Richard D. Jenks

Scratchpad II: an abstract datatype system for mathematical computation

Scratchpad II is an abstract datatype language and system that is under development in the Computer Algebra Group, Mathematical Sciences Department, at the IBM Thomas J. Watson Research Center. Many different kinds of computational objects and data structures are provided. Facilities for computation include symbolic integration, differentiation, factorization, solution of equations and linear algebra. Code economy and modularity is achieved by having polymorphic packages of functions that may create datatypes. The use of categories makes these facilities as general as possible.

The type inference and coercion facilities in the scratchpad II interpreter

The Scratchpad II system is an abstract datatype programming language, a compiler for the language, a library of packages of polymorphic functions and parametrized abstract datatypes, and an interpreter that provides sophisticated type inference and coercion facilities. Although originally designed for the implementation of symbolic mathematical algorithms, Scratchpad II is a general purpose programming language. This paper discusses aspects of the implementation of the interpreter and how it attempts to provide a user friendly and relatively weakly typed front end for the strongly typed programming language.

The SCRATCHPAD language

SCRATCHPAD is an interactive system for symbolic mathematical computation. Its user language, originally intended as a special-purpose non-procedural language, was designed to capture the style and succinctness of common mathematical notations, and to serve as a useful, effective tool for on-line problem solving. This paper describes extensions to the language which enable it to serve also as a high-level programming language, both for the formal description of mathematical algorithms and their efficient implementation.

A pattern compiler

A pattern compiler for the SCRATCHPAD system provides an efficient implementation of sets of user-defined pattern-replacement rules for symbolic mathematical computation such as tables of integrals or summation identities. Rules are compiled together, with common search paths merged and factored out and with the resulting code optimized for efficient recognition over all patterns. Matching principally involves structural comparison of expression trees and evaluation of predicates. Pattern recognizers are “fully compiled” if values of match variables can be determined by solving equations at compile time. Recognition times for several pattern matchers are compared.

The SCRATCHPAD system

SCRATCHPAD is an experimental interactive for symbolic computation in use at the IBM Thomas son Research Center. SCRATCHPAD has facilities manipulation of multi-indexed variables and functions, and infinite sums, products, and sequences, and polynomials of arrays. Among the specific capabilities offered system are the following: Multiple-precision integer arithmetic Rational number arithmetic Polynomial and rational function manipulation Differentiation Symbolic integration Manipulation of formal power series Symbolic solution of systems of polynomial equations Two-dimensional output of mathematical expressions

META/LISP: an interactive translator writing system

META/LISP is a general purpose translator writing system for IBM System/360 currently running on TSS, CP/CMS, and OS/360. The input to the system is a source program which simultaneously describes 1) the syntax of some input data to be translated and 2) algorithms which operate on the input data and a pushdown stack to accomplish a desired translation; the output of the system is a compiled program for translating that input data. In particular when the input data are statements of a higher-level language to be translated into assembly language, META/LISP serves as a compiler-compiler. META/LISP uses the top-down syntax-directed approach which makes the system extremely attractive for the design and implementation of experimental languages; using META/LISP such compilers are easy to write, easy to check out, and - most importantly - easy to modify interactively. The appendices which follow a rather complete description of the system include a self-description of the META/LISP compiler.

Problem #11: generation of Runge-Kutta equations

Generate a set of equations for an explicit k-th order, m stage, Runge-Kutta method for integrating an autonomous system of ordinary differential equations, k and m as large as possible. The number of conditions and variables for various k and m are given in Table 1. Tabulate the costs C(m,k).

SCRATCHPAD/360: reflections on a language design

The key concepts of the SCRATCHPAD language are described, assessed, and illustrated by an example. The language was originally intended as an interactive problem solving language for symbolic mathematics. Nevertheless, as this paper intends to show, it can be used as a programming language as well.

How to make AXIOM into a scratchpad

Scratchpad [GrJe71] was a computer algebra system developed in the early 1970s. Like M&M (Maple [CGG91ab] and Mathematical [W01S92]) and other systems today, Scratchpad had one principal representation for mathematical formulae based on “expression trees”. Its user interface design was based on a pattern-matching paradigm with infinite rewriterule semantics, providing what we believe to be the most natural paradigm for interactive symbolic problem solving. Like M&M, however, user programs were interpreted, often resulting in poor performance relative to similar facilities coded in standard programming languages such as FORTRAN and C. Scratchpad development stopped in 1976 giving way to a new system design ([ JenR79], [JeTr81]) that evolved into AXIOM [JeSu92]. AXIOM has a strongly-typed programming language for building a library of parameterized types and algorithms, and a type-inferencing interpreter that accesses the library and can build any of an infinite number of types for interactive use. We suggest that the addition of an expression tree type to AXIOM can allow users to operate with the same freedom and convenience of untyped systems without giving up the expressive power and run-time efficiency provided by the type system. We also present a design that supports a multiplicity y of programming styles, from the Scratchpad pattern-matching paradigm to functional programming to more conventional procedural programming. The resulting design seems to us to combine the best features of Scratchpad with current AXIOM and to offer a most attractive, flexible, and user-friendly environment for interactive problem solving. Section 2 is a discussion of design issues contrasting AXIOMwith other symbolic systems. Sections 3 and 4 is an assessment of AXIOM’s current design for building libraries and interactive use. Section 5 describes a new interface design for AXIOM, its resulting paradigms, and its underlying semantic model. Section 6 compares this work with others.

A SCRATCHPAD solution to problem #7

The function F(x)= (1/2 - x)(1 - x2)1/2 + x(1 + (1 - (1/2 + x)2)1/2) has a maximum of y=0.674981 at x=0.3437715 (Figure 1). This y value was found [1] to be a root of an irreducible polynomial over the integers of degree 10. The object of problem #7 is to find this polynomial. In addition, we obtain these x and y values to high accuracy and verify that the above y is a global maximum over the domain of interest[2]: 0 < x < 1/2. The SCRATCHPAD conversation below involves the following 10 steps.