The Role of Term Symmetry in E-Unification and E-Completion

A major portion of the work and time involved in completing an incomplete set of reductions using an E-completion procedure such as the one described by Knuth and Bendix [070] or its extension to associative-commutative equational theories as described by Peterson and Stickel [PS81] is spent calculating critical pairs and subsequently testing them for coherence. A pruning technique which removes from consideration those critical pairs that represent redundant or superfluous information, either before, during, or after their calculation, can therefore make a marked difference in the run time and efficiency of an E-completion procedure to which it is applied. The exploitation of term symmetry is one such pruning technique. The calculation of redundant critical pairs can be avoided by detecting the term symmetries that can occur between the subterms of the left-hand side of the major reduction being used, and later between the unifiers of these subterms with the left-hand side of the minor reduction. After calculation, and even after reduction to normal form, the observation of term symmetries can lead to significant savings. The results in this paper were achieved through the development and use of a flexible E-unification algorithm which is currently written to process pairs of terms which may contain any combination of Null-E, C (Commutative), AC (Associative-Commutative) and ACI (Associative-Commutative with Identity) operators. One characteristic of this E-unification algorithm that we have not observed in any other to date is the ability to process a pair of terms which have different ACI top-level operators. In addition, the algorithm is a modular design which is a variation of the Yelick model [Ye85], and is easily extended to process terms containing operators of additional equational theories by simply plugging in a unification module for the new theory

Normalized Rewriting: an Alternative to Rewriting modulo a Set of Equations

AbstractIn the first part of this paper, we introducenormalized rewriting, a new rewrite relation. It generalizes former notions of rewriting modulo a set of equationsE, dropping some conditions onE. For example,Ecan now be the theory of identity, idempotence, the theory of Abelian groups or the theory of commutative rings. We give a new completion algorithm for normalized rewriting. It contains as an instance the usual AC completion algorithm, but also the well-known Buchberger algorithm for computing Gröbner bases of polynomial ideals. In the second part, we investigate the particular case of completion of ground equations. In this case we prove by a uniform method that completion moduloEterminates, for some interesting theoriesE. As a consequence, we obtain the decidability of the word problem for some classes of equational theories, including the AC-ground case (a result known since 1991), the ACUI-ground case (a new result to our knowledge), and the cases of ground equations modulo the theory of Abelian groups and commutative rings, which is already known when the signature contains only constants, but is new otherwise. Finally, we give implementation results which show the efficiency of normalized completion with respect to completion modulo AC

Optimizing mkbTT

We describe performance enhancements that have been added to mkbTT, a modern completion tool combining multi-completion with the use of termination tools

Complete parameterized presentations and almost convex Cayley graphs

This thesis is meant as a contribution to the theory of three classes of groups, those classes being the groups defined by complete parameterized presentations, automatic groups, and groups with almost convex Cayley graphs. Chapter 1 is basically definitions and terminology. Chapter 2 is a short exposition of the theory of automatic groups; we prove only one major result in this chapter (due to (CHEPT)), i.e., that the abelian groups are automatic. In chapter 3 we study presentations of groups and monoids which are complete (with respect to certain orderings of the words in their generators). Such presentations define monoids with fast solutions to their word problems. We define a class of (possibly infinite) presentations which we call r-porameterized, or of type Pr; these presentations are the central theme of this thesis. With the help of the computer program described in chapter 4, we demonstrate that there are group presentations which have infinite r-parameterized completions (i.e. complete supersets), but which have no finite completion with respect to any ShortLex ordering. The 1-parameterized presentations are, arguably, the simplest non finite presentations we can define (at least as far as groups are concerned), but we prove that completeness of such presentations is not in general decidable. Chapter 4 is the description of a (short) program which attempts to complete 1-parameterized group presentations by the Knuth-Bendix method. We conclude the chapter with a short report on its implementation. In chapter 5 we study groups with almost convex Cayley graphs. Such graphs are recursive, but the property of being almost convex does tend to be hard to prove or disprove in practice. We prove that the word length preserving complete groups and the least length bounded automatic groups have almost convex Cayley graphs. We believe that these are strict subclasses because (we shall prove) the group U(3,Z) is almost convex, but is already known not to be automatic and, we conjecture, it has no r-parameterized complete (ShortLex) presentation. We conclude chapter 5 with a slightly generalized, arguably simpler, algebraic proof of J.W. Cannon's theorem that the abelian by finite groups are almost convex