Symmetrization based completion

We argue that most completion procedures for finitely presented algebras can be simulated by term completion procedures based on a generalized symmetrization process. Therefore we present three different constructive definitions of symmetrization procedures that can take the role of the orientation step in a symmetrization based completion procedure. We investigate confluence and compatibility properties of the symmetrized rules computed by the different symmetrization procedures. Based on semicompatibility properties we can present a generic version of the critical pair theorem that specializes to the critical pair theorems of Knuth-Bendix completion procedures and algebraic completion procedures like Buchberger's algorithm respectively. This critical pair theorem also applies to symmetrization based completion procedures using a normalized reduction relation if the result of the symmetrization is both semi-compatible and semi-stable. We conclude our paper showing how a generic Buchberger algorithm for polynomials over arbitrary finitely presented rings can be formulated as a symmetrization based completion procedure

Properties of Monoids That Are Presented By Finite Convergent String-Rewriting Systems - a Survey

In recent years a number of conditions has been established that a monoid must necessarily satisfy if it is to have a presentation through some finite convergent stringrewriting system. Here we give a survey on this development, explaining these necessary conditions in detail and describing the relationships between them. 1 Introduction String-rewriting systems, also known as semi-Thue systems, have played a major role in the development of theoretical computer science. On the one hand, they give a calculus that is equivalent to that of the Turing machine (see, e.g., [Dav58]), and in this way they capture the notion of `effective computability' that is central to computer science. On the other hand, in the phrase-structure grammars introduced by N. Chomsky they are used as sets of productions, which form the essential part of these grammars [HoUl79]. In this way string-rewriting systems are at the very heart of formal language theory. Finally, they are also used in combinatorial semig..