Robert S. Sutor

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.

A first report on the A # compiler

Abstract Machine A major part of the A ] compiler is concerned with producingoptimized intermediate code, or Foam code. “Foam” is anacronym for “First Order Abstract Machine.” The abstractmachine is first order in the sense that it does not treat itstypes as values.Foam is designed to contain only those concepts whichcan have an efficient realization in both Lisp and C. Forexample it is not possible to take an address of a variablebecause that would be inefficient in Lisp (a closure wouldbe created). Nor are dynamic type tests allowed, as thatwould be inefficient in C. We have been asked how the lackof address arithmetic limits the potential performance ofcompiled A ] vs hand-coded C which uses pointers to traversearrays in inner loops. It is our experience that this is a minorconcern on current architectures with optimizing compilers.Foam is not restricted to the precise intersection of Cand Lisp. Some aspects are handled by support libraries.Big integer arithmetic is assumed as part of Foam, and thisis provided as a library for C. Also the memory model differsfrom both C and Lisp in some details: garbage collection isassumed (this is a run time support library in C) and it ispossible to make an explicit request to free storage (in Lispthis is ignored).A Foam program is comprised of a flat sequence of com-mands. Foam types have various sizes and uses. or example,“Char” is a text character whereas “Byte” is a charactersized integer, “DFlo” is a double precision floating point,“Ptr” can point to an array, record, arbitrary sized inte-ger, etc. Reference instructions contain the kind of refer-ence and the position, e.g., “Loc 3” refers to the third localvariable of the current function and “RElt 7 x 2” indicatesthe 2nd field of the record x, using the 7th layout format.Foam operations consist of instructions, such as “If b n,”which indicates that if b is true then proceed to label n, andbuiltin operations, e.g., “HIntLT a b” is a half-word-integerless-than comparison. The builtin operations are type spe-cific and conversion operations are generally provided. Adetailed description of Foam is given elsewhere [26].27

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.

A report on OpenMath: a protocol for the exchange of mathematical information

The proliferation of general purpose mathematical software and specialized mathematical libraries has provided a wealth of computing resources for students, scientists, and engineers. The challenge remains to harness the power of these independent software tools within a single framework. OpenMath will provide formats and a protocol for the exchange of mathematical expressions and objects, thus enabling a unification from a users point of view. This work will also allow the inclusion of mathematical objects in a universal format within databases and electronically published scientific and technical documents.This report will detail the steps taken to date by the OpenMath Consortium to achieve the aforementioned goals. In particular, §2 provides an overview of OpenMath. Emphasis is placed on OpenMaths design goals, initial target applications, and Consortium structure. In §3, an exposition of the levels of the OpenMath model which maps mathematical concepts to their respective concrete representations is provided. Some issues that arise when a mathematical object is transcribed between its visual form and internal representation are presented in §4. Finally, §5 and §6 communicate the current state of the OpenMath prototyping efforts and concomitantly detail the future directions of the OpenMath consortium.

The Scratchpad II Computer Algebra Language and System

Seratchpad II is a computer algebra language and system that features generic operations and abstract datatypes that are parameterized, extensible and dynamically constructible. Algorithms may be expressed at their most natural level of abstraction and defined to operate in the most general setting possible. Although designed specifically for computer algebra, Scratehpad II nevertheless provides a general-purpose programming language and environment.

Introduction to AXIOM

Welcome to the world of AXIOM. We call AXIOM a scientific computation system: a self-contained toolbox designed to meet your scientific programming needs, from symbolics, to numerics, to graphics.

An Overview of AXIOM

Welcome to the AXIOM environment for interactive computation and problem solving. Consider this chapter a brief, whirlwind tour of the AXIOM world. We introduce you to AXIOM’S graphics and the AXIOM language. Then we give a sampling of the large variety of facilities in the AXIOM system, ranging from the various kinds of numbers, to data types (like lists, arrays, and sets) and mathematical objects (like matrices, integrals, and differential equations). We conclude with the discussion of system commands and an interactive “undo.”

User-Defined Functions, Macros and Rules

In this chapter we show you how to write functions and macros, and we explain how AXIOM looks for and applies them. We show some simple one-line examples of functions, together with larger ones that are defined piece-by-piece or through the use of piles.

Some Examples of Domains and Packages

In this chapter we show examples of many of the most commonly used AXIOM domains and packages. The sections are organized by constructor names.

Using Types and Modes

In this chapter we look at the key notion of type and its generalization mode. We show that every AXIOM object has a type that determines what you can do with the object. In particular, we explain how to use types to call specific functions from particular parts of the library and how types and modes can be used to create new objects from old. We also look at Record and Union types and the special type Any. Finally, we give you an idea of how AXIOM manipulates types and modes internally to resolve ambiguities.