Abstract not found

this paper is to make the similarity between Knuth-Bendix completion and the Buchberger algorithm explicit, by describing a general algorithm called S-normalized completion where S is a parameter, such that both algorithms are Normalized Rewriting: an alternative to Rewriting modulo a Set of Equations 3 instances of this general algorithm for a particular choice of S. This has been achieved in two steps.

We introduce normalised rewriting, a new rewrite relation. It generalises former notions of rewriting modulo E, dropping some conditions on E. For example, E can now be the theory of identity, idempotency, the theory of Abelian groups, the theory of commutative rings. We give a new completion algorithm for normalised rewriting. It contains as an instance the usual AC completion algorithm, but also the wellknown Buchberger's algorithm for computing standard bases of polynomial ideals. We investigate the particular case of completion of ground equations, In this case we prove by a uniform method that completion modulo E terminates, for some interesting E. As a consequence, we obtain the decidability of the word problem for some classes of equational theories. We give implementation results which shows the efficiency of normalised completion with respect to completion modulo AC. 1 Introduction Equational axioms are very common in most sciences, including computer science. Equations can ...

Introduction Interest in the theory and applications of rewriting has been growing rapidly, as evidenced in part by four conference proceedings #including this one# #15, 26, 41,66#; three workshop proceedings #33, 47, 77#; #ve special journal issues #5,88, 24, 40, 67#; more than ten surveys #2,7,27, 28, 44, 56,57,76, 82, 81#; one edited collection of papers #1#; four monographs #3, 12,55,65#; and seven books #four of them still in progress# #8,9, 35, 54, 60,75, 84#. To encourage and stimulate continued progress in this area, wehave collected #with the help of colleagues# a number of problems that appear to us to be of interest and regarding whichwe do not know the answer. Questions on rewriting and other equational paradigms have been included; manyhave not aged su#ciently to be accorded the appellation #open problem". Wehave limited ourselves to theoretical questions, though there are certainly many additional interesting questions relating to applications and implementation

Rewriting with associativity, commutativity and identity has been an open problem for a long time. In 1989, Baird, Peterson and Wilkerson introduced the notion of constrained rewriting, to avoid the problem of non-termination inherent to the use of identities. We build up on this idea in two ways: by giving a complete set of rules for completion modulo these axioms; by showing how to build appropriate orderings for proving termination of constrained rewriting modulo associativity, commutativity and identity. 1 Introduction Equations are ubiquitous in mathematics and the sciences. Among the most common equations are associativity, commutativity and identity (existence of a neutral element). Rewriting is an efficient way of reasoning with equations, introduced by Knuth and Bendix [12]. When rewriting, equations are used in one direction chosen once and for all. Unfortunately, orientation alone is not a complete inference rule: given a set of equational axioms E, there may be equal terms...

. This paper presents three approaches dealing with constraints in automated deduction. Each of them illustrates a different point. The expression of strategies using constraints is shown through the example of a completion process using ordered and basic strategies. The schematization of complex unification problems through constraints is illustrated by the example of an equational theorem prover with associativity and commutativity axioms. The incorporation of built-in theories in a deduction process is done for a narrowing process which solves queries in theories defined by rewrite rules with built-in constraints. Advantages of using constraints in automated deduction are emphasized and new challenging problems in this area are pointed out. 1 Motivations Constraints have been introduced in automated deduction since about 1990, although one could find similar ideas in theory resolution [32] and in higher-order resolution [16]. The idea is to distinguish two levels of deduction and t...

. The inefficiency of AC-completion is mainly due to the doubly exponential number of AC-unifiers and thereby of critical pairs generated. We present AC-complete E-unification, a new technique whose goal is to reduce the number of AC-critical pairs inferred by performing unification in a extension E of AC (e.g. ACU, Abelian groups, Boolean rings, ...) in the process of normalized completion [21]. The idea is to represent complete sets of AC-unifiers by (smaller) sets of E-unifiers. Not only the theories E used for unification have exponentially fewer most general unifiers than AC, but one can remove from a complete set of E-unifiers those solutions which have no E-instance which is an AC-unifier. First, we define AC-complete E-unification and describe its fundamental properties. We show how AC-complete E-unification can be done in the elementary case, and how the known combination techniques for unification algorithms can be reused for our purposes. Finally, we give some evidence of t...

of the Thesis Term Rewriting in Associative Commutative Theories with Identities by Martin Joachim Henz Master of Science in Computer Science State University of New York at Stony Brook 1991 Versions of constraint rewriting for completion of rewrite systems in the presence of associative commutative operators with identities have been proposed, in which constraints are used to limit the applicability of rewrite rules. We extend these approaches such that the initially given equations can contain constraints, and such that a suitable version of unification modulo associativity, commutativity and identity can be interleaved with the process of completion. iii To my parents Albert and Klara Henz and my wife Kelly Reedy. Contents Abstract iii Acknowledgements ix 1 Introduction 1 2 Preliminaries 4 2.1 Terms : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.2 Relations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8 2.3 The Associative Commutative Theory wi...

. Automated deduction motivates the introduction of several extensions of rewriting, especially ordered rewriting, class rewriting and rewriting with constraints. This paper is a survey of these three notions, shows the evolution between them and their increasing power of expressivity. 1 Introduction Term rewriting techniques have a wide range of applications in mainly two domains: the first one is the operational semantics of logico-functional programming languages. This area led to various extensions of the rewriting concept, like order-sorted rewriting [29], conditional rewriting [16], priority rewriting [38], concurrent rewriting [18], or graph rewriting [11] : : : . The second domain is automated theorem proving where rewriting techniques are of primarily use in provers using demodulation or simplification inference rules to prune the search space. In this context, it appears that most of interesting proofs in mathematical structures, set and graph theory, or geometry, involv...