Knuth-Bendix Completion with Modern Termination Checking, Master\u27s Thesis, August 2006

Knuth-Bendix completion is a technique for equational automated theorem proving based on term rewriting. This classic procedure is parametrized by an equational theory and a (well-founded) reduction order used at runtime to ensure termination of intermediate rewriting systems. Any reduction order can be used in principle, but modern completion tools typically implement only a few classes of such orders (e.g., recursive path orders and polynomial orders). Consequently, the theories for which completion can possibly succeed are limited to those compatible with an instance of an implemented class of orders. Finding and specifying a compatible order, even among a small number of classes, is challenging in practice and crucial to the success of the method. In this thesis, a new variant on the Knuth-Bendix completion procedure is developed in which no order is provided by the user. Modern termination-checking methods are instead used to verify termination of rewriting systems. We prove the new method correct and also present an implementation called Slothrop which obtains solutions for theories that do not admit typical orders and that have not previously been solved by a fully automatic tool

Equational methods in first order predicate calculus

We show that the application of the resolution principle to a set of clauses can be regarded as the construction of a term rewriting system confluent on valid formulas. This result allows the extension of usual properties and methods of equational theories (such as Birkhoff's theorem and the Knuth and Bendix completion algorithm) to quantifier-free first order theories. These results are extended to first order predicate calculus in an equational theory, as studied by Plotkin (1972), Slagle (1974) and Lankford (1975). This paper is a continuation of the work of Hsiang & Dershowitz (1983), who have already shown that rewrite methods can be used in first order predicate calculus. The main difference is the following: Hsiang uses rewrite methods only as a refutational proof technique, the initial set of formulas being unsatisfiable iff the equation TRUE = FALSE is generated by the completion algorithm. We generalise these methods to satisfiable theories; in particular, we show that the concept of confluent rewriting system, which is the main tool for studying equational theories, can be extended to any quantifier-free first order theory. Furthermore, we show that rewrite methods can be used even if formulas are kept in clausal form

Rewriting Systems and Embedding of monoids in groups

In this paper, a connection between rewriting systems and embedding of monoids in groups is found. We show that if a group with a positive presentation has a complete rewriting system $\Re$ that satisfies the condition that each rule in $\Re$ with positive left-hand side has a positive right-hand side, then the monoid presented by the subset of positive rules from $\Re$ embeds in the group. As an example, we give a simple proof that right angled Artin monoids embed in the corresponding right angled Artin groups. This is a special case of the well-known result of Paris \cite{paris} that Artin monoids embed in their groups

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

Abstract Canonical Inference

An abstract framework of canonical inference is used to explore how different proof orderings induce different variants of saturation and completeness. Notions like completion, paramodulation, saturation, redundancy elimination, and rewrite-system reduction are connected to proof orderings. Fairness of deductive mechanisms is defined in terms of proof orderings, distinguishing between (ordinary) "fairness," which yields completeness, and "uniform fairness," which yields saturation.Comment: 28 pages, no figures, to appear in ACM Trans. on Computational Logi