Based on the fifth Mid-Atlantic Algebra Conference held recently at George Mason University, Fairfax, Virginia. Focuses on both the practical and theoretical aspects of computational algebra. Demonstrates specific computer packages, including the use of CREP to study the representation of theory for finite dimensional algebras and Axiom to study algebras of finite rank.

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Information

Publisher

CRC Press

Year

2018

ISBN

9781351458986

Edition

Topic

Mathematics

Subtopic

Mathematics General

Index

Mathematics

Parallel Computations in *-Semirings

S. KAMAL ABDALI¹ National Science Foundation, Washington, DC 20550

Abstract

Abstract *-semirings are algebraic structures that provide a unified approach to several problem classes in computer science and operations research. For example, *-semirings can be used to describe the algebra related to regular expressions, graph-theoretical path problems, and compiled-code optimization. The theory of matrices over *-semirings has a number of similarities to linear algebra. For example, eliminants and asterates (closures) behave analogously in many ways to determinants and matrix inverses. Matrix computations over *-semirings are interesting in their own right as well as because of their potential applications to linear algebra. This paper uses the eliminant formulation of *-semiring properties to derive parallel algorithms for three kinds of problems involving matrices over *-semirings: eliminant computation, solution of linear systems over *-semirings, and matrix asteration. The algorithms discussed allow the most general computations in which the asteration operation in the base *-semiring is assumed to be non-trivial, and the matrices are assumed to be dense and without any particular structure.

1 Introduction

*-Semirings (also called closed semirings) are algebraic structures that provide a unified approach to a number of problems in computer science and operations research. Examples of such problems include finding shortest or most reliable paths in graphs, finding maximum network flows, cutset enumeration, computing the transitive closure of binary relations, finding the regular expression to describe the language accepted by a finite automaton, solving systems of linear equations, etc. The reader is referred to Conway[6], Carré[2], Tarjan[13], and Gondran & Minoux[8] for some of the most notable formulations of t...

Cover
Half Title
Title Page
Copyright Page
Table of Contents
Preface
Participants
1. Parallel Computations in *-Semirings
2. The Group of Periods of the Sum of Two Functions
3. On Term-Orderings and Rankings
4. Computational Aspects in the Representation Theory of Finite Dimensional Algebras
5. Graphs and the Growth of Monomial Algebras
6. Generating Prime Ideals in the Minkowski Ring of Polytopes
7. Computations in Algebras of Finite Rank
8. An Introduction to Noncommutative Gröbner Bases
9. The Separation Problem of Real Function Partitions and its Significance in Solid Modeling
10. A Survey on the Theory of Weakly-Injective Modules
11. Distributing Symbolic Computations on a Network of Workstations
12. Calculations of Growth and Hilbert Series by Computer