Results 1  10
of
13
ContextSensitive Rewriting Strategies
, 1997
"... Contextsensitive 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 ..."
Abstract

Cited by 43 (30 self)
 Add to MetaCart
Contextsensitive 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 contextsensitive rewriting strategies, i.e., particular, fixed sequences of contextsensitive rewriting steps. We study how to define them in order to obtain efficient computations and to ensure that contextsensitive computations terminate whenever possible. We give conditions enabling the use of these strategies for rootnormalization, normalization, and infinitary normalization. We show that this theory is suitable for formalizing ...
A Sequential Reduction Strategy
 Theoretical Computer Science
, 1996
"... 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 ..."
Abstract

Cited by 22 (2 self)
 Add to MetaCart
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 outermostfairlike 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...
Decidable Call by Need Computations in Term Rewriting (Extended Abstract)
 Proc. of 14th International Conference on Automated Deduction, CADE'97, LNAI 1249:418
, 1997
"... ) 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 rootstable forms in term rewriting. We obtain uniform decidability proofs by making use of elementary ..."
Abstract

Cited by 15 (3 self)
 Add to MetaCart
) 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 rootstable 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...
NarrowingBased Simulation of Term Rewriting Systems with Extra Variables
 In Proc. of the 10th ACM SIGPLAN International Conference on Functional Programming (ICFP’05
, 2005
"... Abstract. Term rewriting systems (TRSs) are extended by allowing to contain extra variables in their rewrite rules. We call the extended systems EVTRSs. They are illnatured since every onestep reduction by their rules with extra variables is infinitely branching and they are not terminating. To s ..."
Abstract

Cited by 7 (5 self)
 Add to MetaCart
Abstract. Term rewriting systems (TRSs) are extended by allowing to contain extra variables in their rewrite rules. We call the extended systems EVTRSs. They are illnatured since every onestep 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 EVTRSs and show that it simulates the reduction sequences of EVTRSs as the narrowing sequences starting from ground terms. We prove the soundness of ground narrowingsequences 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 EVTRSs starting from ground terms. 1
Termination, ACTermination and Dependency Pairs of Term Rewriting Systems
 Ph.D. thesis, JAIST
, 2000
"... 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 ACTRSs, and introduce new methods for eff ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
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 ACTRSs, and introduce new methods for effectively proving ACtermination. Since it is impossible to directly apply the notion of dependency pairs to ACTRSs, 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 ACTRSs. Indeed, this analogy is essential for the extension of dependency pairs to ACTRSs. Based on this analogy, we define ACdependency pairs. To simplify the task of proving termination and ACtermination, 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 ACdependency 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.
Sequentiality, Second Order Monadic Logic and Tree Automata
 IN `PROCEEDINGS 10TH IEEE SYMPOSIUM ON LOGIC IN COMPUTER SCIENCE, LICS'95', IEEE COMPUTER
, 1995
"... 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 ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
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 secondorder monadic logic with k successors, provided P is definable in SkS. We derive several known and new consequences of this remark: 1strong sequentiality, as defined in [8], of a left linear (possibly overlapping) rewrite system is decidable, 2NVsequentiality, 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...
T.: Innermost reductions find all normal forms on rightlinear terminating overlay TRSs
 In Proceedings of the 3rd International Workshop on Reduction Strategies in Rewriting and Programming
, 2003
"... 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, rightlinear and overlay TRSs. This strategy is fairly efficient to calculate all normal forms of a given term by searching reduction ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
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, rightlinear 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
ContextFreeness and Infinitary Normalization
 In Proc. of 14th Japanese Term Rewriting Meeting. Nara Institute of Science and Technology
, 1999
"... In [2] it is shown that contextfree rootnormalizing reduction strategies are infinitary fairnormalizing for confluent TRSs. Lucas [1, Theorem 6.6] shows that every NVstrategy is infinitary fairnormalizing for almost orthogonal NVsequential TRSs. Lucas remarks: “Theorem 6.6 proves that S in Exa ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
In [2] it is shown that contextfree rootnormalizing reduction strategies are infinitary fairnormalizing for confluent TRSs. Lucas [1, Theorem 6.6] shows that every NVstrategy is infinitary fairnormalizing for almost orthogonal NVsequential TRSs. Lucas remarks: “Theorem 6.6 proves that S in Example 6.4 is infinitary fairnormalizing. 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. Contextfree rootnormalizing reduction strategies for R are infinitary fairnormalizing. � Lucas [1, Theorem 6.2] obtained the following result. Theorem 2 NVstrategies are hyper rootnormalizing for almost orthogonal NVsequential TRSs. � Since almost orthogonal TRSs are confluent and hyper rootnormalization implies rootnormalization, a proof of contextfreeness is sufficient to obtain the infinitary fairnormalization of NVstrategies (for almost orthogonal NVsequential TRSs) by Theorem 1. However, Lucas observed that general NVstrategies need not be contextfree by means of the strategy S that contracts the leftmost (rightmost) NVredex if the total number of redexes in the term at hand is odd (even). For instance, with respect to the (almost orthogonal NVsequential) TRS consisting of the single rewrite rule a → f(a, a) all redexes are NVredexes (i.e., occur at NVindex 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 contextfree. It is interesting to note that S will fail to compute any infinite normal form. For instance, the limit of the (unique) Srewrite sequence starting at the term a is the infinite nonnormal form f(f(f( · · ·, a), a), f(a, f(a, · · ·))). This does not contradict infinitary fairnormalization, simply because there are no fair Srewrite sequences! Lucas [1, Theorem 6.6] proved the following result.
Decidable CallbyNeed Computations in Term Rewriting
, 2004
"... 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 ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
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 constructorsharing combinations.
Sequentiality, Monadic SecondOrder Logic and Tree Automata
"... 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 ..."
Abstract
 Add to MetaCart
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. L'evy define in [9] the sequentiality of a predicate P on partially evaluated terms. We show here that the sequentiality of P is definable in SkS, the monadic secondorder logic with k successors, provided P is definable in SkS. We derive several known an new consequences of this remark: 1strong sequentiality, as defined in [9], of a left linear (possibly overlapping) rewrite system is decidable, 2 NVsequentiality, as defined in [17] 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, whi...