Results 1 
5 of
5
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 ...
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...
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...
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.
A Framework for the Analysis of Syntactic Replacement Restrictions
, 1999
"... We formalize the notion of syntactic replacement restriction, which is useful for modeling reductionbased systems which compute with terms and impose restrictions on the possible computations (typically by means of strategies). We emphasize the syntactic flavour of our approach: the restrictions a ..."
Abstract
 Add to MetaCart
We formalize the notion of syntactic replacement restriction, which is useful for modeling reductionbased systems which compute with terms and impose restrictions on the possible computations (typically by means of strategies). We emphasize the syntactic flavour of our approach: the restrictions are associated to components of terms and (in principle) they do not depend on either a particular Term Rewriting System or a computational mechanism (like rewriting, narrowing, residuation, etc.). The replacement restrictions can be used to improve the computational behavior of the unrestricted mechanism. We give a general descriptive and algebraic framework to deal with replacement restrictions. For the descriptive part, we introduce and motivate properties which characterize classes of replacement restrictions. For the algebraic side, the set of replacement restrictions is presented as a complete Boolean algebra. The algebraic operations (and others which we also define) can be used to com...