Abstract not found

Abstract not found

this paper is to show how to solve the constraints that are involved in the deduction mechanism of the first part. This may be interesting in its own since this provides with a unification algorithm for an order-sorted logic with context variables and can be read independently of the first part. This can also be compared with unification of term schemes of various kind (Chen & Hsiang, 1991; Salzer, 1992; Comon, 1995; R. Galbav'y and M. Hermann, 1992). Indeed,

We propose a set of transformation rules for first order formulae whose atoms are either equations between terms or "membership constraints" t 2 i. i can be interpreted as a regular tree language (i is called a sort in the algebraic specification community) or as a tree language in any class of languages which satisfies some adequate closure and decidability properties. This set of rules is proved to be correct, terminating and complete. This provides a quantifier elimination procedure: for every regular tree language L, the first order theory of some structure defining L is decidable. This extends the results of Mal'cev (1971), Maher (1988), Comon and Lescanne (1989). We also show how to apply our results to automatic inductive proofs in equational theories. Introduction To unify two terms s and t means to turn the equation s = t into an equivalent solved form which is either ? (this means that s = t has no solution, or, in other words, that s and t are not unifiable) or else a form...

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.

Contents 1 Introduction 2 2 Semantical methods 3 2.1 Well-founded monotone algebras . . . . . . . . . . . . . . . . . . . . 3 2.2 Polynomial interpretations . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Polynomial interpretations modulo AC . . . . . . . . . . . . . . . . . 13 2.4 Lexicographic combinations . . . . . . . . . . . . . . . . . . . . . . . 14 2.5 Other examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3 A hierarchy of termination 17 3.1 Simple termination . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 Total termination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.3 The hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4 Syntactical methods 25 4.1 Recursive path order . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.2 Justi cation of recursive path order . . . . . . . . . . . . . . . . . . . 30 4.3 Extensions of recursive path order . . . . . . . . . . . . . . . . . . . 36 4.

. We present the first fully syntactic (i.e., non-interpretationbased) AC-compatible recursive path ordering (RPO). It is simple, and hence easy to implement, and its behaviour is intuitive as in the standard RPO. The ordering is AC-total, and defined uniformly for both ground and non-ground terms, as well as for partial precedences. More importantly, it is the first one that can deal incrementally with partial precedences, an aspect that is essential, together with its intuitive behaviour, for interactive applications like Knuth-Bendix completion. 1

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 show that theories presented by a set of ground equations with several associative-commutative (AC) symbols always admit a finite canonical system. This result is obtained through the construction of a reduction ordering which is AC-compatible and total on the set of congruence classes generated by the associativity and commutativity axioms. As far as we know, this is the first ordering with such properties, when several AC function symbols and free function symbols are allowed. Such an ordering is also a fundamental tool for deriving complete theorem proving strategies with built-in associative commutative unification.

. 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...