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. 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 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.

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...

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.

Semantic labelling is a powerful tool for proving termination of term rewrite systems. The usefulness of the extension to equational term rewriting described in Zantema [24] is however rather limited. In this paper we introduce a stronger version of equational semantical labelling, parameterized by three choices: (1) the order on the underlying algebra (partial order vs. quasi-order), (2) the relation between the algebra and the rewrite system (model vs. quasi-model), and (3) the labelling of the function symbols appearing in the equations (forbidden vs.

We present a new technique for proving AC-termination. We show that if certain conditions are met, AC-termination can be reduced to termination, i. e., termination of a TRS S modulo an AC-theory can be inferred from termination of another TRS R with no AC-theory involved. This is a new perspective and opens new possibilities to deal with AC-termination. 1 Introduction Termination of term rewriting systems (TRS's) is crucial for the use of rewriting in proofs and computations, and many theories have been developed in this field, especially for the case where function symbols do not obey any particular law or property. However, many interesting and useful systems have operators which are associative and commutative (AC), and most techniques developed for proving termination of TRS's do not carry over to rewriting modulo equational theories so that the theory developed to study termination of TRS's needs to be adapted to the equational case. Along these lines, a lot of work has been done...

. Like Kapur and Sivakumar in [KS97], we present an ACcompatible simplification ordering total on ground terms that follows the same scheme as the recursive path ordering (RPO). The first improvement with respect to their work is that our ordering has a simpler definition, and as a consequence we can obtain simpler proofs for the properties of the ordering and get a better understanding of the difficulties of finding this kind of orderings. But the main improvement is that, due to the simplicity of our definition, we can provide non-trivial extensions of the ordering to terms with variables, which is crucial in practice for proving termination of term rewriting systems modulo AC and for theorem proving strategies with built-in AC-unification. 1 Introduction Rewrite-based methods with built-in associativity and commutativity (AC) properties for some of the operators are well-known to be crucial in theorem proving and programming. Therefore a lot of work has been done on the development...

We introduce a family of AC-compatible Knuth-Bendix simplification orders which are AC-total on ground terms. Our orders preserve attractive features of the original Knuth-Bendix orders such as existence of a polynomial-time algorithm for comparing terms; computationally e#cient approximations, for instance comparing weights of terms; and preference of light terms over heavy ones. This makes these orders especially suited for automated deduction where e#cient algorithms on orders are desirable.

Mathematical logicians had developed the art of formalizing declarative knowledge long before the advent of the computer age. But they were interested primarily in formalizing mathematics. Because of the important role of nonmathematical knowledge in AI, their emphasis was too narrow from the perspective of knowledge representation, their formal languages were not sufficiently expressive. On the other hand, most logicians were not concerned about the possibility of automated reasoning; from the perspective of knowledge representation, they were often too generous in the choice of syntactic constructs. In spite of these differences, classical mathematical logic has exerted significant influence on knowledge representation research, and it is appropriate to begin this handbook with a discussion of the relationship between these fields. The language of classical logic that is most widely used in the theory of knowledge representation is the language of first-order (predicate) formulas. These are the formulas that John McCarthy proposed to use for representing declarative knowledge in his advice taker paper [176], and Alan Robinson proposed to prove automatically using resolution [236]. Propositional logic is, of course, the most important subset of first-order logic; recent