A quantitative study of reductions in algebraic libraries

Abstract

How much of existing computer algebra libraries is amenable to automatic parallelization? This is a difficult topic, yet of practical importance in the era of commodity multicore machines. This paper reports on a quantitative study of reductions in the AXIOMfamily computer algebra systems. The experiment builds on the introduction of assumptions in OpenAxiom. It identifies a variety of reductions that are candidate for implicit concurrent execution. An assumption is an axiomatic statement of an algebraic property. We hope that this study will encourage wider adoption of axioms, not just for the purpose of expression simplification and provably correct libraries, but also to enable derivation of implicit concurrency in a scalable fashion.

An automatic parallelization framework for OpenAxiom

“…The framework has discovered rich parallelization opportunities implied by reductions in OpenAxiom algebra library [1]. With the framework, we parallelized a software installation test, a set of OpenAxiom algebra library functions, and a user application on homotopy continuation method [2].…”

An automatic parallelization framework for algebraic computation systems

“…Algorithms for detecting reduce call and built-in reduce operator are simpler. More in-depth discussion may be found in our previous work [12].…”

“…We previously reported [12] on the rich opportunity for parallelization in algebraic libraries such as those of the AXIOM family Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee.…”

“…All that is needed is a description of algebraic properties of the input program fragment. The description language is an extension [12] of the "category" subset of the Spad programming language. For example, the content of a univariate polynomial with integer coefficients P = n 0 anX n is the greatest common divisor of all the polynomial coefficients.…”

“…That computation strategy corresponds to the following program: where the function parallelLeftReduce performs reduction in parallel. Notice that while the example uses a polynomial with integer coefficients, all we need is a domain of computation where GCD computation makes sense and is effective, The function gcd remains a monoid operation, which we express in our extension to Spad [12] as: The parameterized assumption statement instructs the reduction detector of the program transformation framework that it can assume (without having to conduct a proof) that gcd is a monoid operation over any GCD domain. The contributions of this paper include:…”