SAINT(ID:153/sai003)A program to do formal integrationSymbolic Automatic INTegrator. J. Slagle, MIT 1961. Written in LISP. A program to do formal integration (Sammett 1961) Related languages
References: Introduction A large high-speed general-purpose digital computer (IBM 7090) was programmed to solve elementary symbolic integration problems at approximately the level of a good college freshman. The program is called SAINT, an acronym for "Symbolic Automatic INTegrator." The SAINT program is written in LISP [5] and most of the work reported here is the substance of a doctoral dissertation at the Massachusetts Institute of Technology [13]. This paper discusses the SAINT program and its performance. Some typical samples of SAINT's external behavior are given so that the reader may think in concrete terms. Let SAINT read into its card reader an IBM card containing (in a suitable notation) the symbolic integration problem, f xe ~ dx. In less than a minute a half, prints out answer, ½e ~2. and SAINT the Except where otherwise noted, every problem mentioned in this paper has been solved by SAINT. Note that SAINT omits the constant of integration, and we, too, shall ignore it throughout our discussion. After working for less than a minute on the problem f e ~ dx (which cannot be integrated in elementary form) SAINT prints out that it cannot solve it. SAINT performs indefinite integration, also called antidifferentiation. In * Received March, 1963. t This work was done in part at the Massachusetts Institute of Technology Computation Center. Operated with support from the U. S. Army, Navy and Air Force. 507 50S JAMES ~. SI,,kGI,E addition, it performs definite and multiple infegration when these are trivial extensions of indefinite integration. SAINT handles integrands that represent explicit elementary functions of a real variable which, for the sake of brevity, will be called elementary functions. The elementary functions are tl~e functions normally encountered in freshman integral caleulus, except that SAINT does not handle hyperbolic notation. The elementary functions are defined recursively as follows: a. Any real constant is an elementary function. b. The variable is an elementary function. c. The finite sum or finite product of elementary functions is an elementary function. d. An elementary function raised to an elementary function power is an elementary function. e. A trigonometric function of an elementary funct, ion is an elementary function. f. A logarithmic or inverse trigonometric function of an elementary function (restricted in range if necessary) is an elementary function. Currently, SAINT uses 26 standard forms. It uses 18 kinds of transformations including integration by parts and various substitution methods. Other methods, including the method of partial fractions, are excluded because of storage limita. tions. (When given two problems requiring the method of partial fractions SAINT found nothing to try and reported failure.) Since the SAINT program uses heuristic methods to search for solution, it is by definition a heuristic program. Although many author have given many definitions, in this paper a heuristic method (or simply a heuristic) is a method which generates a problem's solution by a trial and error process (involving feedback). Extract: Heuristic 10. Heuristic goal list. A list of goals requiring heuristic transformations, or, more briefly, a heuristic goal list, is an ordered list of those goals which are neither of standard form nor amenable to an algorithm-like transformation. A member of the heuristic goal list is called a heuristic goal. New such goals are inserted in order of increasing relative cost estimate. 11. Heuristic transformations. A transformation of a goal is called heuristic when, even though it is applicable and plausible, there is a significant risk that it is not the appropriate next step. In practice, the distinction between heuristic and algorithm-like transformations is largely empirical. The heuristic transformations are analogous to the methods of detachment, forward chaining and backward chaining in the LOGIC THEORIST of Newell, Shaw, and Simon [8]. The ten types of heuristic transformations [13] used by SAINT are designed to suggest plausible transformations of the integrand, substitutions and attempts using the method of integration by parts. Below is given only the most successful heuristic, "substitution for a subexpression whose derivative divides the inte- : grand." Let g (v) be the integrand. For each nonconstant nonlinear subexpression s (v) such that neither its main connective is MINUS nor is it a product with a constant factor, and such that the number of nonconstant factors of the fraction g(v)/ ; s' (v) (after cancellation) is less than the number of factors of g (v), try substituting u = s (v). Thus, in [ x sin x ~ dx, substitute u = X 2. Extract: Heuristic Realization of SAINT on a Computer For t~he most part, the implementation of the integration procedure described above is straightforward though very lengthy. The programming language used is LISP [5], as implemented on the IBM 709 and 7090. LISP manipulates symbolic expressions represented in LIST structures in much the same manner as an IPL language. About a third of the 32,768 register memory of the computer is occupied by the LISP system, which includes many general-purpose programs written by others. Another third is occupied by prerequisite programs. The remaining third is occupied by the SAINT program. Since the program is so large, full details cannot be given here, although more details are given in [13]. Also, since the program is so large, only about 3000 registers are available for working space in the free storage list, despite great effort to make this list as large as possible. The overall procedure is embodied in a LISP program called INTEGRAL with three inputs, namely, the integrand, the variable of integration and the resource allotment. The output of the program is either the integral or an indication of failure. Each goal is represented by an object in the sense of LISP. When a new goal is created, a unique print name such as, G0002, is assigned. In addition to its print name, the association list of a goal contains or may contain: 1. ACTIVE. If ACTIVE Occurs on the association list of a goal, that goal is alive; otherwise it is dead. 2. Two consecutive elements, INTEGRAND and the integrand. 3. Two consecutive elements, SUBGOA~S and the list of subgoals. Some goals have no subgoals, in which case these two elements do not appear. 4. Two consecutive elements, SUPERGOALS and a list of pairs; the first member of each pair is the name of a supergoal. The second member of the pair describes how the solution to the supergoal problem is related to the solution of the problem, and is either: (a) COMPONENT, which denotes that the supergoal integrand is a sum which was decomposed, or (b) an expression which represents a program which, when applied to the solution of this goal, will have for its output the solution of the supergoal. The original problem has no supergoals. The supergoal and subgoal lists fully specify the goal tree (and are the latter's only representation); operations such as pruning on the goal tree are performed by operating on these lists. 5. Four elements, CHARACTER, the character, RELATIVECOSTESTIMATE and tile relative cost estimate. These four elements are associated with the goal only if it is put on the heuristic goal list. 6. Two elements, INTEGRAL and the solution to the problem. These elements are associated with the goal only if it has been achieved. As soon as a new goal g is generated, SAINT uses straightforward methods in an attempt to achieve it. While doing this, SAINT may add g or certain of g's subgoals to the temporary goal list. If g is achieved, an attempt is made to achieve the original goal. This is embodied in the following iterative program, IMSLN [S] where s is some final segment of the goal list. In general, the final segment of a list (gl, g2, ? ? ? , g~) is either the empty list or one of the n lists (gi, g~+l, ? ? ? , g,,) for each i = 1, 2, ? ? ? , n. During the execution of the IMSLN program, any goals appended to the goal list will also be appended to the final segment. The initial value of s is (g) where g is either the original goal or a goal generated by a heuristic transformation. Below is given the iterative procedure IMSLN [8]. a. If s is empty, return with FALSE. b. Let us consider the goal g~, the first member oF s. If gi is the same as some other unachieved goal, h, which precedes gi on the goal list, then make the supergoal of gl an other supergoal of h and calculate I~S~,N of the rest of s, that is, delete g~ the first member of s and go to step "a". e. if g~ is directly achievable either because it is the same as some previously achieved goal or because it is of standard form, achieve it. Then, if pruning with respect to g~ achieves the original goal, terminate with this fact; otherwise calculate I~snN of the rest of s. d. If some algorithm-like transformation is appropriate for g~ , apply it and calculate IMSLN of the rest of s. e. Otherwise, append g~ to the end of the temporary goal list and calculate IMSLN of the rest of s. in [ACM] JACM 10(04) October 1963 view details It is a well known fact learned in freshman calculus that one can formally differentiate almost any expression, while such a statement is definitely not true for integration. Integration of polynomials is quite trivial; integration of an arbitrary function is quite difncult. Thus, it is not surprising that there has been almost no work done in this area, the only major exception being the SAINT system by Slagle, which currently stands as a monument to the handling of integration on a digital computer. In this program he employs heuristic techniques to perform integration; the program does quite well on an MIT freshman calculus examination. Slagle has tried to handle a very wide class of functions and as a result sometimes runs out of time or space in trying to do the integration. in Advances in Computers, Vol. 8 FL Alt and M Rubinoff (Eds.), Academic Press, New York, 1967 view details in Bobrow, D. G. (ed) "Symbol Manipulation Languages and Techniques", Proceedings of the IFIP Working Conference on Symbol Manipulation Languages. North-Holland Publishing Co., Amsterdam, 1968 view details in Bobrow, D. G. (ed) "Symbol Manipulation Languages and Techniques", Proceedings of the IFIP Working Conference on Symbol Manipulation Languages. North-Holland Publishing Co., Amsterdam, 1968 view details in [ACM] CACM 14(08) August 1971 view details [321 programming languages with indication of the computer manufacturers, on whose machinery the appropriate languages are used to know. Register of the 74 computer companies; Sequence of the programming languages after the number of manufacturing firms, on whose plants the language is implemented; Sequence of the manufacturing firms after the number of used programming languages.] in [ACM] CACM 14(08) August 1971 view details The exact number of all the programming languages still in use, and those which are no longer used, is unknown. Zemanek calls the abundance of programming languages and their many dialects a "language Babel". When a new programming language is developed, only its name is known at first and it takes a while before publications about it appear. For some languages, the only relevant literature stays inside the individual companies; some are reported on in papers and magazines; and only a few, such as ALGOL, BASIC, COBOL, FORTRAN, and PL/1, become known to a wider public through various text- and handbooks. The situation surrounding the application of these languages in many computer centers is a similar one. There are differing opinions on the concept "programming languages". What is called a programming language by some may be termed a program, a processor, or a generator by others. Since there are no sharp borderlines in the field of programming languages, works were considered here which deal with machine languages, assemblers, autocoders, syntax and compilers, processors and generators, as well as with general higher programming languages. The bibliography contains some 2,700 titles of books, magazines and essays for around 300 programming languages. However, as shown by the "Overview of Existing Programming Languages", there are more than 300 such languages. The "Overview" lists a total of 676 programming languages, but this is certainly incomplete. One author ' has already announced the "next 700 programming languages"; it is to be hoped the many users may be spared such a great variety for reasons of compatibility. The graphic representations (illustrations 1 & 2) show the development and proportion of the most widely-used programming languages, as measured by the number of publications listed here and by the number of computer manufacturers and software firms who have implemented the language in question. The illustrations show FORTRAN to be in the lead at the present time. PL/1 is advancing rapidly, although PL/1 compilers are not yet seen very often outside of IBM. Some experts believe PL/1 will replace even the widely-used languages such as FORTRAN, COBOL, and ALGOL.4) If this does occur, it will surely take some time - as shown by the chronological diagram (illustration 2) . It would be desirable from the user's point of view to reduce this language confusion down to the most advantageous languages. Those languages still maintained should incorporate the special facets and advantages of the otherwise superfluous languages. Obviously such demands are not in the interests of computer production firms, especially when one considers that a FORTRAN program can be executed on nearly all third-generation computers. The titles in this bibliography are organized alphabetically according to programming language, and within a language chronologically and again alphabetically within a given year. Preceding the first programming language in the alphabet, literature is listed on several languages, as are general papers on programming languages and on the theory of formal languages (AAA). As far as possible, the most of titles are based on autopsy. However, the bibliographical description of sone titles will not satisfy bibliography-documentation demands, since they are based on inaccurate information in various sources. Translation titles whose original titles could not be found through bibliographical research were not included. ' In view of the fact that nany libraries do not have the quoted papers, all magazine essays should have been listed with the volume, the year, issue number and the complete number of pages (e.g. pp. 721-783), so that interlibrary loans could take place with fast reader service. Unfortunately, these data were not always found. It is hoped that this bibliography will help the electronic data processing expert, and those who wish to select the appropriate programming language from the many available, to find a way through the language Babel. We wish to offer special thanks to Mr. Klaus G. Saur and the staff of Verlag Dokumentation for their publishing work. Graz / Austria, May, 1973 in [ACM] CACM 14(08) August 1971 view details A BRIEF HISTORICAL SKETCH ------------------------- The development of systems for symbolic mathematical computation first became an active area of research and implementation during the decade 1961-1971. . . . . . . To put the decade 1961-1971 into perspective, let us recall that FORTRAN appeared about 1958 and ALGOL in 1960. These two languages were designed primarily for numerical mathematical computation. Then in 1960/1961 came the development of LISP, a language for list processing. LISP was a major advancement on the road to languages for symbolic computation. An operation such as symbolic differentiation which is foreign to FORTRAN and ALGOL is relatively easy in LISP. (Indeed this is one of the standard programming assignments for students first learning LISP.) As will be noted later, several computer algebra systems were written in LISP. 1961-1966 --------- In 1961, James Slagle at M.I.T. wrote a LISP program called SAINT for Symbolic Automatic INTegration. This was one of the earliest applications of LISP to symbolic computation and it was the first comprehensive attempt to program a computer to behave like a freshman calculus student. The program was based on a number of heuristics for indefinite integration and it performed about as well as a good calculus student. One of the first systems for symbolic computation was FORMAC, developed by Jean Sammet, Robert Tobey, and others at IBM during the period 1962-1964. It was a FORTRAN preprocessor (a PL/I version appeared later) and it was designed for the manipulation of elementary functions including, of course, polynomials and rational functions. Another early system was ALPAK, a collection of FORTRAN-callable subroutines written in assembly language for the manipulation of polynomials and rational functions. It was designed by William S. Brown and others at Bell Laboratories and was generally available about 1964. A language now referred to as Early ALTRAN was designed at Bell Laboratories during the period 1964-1966. It used ALPAK as its package of computational procedures. There were two other significant systems for symbolic computation developed during this period. George Collins at IBM and the University of Wisconsin (Madison) developed PM, a system for polynomial manipulation, an early version of which was operational in 1961 with improvements added to the system through 1966. The year 1965 marked the first appearance of MATHLAB, a LISP-based system for the manipulation of polynomials and rational functions, developed by Carl Engelman at M.I.T. It was the first interactive system designed to be used as a symbolic calculator. Included among its many firsts was the use of two-dimensional output to represent its mathematical output. The work of this period culminated in the first ACM Symposium on Symbolic and Algebraic Manipulation held in March 1966 in Washington, D.C. That conference was summarized in the August 1966 issue of the Communications of the ACM. 1966-1971 --------- In 1966/1967, Joel Moses at M.I.T. wrote a LISP program called SIN (for Symbolic Integrator). Unlike the earlier SAINT program, SIN was algorithmic in approach and it was also much more efficient. In 1968, Tony Hearn at Stanford University developed REDUCE, an interactive LISP-based system for physics calculations. One of its principal design goals was portability over a wide range of platforms, and as such only a limited subset of LISP was actually used. The year 1968 also marked the appearance of Engelman's MATHLAB-68, an improved version of the earlier MATHLAB interactive system, and of the system known as Symbolic Mathematical Laboratory developed by William Martin at M.I.T. in 1967. The latter was a linking of several computers to do symbolic manipulation and to give good graphically formatted output on a CRT terminal. The latter part of the decade saw the development of several important general purpose systems for symbolic computation. ALTRAN evolved from the earlier ALPAK and Early ALTRAN as a language and system for the efficient manipulation of polynomials and rational functions. George Collins developed SAC-1 (for Symbolic and Algebraic Calculations) as the successor of PM for the manipulation of polynomials and rational functions. CAMAL (CAMbridge Algebra system) was developed by David Barton, Steve Bourne, and John Fitch at the University of Cambridge. It was implemented in the BCPL language, and was particularly geared to computations in celestial mechanics and general relativity. REDUCE was redesigned by 1970 into REDUCE 2, a general purpose system with special facilities for use in high-energy physics calculations. It was written in an ALGOL-like dialect called RLISP, avoiding the cumbersome parenthesized notation of LISP, while at the same time retaining its original design goal of being easily portable. SCRATCHPAD was developed by J. Griesmer and Richard Jenks at IBM Research as an interactive LISP-based system which incorporated significant portions of a number of previous systems and programs into its library, such as MATHLAB-68, REDUCE 2, Symbolic Mathematical Library, and SIN. Finally, the MACSYMA system first appeared about 1971. Designed by Joel Moses, William Martin, and others at M.I.T., MACSYMA was the most ambitious system of the decade. Besides the standard capabilities for algebraic manipulation, it included facilities to aid in such computations as limit calculations, symbolic integration, and the solution of equations. The decade from 1961 to 1971 concluded with the Second Symposium on Symbolic and Algebraic Manipulation held in March 1971 in Los Angeles. The proceedings of that conference constitute a remarkably comprehensive account of the state of the art of symbolic mathematical computation in 1971. 1971-1981 --------- While all of the languages and systems of the sixties and seventies began as experiments, some of them were eventually put into "production use'' by scientists, engineers, and applied mathematicians outside of the original group of developers. REDUCE, because of its early emphasis on portability, became one of the most widely available systems of this decade. As a result it was instrumental in bringing computer algebra to the attention of many new users. MACSYMA continued its strong development, especially with regard to algorithm development. Indeed, many of the standard techniques (e.g. integration of elementary functions, Hensel lifting, sparse modular algorithms) in use today either came from, or were strongly influenced by, the research group at M.I.T. It was by far the most powerful of the existing computer algebra systems. SAC/ALDES by G. Collins and R. Loos was the follow-up to Collins' SAC-1. It was a non-interactive system consisting of modules written in the ALDES (Algebraic DEScription) language, with a translator converting the results to ANSI FORTRAN. One of its most notable distinctions was in being the only major system to completely and carefully document its algorithms. A fourth general purpose system which made a significant mark in the late 1970's was muMATH. Developed by David Stoutemyer and Albert Rich at the University of Hawaii, it was written in a small subset of LISP and came with its own programming language, muSIMP. It was the first comprehensive computer algebra system which could actually run on the IBM family of PC computers. By being available on such small and widely accessible personal computers, muMATH opened up the possibility of widespread use of computer algebra systems for both research and teaching. In addition to the systems mentioned above, a number of special purpose systems also generated some interest during the 1970's. Examples of these include: SHEEP, a system for tensor component manipulation designed by Inge Frick and others at the University of Stockholm; TRIGMAN, specially designed for computation of Poisson series and written in FORTRAN by W. H. Jeffreys at University of Texas (Austin); and SCHOONSCHIP by M. Veltman of the Netherlands for computations in high-energy physics. Although the systems already mentioned have all been developed in North America and Europe, there were also a number of symbolic manipulation programs written in the U.S.S.R. One of these is ANALITIK, a system implemented in hardware by V. M. Glushkov and others at the Institute of Cybernetics, Kiev. 1981-1991 --------- Due to the significant computer resource requirements of the major computer algebra systems, their widespread use remained (with the exception of muMATH) limited to researchers having access to considerable computing resources. With the introduction of microprocessor-based workstations, the possibility of relatively powerful desk-top computers became a reality. The introduction of a large number of different computing environments, coupled with the often nomadic life of researchers (at least in terms of workplace locations) caused a renewed emphasis on portability for the computer algebra systems of the 1980's. More efficiency (particularly memory space efficiency) was needed in order to run on the workstations that were becoming available at this time, or equivalently, to service significant numbers of users on the time-sharing environments of the day. This resulted in a movement towards the development of computer algebra systems based on newer "systems implementation'' languages such as C, which allowed developers more flexibility to control the use of computer resources. The decade also marked a growth in the commercialization of computer algebra systems. This had both positive and negative effects on the field in general. On the negative side, users not only had to pay for these systems but also they were subjected to unrealistic claims as to what constituted the state of the art of these systems. However, on the positive side, commercialization brought about a marked increase in the usability of computer algebra systems, from major advances in user interfaces to improvements to their range of functionality in such areas as graphics and document preparation. The beginning of the decade marked the origin of MAPLE. Initiated by Gaston Gonnet and Keith Geddes at the University of Waterloo, its primary motivation was to provide user accessibility to computer algebra. MAPLE was designed with a modular structure: a small compiled kernel of modest power, implemented completely in the systems implementation language C (originally B, another language in the "BCPL family'') and a large mathematical library of routines written in the user-level MAPLE language to be interpreted by the kernel. Besides the command interpreter, the kernel also contained facilities such as integer and rational arithmetic, simple polynomial manipulation, and an efficient memory management system. The small size of the kernel allowed it to be implemented on a number of smaller platforms and allowed multiple users to access it on time-sharing systems. Its large mathematical library, on the other hand, allowed it to be powerful enough to meet the mathematical requirements of researchers. Another system written in C was SMP (Symbolic Manipulation Program) by Stephen Wolfram at Caltech. It was portable over a wide range of machines and differed from existing systems by using a language interface that was rule-based. It took the point of view that the rule-based approach was the most natural language for humans to interface with a computer algebra program. This allowed it to present the user with a consistent, pattern-directed language for program development. The newest of the computer algebra systems during this decade were MATHEMATICA and DERIVE. MATHEMATICA is a second system written by Stephen Wolfram (and others). It is best known as the first system to popularize an integrated environment supporting symbolics, numerics, and graphics. Indeed when MATHEMATICA first appeared in 1988, its graphical capabilities (2-D and 3-D plotting, including animation) far surpassed any of the graphics available on existing systems. MATHEMATICA was also one of the first systems to successfully illustrate the advantages of combining a computer algebra system with the easy-to-use editing features on machines designed to use graphical user-interfaces (i.e. window environments). Based on C, MATHEMATICA also comes with its own programming language which closely follows the rule-based approach of its predecessor, SMP. DERIVE, written by David Stoutemyer and Albert Rich, is the follow-up to the successful muMATH system for personal computers. While lacking the wide range of symbolic capabilities of some other systems, DERIVE has an impressive range of applications considering the limitations of the 16-bit PC machines for which it was designed. It has a friendly user interface, with such added features as two-dimensional input editing of mathematical expressions and 3-D plotting facilities. It was designed to be used as an interactive system and not as a programming environment. Along with the development of newer systems, there were also a number of changes to existing computer algebra systems. REDUCE 3 appeared in 1983, this time with a number of new packages added by outside developers. MACSYMA bifurcated into two versions, DOE-MACSYMA and one distributed by SYMBOLICS, a private company best known for its LISP machines. Both versions continued to develop, albeit in different directions, during this decade. AXIOM, (known originally as SCRATCHPAD II) was developed during this decade by Richard Jenks, Barry Trager, Stephen Watt and others at the IBM Thomas J. Watson Research Center. A successor to the first SCRATCHPAD language, it is the only "strongly typed'' computer algebra system. Whereas other computer algebra systems develop algorithms for a specific collection of algebraic domains (such as, say, the field of rational numbers or the domain of polynomials over the integers), AXIOM allows users to write algorithms over general fields or domains. As was the case in the previous decade, the eighties also found a number of specialized systems becoming available for general use. Probably the largest and most notable of these is the system CAYLEY, developed by John Cannon and others at the University of Sydney, Australia. CAYLEY can be thought of as a "MACSYMA for group theorists.'' It runs in large computing environments and provides a wide range of powerful commands for problems in computational group theory. An important feature of CAYLEY is a design geared to answering questions not only about individual elements of an algebraic structure, but more importantly, questions about the structure as a whole. Thus, while one could use a system such as MACSYMA or MAPLE to decide if an element in a given domain (such as a polynomial domain) has a given property (such as irreducibility), CAYLEY can be used to determine if a group structure is finite or infinite, or to list all the elements in the center of the structure (i.e. all elements which commute with all the elements of the structure). Another system developed in this decade and designed to solve problems in computational group theory is GAP (Group Algorithms and Programming) developed by J. Neubueser and others at the University of Aachen, Germany. If CAYLEY can be considered to be the "MACSYMA of group theory,'' then GAP can be viewed as the "MAPLE of group theory.'' GAP follows the general design of MAPLE in implementing a small compiled kernel (in C) and a large group theory mathematical library written in its own programming language. Examples of some other special purpose systems which appeared during this decade include FORM by J. Vermaseren, for high energy physics calculations, LiE, by A.M. Cohen for Lie Algebra calculations, MACAULAY, by Michael Stillman, a system specially built for computations in Algebraic Geometry and Commutative Algebra, and PARI by H. Cohen in France, a system oriented mainly for number theory calculations. As with most of the new systems of the eighties, these last two are also written in C for portability and efficiency. Research Information about Computer Algebra ------------------------------------------- Research in computer algebra is a relatively young discipline, and the research literature is scattered throughout various journals devoted to mathematical computation. However, its state has advanced to the point where there are two research journals primarily devoted to this subject area: the "Journal of Symbolic Computation" published by Academic Press and "Applicable Algebra in Engineering, Communication and Computing" published by Springer-Verlag. Other than these two journals, the primary source of recent research advances and trends is a number of conference proceedings. Until recently, there was a sequence of North American conferences and a sequence of European conferences. The North American conferences, primarily organized by ACM SIGSAM (the ACM Special Interest Group on Symbolic and Algebraic Manipulation), include SYMSAM '66 (Washington, D.C.), SYMSAM '71 (Los Angeles), SYMSAC '76 (Yorktown Heights), SYMSAC '81 (Snowbird), and SYMSAC '86 (Waterloo). The European conferences, organized by SAME (Symbolic and Algebraic Manipulation in Europe) and ACM SIGSAM, include the following whose proceedings have appeared in the Springer-Verlag series "Lecture Notes in Computer Science": EUROSAM '79 (Marseilles), EUROCAM '82 (Marseilles), EUROCAL '83 (London), EUROSAM '84 (Cambridge), EUROCAL '85 (Linz), and EUROCAL '87 (Leipzig). Starting in 1988, the two streams of conferences have been merged and they are now organized under the name ISSAC (International Symposium on Symbolic and Algebraic Computation), including ISSAC '88 (Rome), ISSAC '89 (Portland, Oregon), ISSAC '90 (Tokyo), ISSAC '91 (Bonn) and ISSAC '92 (Berkeley). ----------------------------------------------- Professor Keith Geddes Symbolic Computation Group Department of Computer Science University of Waterloo Waterloo ON N2L 3G1 CANADA in [ACM] CACM 14(08) August 1971 view details Resources
|