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 ....

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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 ....

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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 ....

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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.

....debugging. 1 Introduction. We introduce new tools to capture the notions of descendance and residuals that appear regularly in the study of rewriting and reduction systems. These new tools provide an easier encoding of these notions and may have applications in implementing evaluation strategies [1, 5, 8, 13, 14] or debugging algorithms based on rewriting or reduction [3] Most presentations of descendance and residuals use labeling of terms. For any reduction system, one simply produces a labeled version of the system, together with a procedure to transform a derivation on labeled terms into a derivation ....

G. Huet, J.-J. L'evy, "Call by Need Computations in Non-Ambiguous Linear Term Rewriting Systems", IRIA Laboria Report 359, 1979.

....in a term t is reduced if and only if this redex must be reduced by any other sequential strategy computing the normal form of t, and sequential, a rewrite system for which an optimal sequential strategy exists. Huet and L evy have shown that not all orthogonal rewrite systems are sequential [3], and moreover that sequentiality is undecidable 2 . Partially supported by the ESPRIT BRA COMPASS. 2 Huet and L evy s paper was aimed at generalizing the most important syntactic properties of calculus to left linear non overlapping term rewriting systems. Their original paper was ....

....that compute a normal form for every term that posseses one, even in presence of non terminating reductions issuing from the term. Such a strategy is called normalizing. This section is devoted to the introduction of the key concept of strong sequentiality. Most of the material is borrowed from [3, 6], but is applied to left linear systems instead of orthogonal ones. Only those proofs that suffer changes are given here. Let Omega be a new constant symbol representing an unknown part of a term, T Omega the set T (F [ Omega ; V) of Omega terms, and Pos Omega (t) the set of Omega ....

G'erard Huet and Jean-Jacques L'evy. Call by need computations in non-ambiguous linear term rewriting systems. In Jean-Louis Lassez and Gordon Plotkin, editors, Computational Logic: Essays in Honor of Alan Robinson. MIT Press, 1991.

....0 ) assume that one argument evaluates to true and the evaluation of the other one runs into an infinite derivation. This prevents the value true from being produced. Since do not know which is the good argument, we cannot safely evaluate sequentially. The main objective of Huet and Levy s [HL79, HL91] was to give a formal basis for the definition of efficient sequential rewriting strategies, i.e. reduction sequences such that only one redex is reduced in each step. The basic idea is to represent unknown (or unexplored) parts of a term t by using a new symbol Omega Gamma Terms in T ( Sigma [ ....

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.

....Pattern Matching by Term Decomposition 1 1 Introduction We are interested in compiling pattern matching in case of partially evaluated terms in order to do only necessary computations for the match. This is a kind of lazy computation over partially defined terms. In 1979 G. Huet and J J. Levy [5] defined a method for constructing match trees for non ambiguous linear term rewriting systems. However, the application of their results to the problem of compiling pattern matching as in the ML language was not clear until 1988 when A. Laville [6, 7] showed that it is possible to use their ....

..... In this example the value of the variable tree will match the second and the fourth cases but taking the first one as the priority holder, the expression exp2 will be executed. 1. 3 Compilation If patterns are non ambiguous, there is a decision procedure due to Huet and Levy [5] that determines whether an optimal match exists for a set of patterns and, in the case where such a January 1990 Digital PRL Compiling Pattern Matching by Term Decomposition 3 match exists, produces a search tree that allows to compile the match problem. This method can be illustrated with the ....

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G. Huet and J.-J. Levy. Call by need computations in non ambiguous linear term rewriting systems. Rapport IRIA Laboria 359, INRIA, Domaine de Voluceau, Rocquencourt BP105, 78153 Le Chesnay Cedex. FRANCE, 1979.

....described as an orthogonal rewriting system Paul Andr e Melli es Faculteit der Wiskunde en Informatica, VU, Amsterdam Abstract. We show that braids can be described as an orthogonal system in the sense of [1][2] Their computation is therefore confluent, which was already shown by Garside [4] 1 An addition on binary relation We propose a notion of addition on binary relations. Let A be a binary relation on X. We note A : the negation of A, A op its reverse and A = A op ) its dual. Let ....

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

.... which states that always reducing the leftmost outermost redex, i.e. the one with the least occurrence with respect to the lexicographic ordering, leads to the normal form (if one exists) For OE this is not the case due to the fact that patterns can be built , as the next example based on [HL79] shows. Example 6.1. Take the following term F j ( x;I] e) Omega ; II) If we use standard reduction, then we get nowhere F F F . However we can reduce this term to normal form as follows F ( x;I] e) Omega ;I] e This is similar to the situation in term rewriting ....

....Take the following term F j ( x;I] e) Omega ; II) If we use standard reduction, then we get nowhere F F F . However we can reduce this term to normal form as follows F ( x;I] e) Omega ;I] e This is similar to the situation in term rewriting systems investigated in [HL79]. For an overview of the theory of term rewriting systems see e.g. Klo90] As the examples show, has characteristics of both lambda calculus (evidently) and term rewriting systems (where rewriting is based on pattern matching) As shown in the examples term rewriting systems can be coded in ....

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

....has greater potential, but has the drawback of requiring specialized architectures. The second approach is concerned with the trade off between the generality of a class of rewrite systems and the efficiency of a normalizing strategy for systems in the class. Our effort belongs to this category [7, 9, 10, 13, 20, 21, 22, 23]. We introduce a hierarchical structure, called definitional tree, containing rewrite rules and show some of its applications to the control aspects of a rewrite strategy. First, we consider a class of systems whose rules are containable in a simplified, Supported by the National Science ....

....terminology [12] and call orthogonal a left linear, non overlapping rewrite system, and weakly orthogonal a left linear system whose critical pairs are all trivial, that is, if ht; t 0 i is a critical pair, then t is syntactically equal to t 0 . For related concepts and terminology see also [5, 9, 21]. Lemma 5. Any inductively sequential rewrite system R is orthogonal. Lemma 6. If an inductively sequential rewrite system is complete, then a ground term is a normal form if and only if it is a value. 4 Sequential Normalization In this section we use definitional trees to compute normal forms. ....

[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 359, INRIA, Le Chesnay, France, 1979.

.... ) f(oe(M 1 ) oe(M 2 ) oe(M n ) This system is left linear (no variable appears twice in the left hand sides of the rules) and has no critical pairs (the left hand sides of the rules do not overlap with a subterm of another left hand side) It is an orthogonal system in the sense of [9]. 1.2 Compilation to weak calculus with explicit substitution There is no simple definition of weak reduction in the traditional calculus. If reduction under the s is not allowed (i.e. if the ( rule is eliminated) the ChurchRosser property of the calculus no longer holds : y:y ( x:x) ....

....in the framework of the weak oe calculus. In the rest of this paper, we shall investigate the structure of the derivations in conditional TRS. That is, TRS with a restricting domain function. We now use the example of the weak oe calculus to recall some basic notions on terms and TRS. See [9] for a complete introduction to these notions. Each subterm in a term is characterized by its occurrence which is either ffl for the whole term or a sequence of integers n 1 :n 2 : n m representing the access path to it. For instance, in M = 1) s] 2[s] the subterms at occurrences 1:1:1 and ....

[Article contains additional citation context not shown here]

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

....if it generates a normalising sequence from every term which has a normal form. ffl A reduction strategy S is (transfinitely) hypernormalising if quasi (S) is (transfinitely) normalising. For finitary TRSs, Huet and L evy have shown that needed reduction is normalising for orthogonal systems [HL79, HL91], where a needed redex of a term is one such that every reduction of the term to normal form reduces at least one residual of the redex. This does not immediately generalise to the infinitary setting. A simple example is provided by the TRS consisting of the single rule: A B(A; A) The term A ....

....a term t is needed if in every strongly converging reduction of t to normal form some residual of s is rewritten. Theorem 8.5 For orthogonal TRSs, in every term having a normal form but not in normal form, there is at least one needed redex. Proof. Huet and L evy prove this for finite terms in [HL79, HL91]. A study of their proof reveals that it applies equally to infinite terms and strongly convergent reductions to normal form. We only note the few points where the infinitary aspects need some care. Two lemmas need new proofs. Lemma 3.15 of [HL91] Lemma 3.11 of [HL79] stating that every ....

[Article contains additional citation context not shown here]

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

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.

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.

No context found.

G. Huet and J-J. Levy, Call by Need Computations in Non-Ambiguous Linear Term Rewriting Systems, INRIA, Technical Report 359, 1979.

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