Context-sensitive rewriting is a simple restriction of rewriting which is formalized by imposing fixed restrictions on replacements. Such a restriction is given on a purely syntactic basis: it is (explicitly or automatically) specified on the arguments of symbols of the signature and inductively extended to arbitrary positions of terms built from those symbols. Termination is not only preserved but usually improved and several methods have been developed to formally prove it. In this paper, we investigate the definition, properties, and use of context-sensitive rewriting strategies, i.e., particular, fixed sequences of context-sensitive rewriting steps. We study how to define them in order to obtain efficient computations and to ensure that context-sensitive computations terminate whenever possible. We give conditions enabling the use of these strategies for root-normalization, normalization, and infinitary normalization. We show that this theory is suitable for formalizing ...

Kennaway proved the remarkable result that every (almost) orthogonal term rewriting system admits a computable sequential normalizing reduction strategy. In this paper we present a computable sequential reduction strategy similar in scope, but simpler and more general. Our strategy can be thought of as an outermost-fair-like strategy that is allowed to be unfair to some redex of a term when contracting the redex is useless for the normalization of the term. Unlike the strategy of Kennaway, our strategy does not rely on syntactic restrictions that imply conuence. On the contrary, it can easily be applied to any term rewriting system, and we show that the class of term rewriting systems for which our strategy is normalizing properly includes all (almost) orthogonal systems. Our strategy is more versatile; in case of (almost) orthogonal term rewriting systems, it can be used to detect certain cases of nontermination. Our normalization proof is more accessible than Kennaway's. W...

) Ir#ne Durand Universit# de Bordeaux I, France Aart Middeldorp University of Tsukuba, Japan Abstract In this paper we study decidable approximations to call by need computations to normal and root-stable forms in term rewriting. We obtain uniform decidability proofs by making use of elementary tree automata techniques. Surprisingly, by avoiding complicated concepts like index and sequentiality we are able to cover much larger classes of term rewriting systems. 1 Introduction The following theorem of Huet and L#vy [8] forms the basis of all results on optimal normalizing reduction strategies for orthogonal term rewriting systems (TRSs): every reducible term contains a needed redex, i.e., a redex which is contracted in every rewrite sequence to normal form, and repeated contraction of needed redexes results in a normal form, if the term under consideration has a normal form. Unfortunately, needed redexes are not computable in general. Hence, in order to obtain a computable optimal...

Functional logic languages with a complete operational semantics are based on narrowing, a unification-based goal-solving mechanism which subsumes the reduction principle of functional languages and the resolution principle of logic languages. Needed narrowing is an optimal narrowing strategy and the basis of several recent functional logic languages. In this paper, we define a partial evaluator for functional logic programs based on needed narrowing. We prove strong correctness of this partial evaluator and show that the nice properties of needed narrowing carry over to the specialization process and the specialized programs. In particular, the structure of the specialized programs provides for the application of optimal evaluation strategies. This is in contrast to other partial evaluation methods for functional logic programs which can change the original program structure in a negative way. Finally, we present some experiments which highlight the practical advantages of our approach.

Abstract. Term rewriting systems (TRSs) are extended by allowing to contain extra variables in their rewrite rules. We call the extended systems EV-TRSs. They are ill-natured since every one-step reduction by their rules with extra variables is infinitely branching and they are not terminating. To solve these problems, this paper extends narrowing on TRSs into that on EV-TRSs and show that it simulates the reduction sequences of EV-TRSs as the narrowing sequences starting from ground terms. We prove the soundness of ground narrowing-sequences for the reduction sequences. We prove the completeness for the case of rightlinear systems, and also for the case that no redex in terms substituted for extra variables is reduced in the reduction sequences. Moreover, we give a method to prove the termination of the simulation, extending the termination proof of TRSs using dependency pairs, to that of narrowing on EV-TRSs starting from ground terms. 1

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.

Given a term rewriting system R and a normalizable term t, a redex is needed if in any reduction sequence of t to a normal form, this redex will be contracted. Roughly, R is sequential if there is an optimal reduction strategy in which only needed redexes are contracted. More generally, G. Huet and J.-J. Levy define in [8] the sequentiality of a predicate P on partially evaluated terms. We show here that the sequentiality of P is definable in SkS, the second-order monadic logic with k successors, provided P is definable in SkS. We derive several known and new consequences of this remark: 1--strong sequentiality, as defined in [8], of a left linear (possibly overlapping) rewrite system is decidable, 2--NV-sequentiality, as defined in [15] is decidable, even in the case of overlapping rewrite systems 3-- sequentiality of any linear shallow rewrite system is decidable. Then we describe a direct construction of a tree automaton recognizing the set of terms that do have needed redexes, w...

A strategy of TRSs is said to be complete if all normal forms of a given term are reachable from the term. We show the innermost strategy is complete for terminating, right-linear and overlay TRSs. This strategy is fairly efficient to calculate all normal forms of a given term by searching reduction trees. We also discuss the possibilities for weakening the conditions. 1

In [2] it is shown that context-free root-normalizing reduction strategies are infinitary fair-normalizing for confluent TRSs. Lucas [1, Theorem 6.6] shows that every NV-strategy is infinitary fair-normalizing for almost orthogonal NV-sequential TRSs. Lucas remarks: “Theorem 6.6 proves that S in Example 6.4 is infinitary fair-normalizing. This cannot be obtained from Middeldorp’s results. ” In this note we show that Lucas ’ result is a trivial consequence of Middeldorp’s results. The reader is referred to [2, 3, 1] for definitions and explanations of the terminology used above. The following result is due to Middeldorp [2]. Theorem 1 Let R be a confluent TRS. Context-free root-normalizing reduction strategies for R are infinitary fair-normalizing. � Lucas [1, Theorem 6.2] obtained the following result. Theorem 2 NV-strategies are hyper root-normalizing for almost orthogonal NV-sequential TRSs. � Since almost orthogonal TRSs are confluent and hyper root-normalization implies root-normalization, a proof of context-freeness is sufficient to obtain the infinitary fair-normalization of NV-strategies (for almost orthogonal NVsequential TRSs) by Theorem 1. However, Lucas observed that general NVstrategies need not be context-free by means of the strategy S that contracts the leftmost (rightmost) NV-redex if the total number of redexes in the term at hand is odd (even). For instance, with respect to the (almost orthogonal NV-sequential) TRS consisting of the single rewrite rule a → f(a, a) all redexes are NV-redexes (i.e., occur at NV-index positions) and we have f(f(a, a), a) → S f(f(f(a, a), a), a) and f(a, a) → S f(a, f(a, a)), revealing that S is not context-free. It is interesting to note that S will fail to compute any infinite normal form. For instance, the limit of the (unique) S-rewrite sequence starting at the term a is the infinite non-normal form f(f(f( · · ·, a), a), f(a, f(a, · · ·))). This does not contradict infinitary fair-normalization, simply because there are no fair S-rewrite sequences! Lucas [1, Theorem 6.6] proved the following result.

The theorem of Huet and Lévy stating that for orthogonal rewrite systems (i) every reducible term contains a needed redex and (ii) repeated contraction of needed redexes results in a normal form if the term under consideration has a normal form, forms the basis of all results on optimal normalizing strategies for orthogonal rewrite systems. However, needed redexes are not computable in general. In the paper we show how the use of approximations and elementary tree automata techniques allows one to obtain decidable conditions in a simple and elegant way. Surprisingly, by avoiding complicated concepts like index and sequentiality we are able to cover much larger classes of rewrite systems. We also study modularity aspects of the classes in our hierarchy. It turns out that none of the classes is preserved under signature extension. By imposing various conditions we recover the preservation under signature extension. By imposing some more conditions we are able to strengthen the signature extension results to modularity for disjoint and constructor-sharing combinations.