Canonical sets of horn clauses

Abstract

Rewrite rules are oriented equations used to replace equals-by-equals in the speci ed direction. Input terms are repeatedly rewritten according to the rules. When and if no rule applies, the resultant normal form is considered the value of the initial term. If no in nite sequences of rewrites is possible, a rewrite system is said to have t h e termination property. Con uence of a rewrite system is a property that ensures that no term has more than one normal form. A convergent rewrite system is one with both the con uence and termination properties.Let T be a set of rst-order terms, with variables taken from a set X , and G be its subset of ground variable-free terms. If t is a term in T , b y tj we signify the subterm of t rooted at position and by t s or simply t s we denote t with its subterm tj replaced by a term s. W e use the following notations for equational deduction: s ' t stands for the usual sense of equality in logical systems; s $ e t or just s $ t denotes one step of replacement of equals for equals using equation e ; s ! R t or just s ! t stands for one replacement according to the orientation of a rewrite rule in R ; s ! t, f o r a n y n umber including zero of rewrites; s $ t also stands for one rewrite step in either direction. Two terms s and t are said to be joinable if there is a term v such that s ! v t, o r s t for short. For convergent R, an identity s ' t holds in the theory de ned by R each rule viewed as an equation if and only if the normal forms of s and t are identical. Thus, validity of equations is decidable for nite convergent R, since the joinability relation is decidable. A ground convergent rewrite system is one that terminates and de nes unique normal forms for all ground terms. Ground convergent systems can be used to decide validity b y s k olemizing s and t and reducing to normal form. For a survey of the theory of rewriting, see Dershowitz and Jouannaud, 1990 .A conditional equation is a universally-quanti ed de nite Horn clause in which the only predicate symbol is equality. W e write such a clause in the form e 1^ên s ' t n 0 , meaning that equality s ' t holds whenever all the antecedent equations e i , hold. The term s will be called the left-hand side; t is the right-hand side; and the e i are the conditions. I f n = 0 , then the positive unit clause will be called an unconditional equation. Conditional equations are important for specifying abstract data types and expressing logic programs with equations. A conditional rewrite rule is an equational implication in which the equation in the consequent s ' t is oriented. To give operational semantics to such a system, the conditions under which a rewrite may be performed need to made precise. The most popular convention see Dershowitz and Okada, 1990 for conditional rewriting is that the terms in each condition be

Abstract canonical inference

“…The fundamental rôle of proof orderings in automated deduction, and the interpretation of completion as nondeterministic application of inference rules, received systematic treatment in [Bachmair and Dershowitz 1994]. The completion principle can be applied in numerous situations [Dershowitz 1989;Bonacina and Hsiang 1995], including the following: -equational rewriting [Peterson and Stickel 1981;Jouannaud and Kirchner 1986;; -Horn theories [Kounalis and Rusinowitch 1991;Dershowitz 1991a;1991b]; -induction [Kapur and Musser 1987;Fribourg 1989;Bachmair and Dershowitz 1994]; -unification [Doggaz and Kirchner 1991]; and -rewrite programs [Bonacina and Hsiang 1992;Dershowitz and Reddy 1993].…”

Structures for Abstract Rewriting

“…We must check that derivability (i.e. syntactic consequences obtained from ¢ ) coincides with convertibility in rewriting, This property, named logicality, 18 is expressed by: …”

“…To solve this problem, the condition which is imposed on rewrite systems, is decreasingness [23]. Here, this condition is expressed as follows: Decreasingness is essentially the same condition as imposed on conditional equations by theorem-proving procedures in [18,19]. Indeed, in [18,19], a conditional equation c £ u v is decreasing if for every substitution σ, σ u σ v (similar to Point 1. in Definition 9.1) and the proofs of the conditions only involve terms smaller than σ u (similar to Point 3. in Definition 9.1).…”

“…Here, this condition is expressed as follows: Decreasingness is essentially the same condition as imposed on conditional equations by theorem-proving procedures in [18,19]. Indeed, in [18,19], a conditional equation c £ u v is decreasing if for every substitution σ, σ u σ v (similar to Point 1. in Definition 9.1) and the proofs of the conditions only involve terms smaller than σ u (similar to Point 3. in Definition 9.1). The point of decreasingness as defined in [18,19] is to enable one to use equations by using only the decreasing instances of the equations, much like in unfailing Knuth-Bendix completion or Aka ordered completion where one uses the ordered instances of equations to perform simplification.…”

“…Indeed, in [18,19], a conditional equation c £ u v is decreasing if for every substitution σ, σ u σ v (similar to Point 1. in Definition 9.1) and the proofs of the conditions only involve terms smaller than σ u (similar to Point 3. in Definition 9.1). The point of decreasingness as defined in [18,19] is to enable one to use equations by using only the decreasing instances of the equations, much like in unfailing Knuth-Bendix completion or Aka ordered completion where one uses the ordered instances of equations to perform simplification. Here, we do not present the approach developed in [18,19] which defines theorem-proving procedures for conditional equations rather than completion procedures.…”