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 ...

We define a simplification ordering on terms which is AC-compatible and total on nonAC -equivalent ground terms, without any restrictions on the signature like the number of AC-symbols or free symbols. Unlike previous work by Narendran and Rusinowitch [NR91], our AC-RPO ordering is not based on polynomial interpretations, but on a simple extension of the well-known RPO ordering (with a total (arbitrary) precedence on the function symbols). This solves an open question posed e.g. by Bachmair [Bac92]. A second difference is that this ordering is also defined on terms with variables, which makes it applicable in practice for complete theorem proving strategies with built-in AC-unification and for orienting non-ground rewrite systems. The ordering is defined in a simple way by means of rewrite rules, and can be easily implemented, since its main component is RPO. 1 Introduction Automated termination proofs are well-known to be crucial for using rewriting-like methods in theorem proving an...

It is shown that for sets of Horn clauses saturated under basic paramodulation, the word and unifiability problems are in NP, and the number of minimal unifiers is simply exponential (i). For Horn sets saturated wrt. a special ordering under the more restrictive inference rule of basic superposition, the word and unifiability problems are still decidable and unification is finitary (ii). These two results are applied to the following languages. For shallow presentations (equations with variables at depth at most one) we show that the closure under paramodulation can be computed in polynomial time. Applying result (i), it follows that shallow unifiability is in NP, which is optimal since unifiability in ground theories is already NP-hard. The shallow word problem is even shown to be polynomial. Generalizing shallow theories to the Horn case, we obtain (two versions of) a language we call Catalog, a natural extension of Datalog to include functions and equality. The closure under paramo...

) Robert Nieuwenhuis Technical University of Catalonia Pau Gargallo 5, 08028 Barcelona, Spain E-mail: roberto@lsi.upc.es. Abstract We prove that for sets of Horn clauses saturated under basic paramodulation, the word and unifiability problems are in NP, and the number of minimal unifiers is simply exponential (i). For Horn sets saturated wrt. a special ordering under the more restrictive inference rule of basic superposition, the word and unifiability problems are still decidable and unification is finitary (ii). We define standard theories, which include and significantly extend shallow theories. Standard presentations can be finitely closed under superposition and result (ii) applies. Generalizing shallow theories to the Horn case, we obtain (two versions of) a language we call Catalog, a natural extension of Datalog to include functions and equality. The closure under paramodulation is finite for Catalog sets, hence (i) applies. Since for shallow sets this closure is even polynom...

We introduce the notion of an associative-commutative congruence closure and show how such closures can be constructed via completion-like transition rules. This method is based on combining completion algorithms for theories over disjoint signatures to produce a convergent rewrite system over an extended signature. This approach can also be used to solve the word problem for ground AC-theories without the need for AC-simplification orderings total on ground terms. Associative-commutative congruence closure provides a novel way to construct a convergent rewrite system for a ground AC-theory. This is done by transforming an AC-congruence closure, which is described by rewrite rules over an extended signature, to a rewrite system over the original signature. The set of rewrite rules thus obtained is convergent with respect to a new and simpler notion of associative-commutative reduction.

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...

We introduce a generic definition of reduction orderings on term algebras containing associative-commutative (hereafter denoted AC) operators. These orderings are compatible with the AC theory hence makes them suitable for use in deduction systems where AC operators are built-in. Furthermore, they have the nice property of being total on AC classes of ground terms, a required property for example to avoid failure in ACcompletion, or to insure completeness of ordered strategies in first-order theorem proving with built-in AC operators. We show that the two definitions already known of such total and AC-compatible orderings [24, 25] are actually instances of our definition. Finally, we find new such orderings which have more properties, first an ordering based on an integer polynomial interpretation, answering positively to a question left open by Narendran and Rusinowitch, and second an ordering which allow to orient the distributivity axiom in the usual way, answering positively to a ...

Franz Baader LuFg Theoretical Computer Science, RWTH Aachen Ahornstraße 55, 52074 Aachen, Germany e-mail: baader@informatik.rwth-aachen.de 1 Introduction Reduction orderings that are total on ground terms play an important role in many areas of automated deduction. For example, unfailing completion [4]---a variant of Knuth-Bendix completion that avoids failure due to incomparable critical pairs---presupposes such an ordering. In addition, using a reduction ordering that is total on ground terms, one can show that any finite set of ground equations has a decidable word problem [13, 20]. It is very easy to obtain such orderings. Indeed, many of the standard methods for constructing reduction orderings yield orderings that are total on ground terms: both Knuth-Bendix orderings [12] and lexicographic path orderings [10] are total on ground terms if they are based on a total precedence ordering on the set of function symbols. Things become more complex if one is interested in reduction or...

Abstract. Bit vectors and bit operations are proposed for efficient propositional inference. Bit arithmetic has efficient software and hardware implementations, which can be put to advantage in Boolean satisability procedures. Sets of variables are represented as bit vectors and formulæ as matrices. Symbolic operations are performed by bit arithmetic. As examples of inference done in this fashion, we describe ground resolution and ground completion. “It does take a little bit of inference.” – Tony Fratto, Deputy Press Secretary, USA 1

Abstract. AC-completion efficiently handles equality modulo associative and commutative function symbols. When the input is ground, the procedure terminates and provides a decision algorithm for the word problem. In this paper, we present a modular extension of ground ACcompletion for deciding formulas in the combination of the theory of equality with user-defined AC symbols, uninterpreted symbols and an arbitrary signature disjoint Shostak theory X. Our algorithm, called AC(X), is obtained by augmenting in a modular way ground AC-completion with the canonizer and solver present for the theory X. This integration rests on canonized rewriting, a new relation reminiscent to normalized rewriting, which integrates canonizers in rewriting steps. AC(X) is proved sound, complete and terminating, and is implemented to extend the core of the Alt-Ergo theorem prover. Key words: decision procedure, associativity and commutativity, rewriting, AC-completion, SMT solvers, Shostak’s algorithm. 1