Abstract not found

Solving an equation in an algebra of terms is known as unification. Solving more complex formulas combining equations and involving in particular negation is called disunification. With such a broad definition, many works fall into the scope of disunification. The goal of this paper is to survey these works and bring them together in a same framework. R'esum'e On appelle habituellement (algorithme d') unification un algorithme de r'esolution d'une 'equation dans une alg`ebre de termes. La r'esolution de formules plus complexes, comportant en particulier des n'egations, est appel'ee ici disunification. Avec une d'efinition aussi 'etendue, de nombreux travaux peuvent etre consid'er'es comme portant sur la disunification. L'objet de cet article de synth`ese est de rassembler tous ces travaux dans un meme formalisme. Laboratoire de Recherche en Informatique, Bat. 490, Universit'e de Paris-Sud, 91405 ORSAY cedex, France. E-mail: comon@lri.lri.fr i Contents 1 Syntax 5 1.1 Basic Defini...

Previous work on combination techniques considered the question of how to combine unification algorithms for disjoint equational theories E 1 ; : : : ; E n in order to obtain a unification algorithm for the union E 1 [ : : : [ E n of the theories. Here we want to show that variants of this method may be used to decide solvability and ground solvability of disunification problems in E 1 [ : : : [E n . Our first result says that solvability of disunification problems in the free algebra of the combined theory E 1 [ : : : [E n is decidable if solvability of disunification problems with linear constant restrictions in the free algebras of the theories E i (i = 1; : : : ; n) is decidable. In order to decide ground solvability (i.e., solvability in the initial algebra) of disunification problems in E 1 [ : : : [ E n we have to consider a new kind of subproblem for the particular theories E i , namely solvability (in the free algebra) of disunification problems with linear constant restricti...

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 ...

. In a previous paper we have introduced a method that allows one to combine decision procedures for unifiability in disjoint equational theories. Lately, it has turned out that the prerequisite for this method to apply---namely that unification with so-called linear constant restrictions is decidable in the single theories---is equivalent to requiring decidability of the positive fragment of the first order theory of the equational theories. Thus, the combination method can also be seen as a tool for combining decision procedures for positive theories of free algebras defined by equational theories. Complementing this logical point of view, the present paper isolates an abstract algebraic property of free algebras--- called combinability---that clarifies why our combination method applies to such algebras. We use this algebraic point of view to introduce a new proof method that depends on abstract notions and results from universal algebra, as opposed to technical manipul...

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...

In a recent paper [BS91] we introduced a new unification algorithm for the combination of disjoint equational theories. Among other consequences we mentioned (1) that the algorithm provides us with a decision procedure for the solvability of general A- and AI-unification problems and (2) that Kapur and Narendran's result about the NP-decidability of the solvability of general AC- and ACI-unification problems (see [KN91]) may be obtained from our results. In [BS91] we did not give detailled proofs for these two consequences. In the present paper we will treat these problems in more detail. Moreover, we will use the two examples of general A- and AI-unification for a case study of possible optimizations of the basic combination procedure.

this paper. On the one hand, dening a semantics for the combined system may depend on methods and results from formal logic and universal algebra. On the other hand, an ecient combination of the actual constraint solvers often requires the possibility of communication and cooperation between the solvers.

Unification modulo the theory of Boolean algebras has been investigated by several autors. Nevertheless, the exact complexity of the decision problem for unification with constants and general unification was not known. In this research note, we show that the decision problem is \Pi p 2 - complete for unification with constants and PSPACE-complete for general unification. In contrast, the decision problem for elementary unification (where the terms to be unified contain only symbols of the signature of Boolean algebras) is "only" NP-complete.