Abstract not found

We prove the completeness of (basic) deduction strategies with constrained clauses modulo associativity and commutativity (AC). Here each inference generates one single conclusion with an additional equality s = AC t in its constraint (instead of one conclusion for each minimal AC-unifier, i.e. exponentially many). Furthermore, computing AC-unifiers is not needed at all. A clause C [[ T ]] is redundant if the constraint T is not AC-unifiable. If C is the empty clause this has to be decided to know whether C [[ T ]] denotes an inconsistency. In all other cases any sound method to detect unsatisfiable constraints can be used. 1 Introduction Some fundamental ideas on applying symbolic constraints to theorem proving were given in [KKR90], where a constrained clause is a shorthand for its (infinite) set of ground instances satisfying the constraint. In a constrained equation f(x) ' a [[ x = g(y) ]], the equality `=' of the constraint is usually interpreted in T (F) (syntactic equality), ...

. We survey recent results about ordering constraints on trees and discuss their applications. Our main interest lies in the family of recursive path orderings which enjoy the properties of being total, well-founded and compatible with the tree constructors. The paper includes some new results, in particular the undecidability of the theory of lexicographic path orderings in case of a non-unary signature. 1 Symbolic Constraints Constraints on trees are becoming popular in automated theorem proving, logic programming and in other fields thanks to their potential to represent large or even infinite sets of formulae in a nice and compact way. More precisely, a symbolic constraint system, also called a constraint system on trees, consists of a fragment of first-order logic over a set of predicate symbols P and a set of function symbols F , together with a fixed interpretation of the predicate symbols in the algebra of finite trees T (F) (or sometimes the algebra of infinite trees I(F)) ov...

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 ...

. A new path ordering for showing termination of associativecommutative (AC) rewrite systems is defined. If the precedence relation on function symbols is total, the ordering is total on ground terms, but unlike the ordering proposed by Rubio and Nieuwenhuis, this ordering can orient the distributivity property in the proper direction. The ordering is defined in a natural way using recursive path ordering with status as the underlying basis. This settles a longstanding problem in termination orderings for AC rewrite systems. The ordering can be used to define an ordering on nonground terms. 1 Introduction Rewriting techniques reduce the search space for finding proofs substantially because of the ability to orient equality, which is symmetric, into a terminating directed rewrite relation (!), which is anti-symmetric, using well founded orderings. Rules are used for "simplifying" expressions by repeatedly replacing instances of left-hand sides by the corresponding right-hand s...

A new criterion for termination of rewriting has been described by Arts and Giesl in 1997. We show how this criterion can be generalized to rewriting modulo associativity and commutativity. We also show how one can build weak AC-compatible reduction orderings which may be used in this criterion.

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...

We define a superposition calculus specialized for abelian groups represented as integer modules, and show its refutational completeness. This allows to substantially reduce the number of inferences compared to a standard superposition prover which applies the axioms directly. Specifically, equational literals are simplified, so that only the maximal term of the sums is on the left-hand side. Only certain minimal superpositions need to be considered; other superpositions which a standard prover would consider become redundant. This not only reduces the number of inferences, but also reduces the size of the AC-unification problems which are generated. That is, AC-unification is not necessary at the top of a term, only below some non-AC-symbol. Further, we consider situations where the axioms give rise to variable overlaps and develop techniques to avoid these explosive cases where possible. 1 Introduction Historically, starting from plain resolution, more and more problematic axioms ha...

Abstract. We present a method of integrating linear rational arithmetic into superposition calculus for first-order logic. One of our main results is completeness of the resulting calculus under some finiteness assumptions. 1

. We develop a proof technique for dealing with narrowing and refutational theorem proving in a uniform way, clarifying the exact relationship between the existing results in both fields and allowing us to obtain several new results. Refinements of narrowing (basic, LSE, etc.) are instances of the technique, but are also defined here for arbitrary (possibly ordering and/or equality constrained or not yet convergent or saturated) Horn clauses, and shown compatible with simplification and other redundancy notions. By narrowing modulo equational theories like AC, compact representations of solutions, expressed by AC-equality constraints, can be obtained. Computing AC-unifiers is only needed at the end if one wants to "uncompress" such a constraint into its (doubly exponentially many) concrete substitutions. 1 Introduction Answer computation in some logic is at the heart of many applications in computer science, such as (functional) logic programming, automated theorem proving /discoverin...