Anne Heyworth

In this paper we show how string rewriting methods can be applied to give a new method of computing double cosets. Previous methods for double cosets were enumerative and thus restricted to finite examples. Our rewriting methods do not suffer this restriction and we present some examples of infinite double coset systems which can now easily be solv...

A rewriting system is a set of equations over a given set of terms called rules that characterize a system of computation and is a powerful general method for providing decision procedures of equational theories, based upon the principle of replacing subterms of an expression with other terms. In particular, a string rewriting system is usually ass...

One problem in computational group theory is to find a presentation of the subgroup generated by a set of elements of a group. The Reidemeister-Schreier algorithm was developed in the 1930’s and gives a solution based upon enumerative techniques. This however means the algorithm can only be applied to finite groups. This paper proposes a rewriting...

Kan extensions over the category of Sets provide a unifying framework for computation of group, monoid and category actions allowing a number of diverse problems to be solved with a generalised form of string rewriting. This paper extends these techniques to K-algebras and K-categories by using Gröbner basis techniques to compute Kan extensions ove...

Standard noncommutative Gröbner basis procedures are used for computing ideals of free noncommutative polynomial rings over fields. This paper describes Gröbner basis procedures for one-sided ideals in finitely presented noncommutative algebras over fields. The polynomials defining a K-algebra A as a quotient of a free K-algebra are combined with t...

As autonomous mobile robots grow increasingly complex, the need for a method of modeling and testing their control systems becomes greater. This paper discusses the use of Petri nets as a means of modeling and testing the control of a mobile robot, concentrating specifically on the reachability testing of the Petri net model through the use of Grbn...

This paper involves categories and computer science. The paper is motivated by a question which arises from two pieces of research. Firstly, the work of Brown and Heyworth which extends rewriting techniques to enable the computation of left Kan extensions over the category of sets. It is well known that left Kan extensions can be defined over categ...

This paper contains introductory material on Petri nets and Groebner basis theory and makes some observations on the relation between the two areas. The aim of the paper is to show how Groebner basis procedures can be applied to the problem of reachability in Petri nets, and to give details of an application to testing models of navigational system...

The basic method of rewriting for words in a free monoid given a monoid presentation is extended to rewriting for paths in a free category given a 'Kan extension presentation'. This is related to work of Carmody-Walters on the Todd-Coxeter procedure for Kan extensions, but allows for the output data to be infinite, described by a language. The resu...

The key idea is that rewriting procedures can be enhanced so that they not only rewrite words but record (log) how the rewriting has taken place. We introduce logged rewrite systems and present a variation on the Knuth-Bendix algorithm for obtaining (where possible) complete logged rewrite systems. This procedure is then applied to work of Brown an...

Presentations of Kan extensions of category actions provide a natural framework for expressing induced actions, and therefore a range of different combinatorial problems. Rewrite systems for Kan extensions have been defined and a variation on the Knuth-Bendix completion procedure can be used to complete them -- when possible. Regular languages and...

Introduction This is a brief account of work of Brown and Heyworth [1] on extensions of rewriting methods. The standard expression of such methods is in terms of words w in a free monoid on a set . This may be extended to terms xjw where x belongs to a set X and the link between x and w is in terms of an action. More precisely, we suppose a monoid...

Rewriting for semigroups is a special case of Groebner basis theory for noncommutative polynomial algebras. The fact is a kind of folklore but is not fully recognised. The aim of this paper is to elucidate this relationship, showing that the noncommutative Buchberger algorithm corresponds step-by-step to the Knuth-Bendix completion procedure.

This thesis concentrates on the development and application of rewriting and Groebner basis methods to a range of combinatorial problems. Chapter Two contains the most important result, which is the application of Knuth-Bendix procedures to Kan extensions, showing how rewriting provides a useful method for attempting to solve a variety of combinato...

Abstract The kan package was originally implemented in 1997 using the GAP 3 language, to compute induced actions of categories, when the first author was studying for a Ph.D. in Bangor. This version only provides functions for the computation,of normal forms of representatives of double cosets of finitely presented groups, Bug reports, suggestions...