G. Huet, J.-J. L evy. Call by Need Computations in Non-Ambiguous Linear Term Rewriting Systems. Rapport de recherche INRIA 359, 1979. Reprinted as: Computations in orthogonal rewriting systems. In J.-L. Lassez and G. D. Plotkin, editors, Computational Logic; Essays in Honor of Alan Robinson, pages 394--443. MIT Press, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....or untoark any node of the tree. The marked node has little box with mark number inside on the top left corner. Dragging the vertical or horizontal scrollbar changes the viewport of the visualized term. Marks and subtree nodes propagate through rewriting process following the concept of residual[3]. So one can see the movement of a subterm through rewriting. 4.3 Sequence Viewer From the option menu of Term Viewer, Sequence Viewer in Figure 8 can be invoked. Step button rewrites the term and displays the new tree. With this viewer, one can intuitively obtain the structural transition ....

Huet, G., Levy, J.J.: "Call by need computation in non-ambiguous linear term rewriting systems", Rap- port Laboria 359, IRIA(1979).

....whose vertices and edges are the terms P and reduction steps P Q of . Rewriting theory is concerned with that graph, its properties and the dynamics of syntax it re ects. The fundamental technique to analyse the dynamics is to trace reduction steps in the course of computation, see (L evy 1978; Huet and L evy 1979; Klop 1992; Gonthier and L evy and Melli es 1992) for a theory of residuals and its applications. P 1 P 2 P P P 5 3 4 P Semantics, or what a program means to its environment. A program P ( term) is the combination of di erent procedures m, n or p ( subterms) whose interaction is the ....

.... . On double categories and multiplicative linear logic 17 A pairing P is declared correct when it performs a sequence D : D which cuteliminates all its paired links, i.e. such that P n D is the empty pairing ; In that case, a simple permutation argument (adapted from the cube lemma in (Huet and L evy 1979)) shows that D = E and P 0 n D = P 0 n E for every sequence E : E performed by P such that P n E = and every pairing P 0 P of . We write n P for D and say that n P is the net obtained by cut eliminating all the paired links in P. We also write P 0 n P for P 0 n D. Real ....

G. Huet, J.-J. Levy, Call by Need Computations in Non-Ambiguous Linear Term Rewriting Systems. Rapport de recherche INRIA 359, 1979.

.... it is necessary (1) to provide a method to decide whether a redex is root needed and (2) to identify the class of TRSs ensuring that every non root stable term has a redex for which the previous method succeeds [DM97] Decidable approximations to standard Huet and L evy s notion of neededness 1 [HL79,HL91] have been extensively explored (see, for instance Work partially supported by Spanish CICYT under grant TIC 98 0445 C03 01. 1 A redex in a term is needed if the redex (itself or one of its descendants) is reduced in each rewriting sequence leading to a normal form. ....

....investigated the use of these approximations to capture rootneededness in almost orthogonal TRSs [Luc98] We have demonstrated that NVsequentiality [Oya93] is the most general approximation to root neededness and the only one which is adequate for infinitary normalization. Strong sequentiality [HL79,HL91] is a particular case of NV sequentiality. Every NV sequential TRS admits a rewriting strategy which reduces redexes placed on special positions called nv indices. Such kinds of strategies have been proved useful for infinitary normalization in (almost) orthogonal TRSs [Luc98,Mid99] Constructor ....

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G. Huet and J.J. L'evy. Call by need computations in nonambiguous linear term rewriting systems. Technical Report 359, IRIA Laboria, LeChesnay, France, 1979.

....leads to the above result also suggests an efficient sequential algorithm for the evaluation of a class functional programs satisfying certain constraints, an algorithm which respects the mathematical semantics of the program considered as a term rewrite system. 1 Introduction Huet and L evy [Huet and L evy, 1979, Huet and L evy, 1991] have considered the problem of call by need computation of normal forms in orthogonal term rewrite systems. Call by need here means that no redex is ever reduced unless it must be reduced in order to compute the normal form. In general, such a redex cannot be effectively ....

....An actual algorithm must use computable approximations to these properties. 3 Strongly root needed redexes 3.1 Definitions We assume familiarity with the basic concepts of term rewriting, and in particular, with the notions of orthogonality and residuals. For an introduction, see [Klop, 1991] [Huet and L evy, 1979] is the fundamental reference for call by need computation in orthogonal (there called regular) term rewrite systems; Huet and L evy, 1991] is a revised and more easily accessible version. We restrict attention throughout to orthogonal term rewrite systems. In order to be precise about concepts ....

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G. Huet and J.-J. L'evy. Call by need computations in nonambiguous linear term rewriting systems. Technical Report Rapport de Recherche 359, INRIA, 1979.

.... normalizing for terminating TRSs. However, whenever a replacement map cannot ensure termination (or we fail to prove that it can) we need to provide normalizing strategies. A well known theory for defining (efficient) normalizing strategies is Huet and L evy s theory of needed reductions [HL79, HL91] A needed redex is a redex which is contracted (either itself or some of its descendants) in every 4 rewrite sequence to a normal form. For orthogonal TRSs, every term which is not a normal form contains a needed redex. Moreover, repeated reduction of needed redexes is normalizing [HL91] ....

....normalizing, etc. Efficiency: To ensure that computations achieved by using the strategy satisfy some criterion for efficiency; for instance minimality of reduction steps. For orthogonal TRSs, Huet and Levy s notion of needed reduction provides a framework for defining normalizing strategies [HL79, HL91] A needed redex in a term t is a redex which must be reduced (either itself or some descendant) in any normalizing derivation starting from t [HL91] Reduction sequences which only contract needed redexes are called needed reductions. Neededness has two main theoretical aspects: 1. It ....

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G. Huet and J.J. L'evy. Call by need computations in nonambiguous linear term rewriting systems. Technical Report 359, IRIA Laboria, LeChesnay, France, 1979.

....that contracts only needed redexes is normalizing. They also show that it is undecidable, in general, whether a redex is needed in a term; however, the left most redex is always needed, and this yields another proof of the normalization theorem. Similar results were shown by Huet and L evy [26] in their early study of neededness in the context of orthogonal term rewriting systems, and much has been done since in various contexts see [36] for references to some papers. Similar results were discovered independently by Khasidashvili [31] see also [33, 35] in particular, the proof of ....

G. Huet and J.-J. L'evy. Call by need computations in non-ambiguous linear term rewriting systems. Preprint 359, INRIA, 1979.

....sides. Letting Omega be an extra constant, we represent prefixes of terms by Omega terms in T 6[f Omega g (also denoted by T Omega simply) t Omega denotes the Omega term obtained from a term t by replacing each variable with Omega . The prefix ordering on T Omega is defined as follows [HL79, Klo92, KM91]: i) t Omega , ii) t t, iii) f(t 1 ; t n ) f(s 1 ; s n ) if t i s i for i = 1; n. t and s are compatible, written by t s, if u t and u s for some u; otherwise they are incompatible, denoted by t#s. s t t denotes a minimal Omega term u such that u s and u ....

....orthogonal PRSs can be easily proven in the same way as for orthogonal TRSs, by tracing the classical proof in [Hue80] Theorem 8. A orthogonal PRS is confluent. 4 Strong Sequentiality of PRSs The fundamental concept of strong sequentiality for orthogonal TRSs was introduced by Huet and L evy[HL79]. In this section, we first explain the basic notions related to strong sequentiality, according to [HL79] for orthogonal PRSs. We next describe a useful decision procedure that determines the index of a given Omega term with respect to R . Definition 9 (arbitrary systems) Let R be a PRS. ....

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G. Huet and J.-J. L'evy. Call by need computations in non-ambiguous linear term rewriting systems. Technical Report 359, INRIA, 1979.

....terms and other steps which are blind to the internal structure of those terms. In this section we develop some machinery enabling us to track the progress of subterms during a reduction. We use the notions of descendant of an occurrence with respect to an algebraic reduction (essentially as in [HL79]) Definition 2.1 Let ae : M R Gamma N have redex u and source term S. For an occurrence d of M , the set d=ae of descendants of d with respect to ae is the set of occurrences in N defined as follows. If d does not extend u then d=ae = fdg. If d is uw, w a non variable address of S, then ....

G. Huet, J.J. L'evy. Call by need computations in non-ambiguous linear term rewriting systems, Rapport Laboria 359, INRIA, 979.

....will also show that it is sometimes even easier to derive properties of term graph rewriting systems than to derive similar properties of term rewriting systems. In section 4, the concept of strong sequentiality is incorporated in TGRS s. This concept has been introduced in TRS s by Huet and L evy (Huet and L evy (1979)) Strong sequentiality is a decidable property which admits an efficiently normalizing reduction strategy for the corresponding class of strongly sequential TRS s. The key idea of the strongly sequential TRS s is that in these systems it always possible to indicate at least one needed redex . It ....

....corresponding class of strongly sequential TRS s. The key idea of the strongly sequential TRS s is that in these systems it always possible to indicate at least one needed redex . It should be noted that for ordinary orthogonal TRS s this in general not possible. Strong sequentiality as defined by Huet and L evy (1979) and Klop and Middeldorp (1991) is based on the notion of Omega reduction. In this paper a similar notion for graph rewriting is presented, called reduction (section 4) The reduction of a graph leads to its reduct . reducts are used to introduce the concept of index with which the so ....

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Huet, G. and J.J. L'evy (1979). Call by need computations in non-ambiguous linear term rewriting systems, Technical Report 359 , INRIA.

....process. Keywords: Functional logic languages, Implementation, Incrementality, Needed narrowing. 1 Introduction For orthogonal term rewriting systems (TRSs) Huet and L evy s strong sequentiality provides a formal basis for the mechanization of sequential, normalizing rewriting computations [HL79, HL91] In [HL79, HL91] Huet and L evy de ned the notion of strongly needed redex and showed that, for the class of strongly sequential TRSs (SS) the steady reduction of strongly needed redexes is normalizing. Huet and L evy showed that the strong sequentiality of a TRS is decidable. They also ....

....Functional logic languages, Implementation, Incrementality, Needed narrowing. 1 Introduction For orthogonal term rewriting systems (TRSs) Huet and L evy s strong sequentiality provides a formal basis for the mechanization of sequential, normalizing rewriting computations [HL79, HL91] In [HL79, HL91] Huet and L evy de ned the notion of strongly needed redex and showed that, for the class of strongly sequential TRSs (SS) the steady reduction of strongly needed redexes is normalizing. Huet and L evy showed that the strong sequentiality of a TRS is decidable. They also provided an ....

[Article contains additional citation context not shown here]

G. Huet and J.-J. Levy. Call by need computations in nonambiguous linear term rewriting systems. Technical report, IRIA Laboria, LeChesnay, France, 1979.

....We believe that this approach is worth while as a first step since by this restriction we can rely upon the well known concept of indexes when we try to explain why the functional strategy works well for a wide class of orthogonal term rewriting systems. The concept of indexes was proposed by Huet and L evy (1979). They introduced the subclass of strongly sequential orthogonal term rewriting systems for which index reduction is normalizing. However, for reasons of efficiency their approach is not very feasible in a practical sense. An important problem they had to cope with is the fact that indexes in ....

....a subclass of transitive term rewriting systems: so called left incompatible term rewriting systems. 2. Preliminaries In the sequel we will assume that the reader is familiar with the basic concepts concerning term rewriting systems as introduced by Dershowitz and Jouannaud (1990) Klop (1992) or Huet and L evy (1979). 2.1. Term Rewriting Systems The following definitions are based on definitions given in Klop (1992) In contrast with Klop (1992) we use the notion constant symbol for a symbol that cannot be rewritten, instead of for a function symbol with arity 0. 2.1. Definition. A Term Rewriting ....

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Huet, G. and J.J. L'evy (1979). Call by need computations in non-ambiguous linear term rewriting systems, Technical Report 359 , INRIA.

....possible. Thus, a lazy compiler should compile function F as F 2 , not as F 1 . It is essential for a lazy ML compiler to produce a correct compilation of pattern matching whenever there exists one. This problem has first been solved in the case of non overlapping patterns by Huet and L evy [2]. Given a set of (possibly) overlapping patterns, A. Laville [5] shows how to replace them, when possible, by an equivalent set of non overlapping patterns, compiled using Huet and L evy s technique. A. Su arez and L. Puel [8] translate the initial set of overlapping patterns into an equivalent ....

....Pattern matching is modeled as a function over the set of partial values. This function is compiled into multiway branches represented by simple pattern matching expressions, in the spirit of [1] 3. 1 The matching function Pattern matching is usually formalized as a predicate on partial values [2, 5]. We prefer a representation as a function over partial values, closer to pattern matching in ML. A clause is a three tuple (i; p; e) where i is an integer, p is a pattern and e is a partial term, such that all variables in term e are variables of pattern p. Integer i is the number of the ....

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G. Huet, J.-J. L'evy, "Call by Need Computations in Non-Ambiguous Linear Term Rewriting Systems ". INRIA, technical report 359, 1979.

....Suarez [10] devised a clever compilation scheme to generate statically a PMT in lazy languages. Such a tree is then used at run time for fast rule indexing and takes full advantage of the nature of the LHS terms in a definition. Their work simplified and generalized seminal ideas by Huet and Levy [7] that were in turn sharpened by Laville [9] The gist of the Puel Suarez method rests on generalized notions of constructor terms and sequentiality. They called the new terms constrained terms. Although partially ordered sorts provide a substantially improved expressiveness over manysorted ....

....for which every PMT associated to S will fail to terminate and an optimal PMT is a PMT that will only fail to terminate on the strict set of S. We show that optimality of an order sorted PMT is a decidable property equivalent to a generalization of the notions of strong sequentiality presented in [7] and [10] Sequentiality of a pattern matching problem S is the possibility of systematically expanding any term step by step until either it matches a pattern of S or it is clear that a positive 1 See [8] for a discussion of unification and matching in equational theories. May 1991 Digital ....

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Gerard Huet and Jean-Jacques Levy. Call by need computations in non ambiguous linear term rewriting systems. Rapport INRIA Laboria 359, INRIA, Le Chesnay, France (1979).

....Orthogonal term rewriting systems are defined to have linear and non ambiguous left hand sides of rules (LHS) In this class of systems, the parallel outermost strategy, a normalization strategy which reduces all outermost redexes in each step, has been shown to be complete. Huet and Levy [HL79] showed that a restricted notion of sequentiality, called strong sequentiality, that only considers the LHS s, seems to be much more useful in order to perform efficient implementations of orthogonal rewriting systems. They developed a normalization algorithm which eliminates useless ....

....u 2 O(t) is a structure direction of P in t[ Omega oe ] u if and only if Accept(P; t) StrucAccept(P; t; u) i. e, SortAccept(P; t; u) In the untyped case, a position is a potential direction if and only if it is a structure one and the definition coincides with that of direction in [HL79]. Finally, a position u 2 O(t) is a sort direction when it is a potential but not a structure direction, i.e, t=u = Omega oe , u is a potential direction and SortAccept(P; t; u) 6= Sequentiality) The predicate P is said to be sequential at t if and only if, whenever P (t) false and ....

[Article contains additional citation context not shown here]

G'erard Huet and Jean-Jacques L'evy. Call by need computations in non ambiguous linear term rewriting systems. Technical Report IRIA Laboria 359, INRIA, Le Chesnay, France, 1979.

....similarities to that derived here; in particular, it naturally detects inexhaustive matches and redundant rules. Petterson compares the use of memoization (Section 7.5) to minimization of automata. The top down, left right matching order does not produce optimal compiled matchers. Huet and L evy [7] studied optimality for unambiguous matches (nonoverlapping patterns only) However, programming languages usually allow patterns to overlap, but impose an order on them, and make the match deterministic by selecting the first matching one. To adapt Huet and L evy s work to programming practice, ....

G. Huet and J.-J. L'evy. Call by need computations in non-ambiguous linear term rewriting systems. Rapport de Recherche 359, IRIA Rocquencourt, France, 1979.

....given a pattern matching de nition, it must both determine whether this de nition can be compiled correctly Two Techniques for Compiling Lazy Pattern Matching 3 or not and produce a correct compilation when possible. This problem has rst been solved in the case of non overlapping patterns by Huet and L evy [1991]. Given a set of possibly overlapping patterns, Laville [1991] shows how to replace them, when possible, by an equivalent set of non overlapping patterns, compiled using Huet and L evy s technique. Laville s method is not complete, since it cannot treat the case of datatypes with in nite ....

....q. If p and q are compatible, then they are also said to be ambiguous or overlapping. Indeed, as a consequence of lemma 2.1, two patterns are compatible if and only if they admit a common instance. 2. 3 The matching function Pattern matching is usually formalized as a predicate on partial values [Huet and L evy 1991; Laville 1991] I prefer a representation as a function over partial values, closer to pattern matching in ML. A clause is a triple (i; p; e) where i is an integer, p is a pattern and e is a partial term, such that all variables in term e are variables of pattern p. Integer i is the number of ....

[Article contains additional citation context not shown here]

Huet, G. and L evy, J.-J. 1991. Call by need computations in non-ambiguous linear term rewriting systems. In J.-L. Lassez and G. D. Plotkin (Eds.), Computational Logic, Essays in Honor of Alan Robinson. The MIT Press.

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G. Huet, J.-J. L evy. Call by Need Computations in Non-Ambiguous Linear Term Rewriting Systems. Rapport de recherche INRIA 359, 1979. Reprinted as: Computations in orthogonal rewriting systems. In J.-L. Lassez and G. D. Plotkin, editors, Computational Logic; Essays in Honor of Alan Robinson, pages 394--443. MIT Press, 1991.

No context found.

Huet, G., L'evy, J-J.: "Call by Need Computations in Non-Ambiguous Linear Term Rewriting Systems". Rapport de recherche INRIA 359, 1979.

No context found.

G. Huet, J.-J. L evy. Call by Need Computations in Non-Ambiguous Linear Term Rewriting Systems. Rapport de recherche INRIA 359, 1979. Reprinted as: Computations in orthogonal rewriting systems. In J.-L. Lassez and G. D. Plotkin, editors, Computational Logic; Essays in Honor of Alan Robinson, pages 394--443. MIT Press, 1991.

No context found.

G. Huet, J-J. L'evy, "Call by Need Computations in Non-Ambiguous Linear Term Rewriting Systems". Rapport de recherche INRIA 359, 1979. Republished as "Computations in orthogonal rewriting systems, I and II", in Jean-Louis Lassez and Gordon Plotkin, editors, Computational logic, essays in honor of Alan Robinson, pages 395---443. MIT Press, Cambridge, Massachussets, 1991.

No context found.

G. Huet, J.-J. Levy, \Call by Need Computations in Non-Ambiguous Linear Term Rewriting Systems". Rapport de recherche INRIA 359, 1979.

No context found.

G. Huet, J.-J. Levy. Call by Need Computations in Non-Ambiguous Linear Term Rewriting Systems. Rapport de recherche INRIA 359, 1979.

No context found.

G. Huet, J.J. Levy, "Call-by-need computations in nonambiguous linear term rewriting systems", INRIA Reports 359, 1979.

No context found.

G. Huet, J.-J. Levy. \Call by Need Computations in Non-Ambiguous Linear Term Rewriting Systems". Rapport de recherche INRIA 359, 1979.

No context found.

G. Huet and J.-J. L#evy. Call by need computations in non-ambiguous linear term rewriting systems. Rapport Laboria 359, Institut National de Recherche en Informatique et en Automatique, Le Chesnay,France, August 1979.

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