Showing papers presented at "Symposium on Symbolic and Algebraic Manipulation in 1966"

The symbolic manipulation of poisson series

Abstract: Poisson series of three variables are manageable symbolically through a set of formal subroutines written partially in the IBM 7094 machine language, but to be called in the FoRTRAN language for use in FoRTRAN programs. An effort has been made to supply those operations which are most required by celestial mechanics. The routines are entirely self-contained subroutines and require only standard FoRTRAN input/output units 5 and 6; they are designed to avoid waste and overflow of core storage space.

A language and system for symbolic algebra on a digital computer

Abstract: This paper describes the ALPAK system and the ALTRAN language for symbolic algebra on a digital computer. The ALPAK system is specifically designed for the efficient handling of large scale algebraic computations, and has been applied to a wide variety of practical problems. The ALTRAN language is a dialect of FORTRAN for describing algebraic manipulations. Although ALTRAN is still being developed, a useful subset has been implemented. A programming system called OEDIPUS, which serves as a foundation for the second and newest version of ALPAK, is briefly described. The past investment in, present availability of, and future plans for ALPAK and ALTRAN are outlined.

An on-line program for non-numerical algebra

Abstract: The paper describes a program for on line manipulation and simplification of non-numerical algebra. The program is written in LISP 1,5 for use under the time-sharing system of the Q-32 computer at System Development Corporation in Santa Monica, California.The structure of the program makes the program easy to be expanded, and the user is allowed fairly extensive control over the course of the program. Intermediate results can be printed at will, and changes to the simplification rules can be done rather easily at several stages of the program.The program has been written and debugged from a remote Teletype consol at Stanford University.

PANON-1B: a programming language for symbol manipulation

Abstract: PANON-1B is a Universal Programming Language for Symbol Manipulation, base on a particular extension of Markov Normal Algorithms. It consists essentially of a sequence of Labelled Structural Transformation Rules to be applied to an arbitrary argument string, according to appropriate sequencing rules. PANON-1B includes also specifications for input-output operations. The basic principles are discussed. Some details of a particular complete hardware representation are given.

Application of LISP to sequence prediction

Abstract: This paper describes a LISP program that analyzes sequences of letters and numbers. The program, which operates interactively, takes as input a sequence such as ACEGI...It produces as output the next element of the sequence, in this case K, together with a LISP program that predicts the n'th element of the sequence. The program is organized into an executive and a set of workers. The executive calls upon each worker in turn. The worker will respond if it recognizes the sequence as the kind that it knows about; since the workers may themselves call upon the executive, the recognition process is recursive. Workers exist for cyclic sequences, linear sequences, letter sequences, polynomials, interwining of sequences, sequences with predictable first differences, sequences with predictable first ratios, and standard sequences such as the primes, the squares, and the cubes. New workers can be added easily. It is concluded that the program can operate efficiently if somewhat slowly on a significantly large collection of sequences, and that the organizational scheme used has application to other kinds of pattern recognition.

A FORMAC program for the solution of linear boundary and initial value problems

Abstract: A computer program is described which has been developed for obtaining approximate solutions to linear initial and boundary value problems involving differential equations. The program can be used for ordinary curve fitting as well. For each problem, input to the program includes:1. The equations (in symbolic form) to be satisfied - the differential equations, equations describing auxiliary conditions such as boundary conditions, etc.2. A numerical description of the regions in which each of the equations are to be satisfied.3. Sets of functions (in symbolic form) to be used in linear combinations to approximate the solution functions.Given the above input, the program generates an approximation to the solutuon of the specified problem in terms of the specified functions which is optimum in the least squares sense. A number of examples illustrating the use of the program are included in the paper.

Models for mathematical systems

Abstract: A program, written for the IBM 7090, which defines a model for a finite collection of associative algebras for the computer, is described and illustrated. The program contains a structure broad enough in scope to allow one to perform operations on such diverse mathematical concepts as differential equations, infinite series and differential forms in a simple yet comprehensive manner, while also serving in a foundation upon which a variety of higher level symbol manipulation languages can be developed.The development of the structure of this model is given. In particular, it is assumed that a family of not necessarily commutative algebras defined over a collection of commutative rings is given, where the algebras are algebras of modules generated by recursive sets. A brief discussion is given on how the vectors and scalars of these algebras should be represented, and how their basic algebraic operations (addition and multiplication) can be handled in a systematic manner without regard to the specific underlying scalar rings involved. Functional expressions are treated by developing a procedure for assigning an arbitrary list of operators to an algebra, and then assigning to each operator a list of assumptions that it is to satisfy. The latter is accomplished through the use of what are called "axiom" or "partial evaluation" functions. The resulting structure, which is called ALGEBRA, is shown to serve as a model for a family of associative algebras, where the resulting families of equivalence classes of representations for the vectors are uniquely representable. Various related notions, such as topological and decision considerations, are also briefly discussed.After having outlined the development of the system ALGEBRA, polynomial, exterior and series algebras are given as examples to illustrate how the model can be used. It is observed that a wide variety of mathematical data can be handled fairly simply, depending only on the formulation of the problem under consideration. The possibility of constructing various higher level symbol manipulation languages that have a mathematical structure, broad enough in scope to allow an analyst to manipulate various problems in a variety of ways, is also discussed. A programming language called FLAP has already been developed, demonstrating that such a language can indeed be studied and formulated in a consistent manner.

Format-directed list processing in LISP

Abstract: This article describes a notation and a programming language for expressing, from within a LISP system, string transformations such as those performed in COMIT or SNOBOL. A simple transformation (or transformation rule) is specified by providing a pattern which must match the structure to be transformed and a format which specifies how to construct a new structure according to the segmentation specified by the pattern. The patterns and formats are greatly generalized versions of the left-half and right-half rules of COMIT and SNOBOL. For example, elementary patterns and formats can be variable names, results of computations, disjunctive sets, or repeating subpatterns; predicates can be associated with elementary patterns which check relationships among separated elements of the match; it is no longer necessary to restrict the operations to linear strings since elementary patterns can themselves match structures. The FLIP language has been implemented in LISP 1.5, and has been successfully used in such disparate tasks as editing LISP functions and parsing Kleene regular expressions.

MANIP: a computer system for algebra and analytic differentiation

Abstract: A mathematical expression written in FORTRAN notation is stored in the computer as a string of BCD characters. The algebra program operates exhaustively on the expression according to standard laws of algebra which have been translated into a complicated set of logical decision rules for manipulating character sequences. Similarly, the differentiation program finds partial and mixed partial derivatives of any order. Most numeric and algebraic operations are performed automatically. A number of pseudo instructions is available and provides additional flexibility.

GRAD Assistant: a program for symbolic algebraic manipulation and differentiation

Abstract: The General Recursive Algebra and Differentiation Assistant (GRAD Assistant) now under development, is a LISP-based system which can perform symbolic algebraic manipulation and differentiation. It is designed to produce a simplified expression of the result of any, sequence of symbolic manipulations involving addition, multiplication, exponentiation, and their inverses, and differentiation. In particular, it will never over- look that an expression is in fact independent of a particular variable. It does these things without requiring any externally supplied "hints" about difficult expressions. It can read and write a Fortran-like infix notation. It has been used in connection with a heuristic ordinary-differential-equation-solving program and a program that performs calculations arising from Ricmannian geometry.

Programming languages, logic and cooperative games

Abstract: The author presents an application of computers which is not restricted to a single problem area but is perhaps not general purpose enough to be considered a programming language, even a problem-oriented one. The application is a program to eliminate quantifiers and bound variables from logical statements in the theory of addition on the real numbers. These statements differ from work in linear inequalities by formal use of quantifiers. The first, introductory, part of the paper argues that from one point of view, this application does provide a higher level programming language. The second part is a description of the method, due basically to Langford, that has been programmed. The last section describes a subapplication to the problem area of bargaining sets of cooperative games where other interesting non-numerical problems arise.

A syntax-directed approach to automated aids for symbolic mathematics

Abstract: Designing a system for the analytic processing of mathematical functions around a central syntax processor and permitting the user to work within the context of mathematical syntax leads to a flexible system with a broad range of capabilities. One advantage to this approach is that the basic system can be developed without many a priori restrictions on the nature of the mathematical entities to be processed. Once the basic structure has been developed, the user is free to define and operate within his own system of mathematics by modifying or extending the syntax definitions. After the basic system has been developed, higher level operations such as those involving matrices or functions of a complex variable can be added to the existing structure, with a minimum of effort because of the flexibility inherent in the syntax-directed approach. A potential disadvantage to the syntax-directed approach is the length of time required to perform an operation if the processing is handled interpretively.

History, features, and commentary on FORMAC

Abstract: The purpose of this paper is to summarize significant characteristics and features of the FORMAC system. FORMAC (FORmula MAnipulation Compiler), an extension of IBM 7090/94 FORTRAN IV, permits algebraic manipulation of mathematical expressions. The FORMAC system provides such capabilities as symbolic differentiation, substitution, evaluation, expansion, and finding coefficients.This paper comments on the project formation, FORMAC capability and language, overall approach to implementation, application problems for which the system has proved of value, and the results of an experimental venture to develop a time-sharing FORMAC algebraic desk calculator.

An algebraic manipulator using SLIP

Abstract: A notation for representing algebraic expressions to be manipulated by a SLIP type list processor is described in detail. This notation allows Fortran type algebraic expressions to be represented in the form of list structures which will be used in a set of programs which are being developed.

Operators for manipulating language structures

Abstract: The manipulation of symbolic expressions, optimization of computer programs by a compiler, and the use of graphical or pictorial input-output have heretofore been considered to be unrelated problems. The Algorithmic Theory of Language provides a language structure capable of representing both the syntactic and semantic structure of statements in algebraic, procedural, or graphical languages. Utilizing the semantic sequencing information in the structure, “operators” defined for atomic forms may be applied to arbitrarily complex structures to provide a uniform and powerful manipulation capability for these and other areas of application. This paper describes an on-line, experimental version of a system constructed in this manner.