. A usual technique in symbolic constraint solving is to apply transformation rules until a solved form is reached for which the problem becomes simple. Ordering constraints are well-known to be reducible to (a disjunction of) solved forms, but unfortunately no polynomial algorithm deciding the satisfiability of these solved forms is known. Here we deal with a different notion of solved form, where fundamental properties of orderings like transitivity and monotonicity are taken into account. This leads to a new family of constraint solving algorithms for the full recursive path ordering with status (RPOS), and hence as well for other path orderings like LPO, MPO, KNS and RDO, and for all possible total precedences and signatures. Apart from simplicity and elegance from the theoretical point of view, the main contribution of these algorithms is on efficiency in practice. Since guessing is minimized, and, in particular, no linear orderings between the subterms are guessed, ...

. A new decision procedure for the existential fragment of ordering constraints expressed using the recursive path ordering is presented. This procedure is nondeterministic and checks whether a set of constraints is solvable over the given signature, i.e., the signature over which the terms in the constraints are defined. It is shown that this nondeterministic procedure runs in polynomial time, thus establishing the membership of this problem in the complexity class NP for the first time. 1 Introduction Reduction orderings have been developed for testing the termination of term rewrite systems. Among these orderings the Recursive Path Ordering (RPO) [3] is one of the most used since it gives simple syntactic conditions for orienting equations as rules. An optimal way to apply RPO is to orient an equation u = v as u ! v if for all ground substitutions `, u` Ø rpo v`. When the precedence is total this amounts to checking the unsatisfiability of the ordering constraint v Ø rpo uu ¸ rpo...

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.

. A conditional term rewriting system (CTRS) is called simplifying if there exists a simplification ordering ? on terms such that the left-hand side of any rewrite rule is greater than the right-hand side and the terms occurring in the conditions of that rule. If a simplifying join CTRS consists of finitely many rules, it is terminating and the applicability of a rewrite rule is decidable by recursively reducing the terms in the conditions. Consider two finite CTRSs R1 and R2 which may share constructors (symbols that do not occur at the root position of the lefthand side of any rewrite rule) but no other function symbols. It will be shown that the combined CTRS R = R1[R2 is simplifying if and only if R1 and R2 are simplifying. Moreover, confluence is a modular property of finite simplifying join CTRSs. 1 Introduction During the past decade, term rewriting has gained an enormous importance in fields of computer science concerned with symbolic manipulation. Among others, it may be vie...

theory of the recursive path ordering is decidable in the case of unary signatures with total precedence. This solves a problem that was mentioned as open in [6]. The result has to be contrasted with the undecidability results of the lexicographic path ordering [6] for the case of symbols with arity 2 and total precedence and for the case of unary signatures with partial precedence. We recall that lexicographic path ordering (lpo) and the recursive path ordering and many other orderings such as [13, 10] coincide in the unary case. Among the positive results it is known that the existential theory of total lpo is decidable [3, 17]. The same result holds for the case of total rpo [8, 15]. The proof technique we use for our decidability result might be interesting by itself. It relies on encoding of words as trees and then on building a tree automaton to recognize the rpo relation. Key words: Recursive path ordering, first-order theory, ground reducibility

. A term rewrite system (TRS) terminates if, and only if, its rules are contained in a reduction ordering ?. In order to deal with any set of equations, including inherently non-terminating ones (like commutativity) , TRS have been generalized to ordered TRS (E; ?), where equations of E are applied in whatever direction agrees with ?. The confluence of terminating TRS is well-known to be decidable, but for ordered TRS the decidability of confluence has been open. Here we show that the confluence of ordered TRS is decidable if ? belongs to a large class of path orderings (including most practical orderings like LPO, MPO, RPO (with status), KNS and RDO), since then ordering constraints for ? can be solved in an adequate way. For ordered TRS (E; ?) where E consists of constrained equations, confluence is shown to be undecidable. Finally, also ground reducibility is proved undecidable for ordered TRS. 1

this paper we introduce some new notions of solved form, where, in addition to the closure under the classical RPO decomposition rules, a restricted form of transitivity through variables is applied. It is proved that if C is a solved form in this sense, then it is satisfiable under extended signatures if, and only if, it has no cycle (Section 5)

Various methods for proving the termination of term rewriting systems have been suggested. Most of them are based on the notion of simplification ordering. In this paper, the theoretical time complexities (of the worst cases) of a collection of well-known simplification orderings will be presented. 1 Introduction and Summary Term rewriting systems (TRSs, for short) provide a powerful tool for expressing nondeterministic computations and as a result they have been widely used as, for example, in theorem provers. Moreover, they can usefully be applied in many other areas of computer science and mathematics such as abstract data type specifications and program verification. A main requirement of TRSs is expressed by the termination property. There exist various methods of proving the termination of TRSs. Most of these are based on reduction orderings which are well-founded, compatible with the structure of terms 1 and stable with respect to (w.r.t., for short) substitutions. The notion...

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