble producing the output they lowing features... want. They are of course less critical of the esthetics of The learning time is their output than users short. A few minutes gives the trained in mathematics. After general flavor, and typing a a transition period, most find page or two of a paper gen- using a computer more erally uncovers most of the interesting than a regular misconceptions about how it typewriter. works. The main difficulty that The second user group is users have seems to be much larger, the secretaries remembering that a blank is a and mathematical typists who October 25, 2015 A System for Typesetting Mathematics USD:26-29 delimiter; even experienced The language is somewhat users use blanks where they prolix, but this doesn't seem shouldn't and omit them when excessive considering how much they are needed. A common is being done, and it is cer- instance is typing tainly more compact than the corresponding TROFF commands. f(x sub i) For example, here is the source for the continued frac- which produces tion expression in Section 1 f(xi) of this paper: instead of f(xi) Since the EQN language knows no mathematics, it cannot deduce that the right parenthesis is not part of the subscript. October 25, 2015 USD:26-30 A System for Typesetting Mathematics a sub 0 + b sub 1 over define emx "{e sup mx}" {a sub 1 + b sub 2 over define mab "{m sqrt ab}" {a sub 2 + b sub 3 over define sa "{sqrt a}" {a sub 3 + ... }}} define sb "{sqrt b}" int dx over {a emx - be sup -mx} ~=~ This is the input for the large integral of Section 1; left { lpile { notice the use of definitions: 1 over {2 mab} ~log~ {sa emx - sb} over {sa emx + sb} above October 25, 2015 A System for Typesetting Mathematics USD:26-31 1 over mab ~ tanh sup -1 ( safraction?),x ) and changing things found deficient by our above users (shouldn't a tilde be a delimiter?). -1 over mab ~ coth sup -1 ( sa over sb emx ) The program consists of a number of small, essentially } unconnected modules for code generation, a simple lexical analyzer, a canned parser As to ease of construc- which we did not have to tion, we have already men- write, and some miscellany tioned that there are really associated with input files only a few person-months and the macro facility. The invested. Much of this time program is now about 1600 has gone into two lines of C [6], a high-level things-fine-tuning (what is language reminiscent of BCPL. the most esthetically pleasing About 20 percent of these space to use between the lines are ``print'' state- numerator and denominator of a October 25, 2015 USD:26-32 A System for Typesetting Mathematics ments, generating the output ics, this provides a way to code. get a typed version of the final output which is close The semantic routines enough for debugging purposes, that generate the actual TROFF and sometimes even for ulti- commands can be changed to mate use. accommodate other formatting languages and devices. For7. Conclusionsexample, in less than 24 hours, one of us changed the We think we have shown entire semantic package to that it is possible to do drive NROFF, a variant of acceptably good typesetting of TROFF, for typesetting mathematics on a photo- mathematics on teletypewriter typesetter, with an input devices capable of reverse language that is easy to learn line motions. Since many and use and that satisfies potential users do not have many users' demands. Such a access to a typesetter, but package can be implemented in still have to type mathemat- short order, given a October 25, 2015 A System for Typesetting Mathematics USD:26-33 compiler-compiler and a decent grammar, we can change our typesetting program under- minds readily and still be neath. reasonably sure that if a con- struction works in one place Defining a language, and it will work everywhere. building a compiler for it with a compiler-compiler seemsAcknowledgementslike the only sensible way to do business. Our experience We are deeply indebted to with the use of a grammar and J. F. Ossanna, the author of a compiler-compiler has been TROFF, for his willingness to uniformly favorable. If we had modify TROFF to make our task written everything into code easier and for his continuous directly, we would have been assistance during the develop- locked into our original ment of our program. We are design. Furthermore, we would also grateful to A. V. Aho for have never been sure where the help with language theory, to exceptions and special cases S. C. Johnson for aid with the were. But because we have a compiler-compiler, and to our October 25, 2015 USD:26-34 A System for Typesetting Mathematics early users A. V. Aho, S. I. (July 1974), 365-375. Feldman, S. C. Johnson, R. W. [4] Ossanna, J. F., TROFF Hamming, and M. D. McIlroy for User's Manual. Bell their constructive criticisms. Laboratories Computing Science Technical ReportReferences54, 1977. [1]A Manual of Style, 12th [5] Aho, A. V., and Johnson, Edition. University of S. C., ``LR Parsing.'' Chicago Press, 1969. pComp. Surv. 6, 2 (June 295. 1974), 99-124. [2]Model C/A/T Photo-[6] B. W. Kernighan and D. M.typesetter. Graphic Sys- Ritchie,The C Program-tems, Inc., Hudson, N. H.ming Language. Prentice- [3] Ritchie, D. M., and Hall, Inc., 1978. Thompson, K. L., ``The UNIX time-sharing sys- tem.''Comm. ACM 17, 7 October 25, 2015

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