. 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 ...

. 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...

Recently, the framework of rewriting modules was proposed and provided modular and incremental termination criteria. In this paper, we extend these results to the important case of Associative and Commutative rewriting by means of AC-dependency pairs.

The dependency pair technique of Arts and Giesl [1-3] for termination proofs of term rewrite systems (TRSs) is extended to rewriting modulo equations. Up to now, such an extension was only known in the special case of AC-rewriting [16, 18]. In contrast to that, the proposed technique works for arbitrary non-collapsing equations (satisfying a certain linearity condition). With the proposed approach, it is now possible to perform automated termination proofs for many systems where this was not possible before. In other words, the power of dependency pairs can now also be used for rewriting modulo equations.

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 propose a new technique for dealing with an equational theory E in the clausal framework. This consists of the definition of two inference rules called contextual superposition and extended superposition, and of an algorithm for computing the only needed applications of these last inference rules only by examining the axioms of E. We prove the refutational completeness of this technique for a class of theories E that include all the regular theories, i.e. any theory whose axioms preserve variables. This generalizes the results of Wertz [31] and Paul [17] who could not prove the refutational completeness of their superpositionbased systems for any regular theory. We also combine a collection of strategies that decrease the number of possible deductions, without loss of completeness: the superposition strategy, the positive ordering strategy, and a simplification strategy. These results have been implemented in a theorem prover called DATAC, for the case of commutative...

Copyright c ○ 2000 by Keiichirou KUSAKARI Recently, Arts and Giesl introduced the notion of dependency pairs, which gives effective methods for proving termination of term rewriting systems (TRSs). In this thesis, we extend the notion of dependency pairs to AC-TRSs, and introduce new methods for effectively proving AC-termination. Since it is impossible to directly apply the notion of dependency pairs to AC-TRSs, we introduce the head parts in terms and show an analogy between the root positions in infinite reduction sequences by TRSs and the head positions in those by AC-TRSs. Indeed, this analogy is essential for the extension of dependency pairs to AC-TRSs. Based on this analogy, we define AC-dependency pairs. To simplify the task of proving termination and AC-termination, several elimination transformations such as the dummy elimination, the distribution elimination, the general dummy elimination and the improved general dummy elimination, have been proposed. In this thesis, we show that the argument filtering method combined with the AC-dependency pair technique is essential in all the elimination transformations above. We present remarkable simple proofs for the soundness of these elimination transformations based on this observation. Moreover, we propose a new elimination transformation, called the argument filtering transformation, which is not only more powerful than all the other elimination transformations but also especially useful to make clear an essential relationship among them.

Equation solving is the process of nding a substitution of terms for variables that makes two terms equal in a given theory, while semantic uni cation is the process that generates a basis set of such unifying substitutions. A simpler variant of the problem is semantic matching, where the substitution is made in only one of the terms. Semantic uni cation and matching constitute an important component of theorem proving and programming language interpreters. In this thesis we formulate a uni cation procedure based on a system of transformation rules that looks at goals in a lazy, top-down fashion, and prove its soundness and completeness for equational theories described by convergent rewrite systems ( nite sets of equations that compute unique output values when applied from left-to-right to input values). We consider di erent variants of the system of transformation rules. We describe syntactic restrictions on the equations under which simpler sets of transformation rules are su cient for generating a complete set of semantic matchings. We show that our rst-order uni cation pro-cedure, with slight modi cations, can be used to solve the satis ability problem in combinatory logic together with a convergent set of algebraic axioms, resulting in a complete higher-order uni cation procedure for the given algebra. We also provide transformation rules to handle sit-