G. Huet, J.-J. L'evy, "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

....if none of the descendants of 8 appear in redex arguments of terms in P, and is absorbed in P otherwise; s is unabsorbed in t if it is unabsorbed in any reduction starting from t and absorbed in t otherwise [Kha93] Remark 3. 1 It is easy to see that a redex u C t is unabsorbed iff u is ezternal [HuL91] in t. Clearly, unabsorbedness implies essentiality, and Huet Levy and Maranget neededness coincide for norrealizable terms. Definition 3.4 Let P: t s and o be a subterm or a component in t. Then we say that P deletes o if o doesn t have P descendants. Definition 3.5 (1) Let P: t tl ....

....reduction, since, e.g. g(a, a) g(b, a) g(a, a) G 5 , but g(b, a) 5 . Note that the second occm rence of a in g(a, a) is 5, unneeded, but its residual in g(b, a) is 5, needed. f(a) g(a,a) f(b) g(b, b) The most appealing examples of stable sets, for an OERS, are the set of normal forms [HuL91], the set of head normal forms [BKKS87] the set of weakhead normal forms (a partial result is in [Mar92] and the set of constructorhead normal forms fox constructor TRSs [NSk94] The sets of terms having (resp. not having) head , constructor head ) normal forms are stable as well. The graph ....

Huet G., Lvy J.-J. Computations in Orthogonal Rewriting Systems. In: Computational Logic, Essays in Honor of Alan Robinson, J.-L. Lassez and G. Plotkin, eds. MIT Press, 1991.

....proposed a normalisation procedure based on lazy rewriting. A preliminary transformation that allows extending the application scope of thunki cation while preserving a nice correspondence between normalisation traces was also presented. Finding optimal derivations is undecidable in general [16] [11] and even when it is decidable, the decision procedures are often dicult to implement. In practice, most of interesting results only involve orthogonal constructorbased TRSs [20] 2] 21] We think that our normalisation procedure is helpful since the normalisation procedure is reasonably ecient ....

G. Huet and J-J. Levy. Computations in orthogonal rewriting systems, Part I + II. In J-L. Lassez and G. D. Plotkin, editors, Computational Logic { Essays in Honor of Alan Robinson, pages 395-443, 1991.

....unification at nodes 1 and 2, we can conclude that unification at nodes 5, 6 and 7 is bound to fail. Early detection of such non unifiable subterms can lead to further savings in time. There has been considerable research in factoring out common computations for pattern matching 1 (e.g. [3, 4, 9]) In [10] we gave an algorithm for factoring out common computations that arise in indexing of Prolog clauses 2 . However the design of a similar efficient algorithm for the subterm unification problem has remained open and forms the topic of this paper. 1 In pattern matching p is always ....

G. Huet and J.-J. Levy, Computations in Orthogonal Rewriting Systems, In Essays in Computational Logic (in honor of Alan Robinson), J.-L. Lassez and G. Plotkin, ed., MIT Press, 1991. 41

....of a good narrowing strategy, i.e. a restriction on the narrowing steps issuing from t, without loosing completeness. Needed narrowing [5] is currently the best known narrowing strategy due to its optimality properties. Needed narrowing extends the Huet and L evy s notion of a needed reduction [23]. The definition of needed narrowing [5] uses the notion of definitional tree [4] Roughly speaking, a definitional tree for a function symbol f is a tree whose leaves contain all (and only) the rules used to define f and whose inner nodes contain information to guide the (optimal) pattern ....

....redexes are contracted in such derivations. This restriction is necessary in order to ensure that the rank of a term is never infinite (hence it is well defined) since it suffices to reduce a finite number of needed redexes in order to obtain the constructor normal form (if it exists) of a term [23]. We have considered the longest needed reduction sequence, since (outermost) needed narrowing based unfolding can increase in general the length of (arbitrary) needed reductions, and thus the notion of rank in terms of the shortest sequence would not be preserved by unfolding. Also, folding steps ....

[Article contains additional citation context not shown here]

G. Huet and J.J. L'evy. Computations in orthogonal rewriting systems, Part I + II. In J.L. Lassez and G.D. Plotkin, editors, Computational Logic -- Essays in Honor of Alan Robinson, pages 395--443. The MIT Press, Cambridge, MA, 1992.

....a good narrowing strategy, i.e. a restriction on the narrowing steps issuing from t, without losing completeness. In the following, we brie y outline the needed narrowing strategy (a formal description can be found in [7] Needed narrowing extends Huet and L evy s notion of a needed reduction [23] and is de ned on inductively sequential programs [3] Roughly speaking, in an inductively sequen leq( X ,Y) leq(zero,Y) true leq(succ(M) Y ) leq(succ(M) zero) false leq(succ(M) succ(N) leq(M,N) Q Q Q Q Q Q Q Q Q Q Q Q Figure 1: De nitional tree for the ....

....several replacements. In these situations, there cannot be a single in place replacement. Furthermore, the steps that we compute in overlapping inductively sequential rewrite systems are needed, but only modulo nondeterministic choices [5] Hence, some step may not be needed in the strict sense of [7, 23], but we may not be able to know by feasible means which steps. The architecture of our implementation is characterized by terms and computations. Both terms and computations are organized into tree like linked (dynamic) structures. A term consists of a root symbol applied to zero or more ....

G. Huet and J.-J. Levy. Computations in orthogonal rewriting systems. In J.-L. Lassez and G. Plotkin, editors, Computational Logic: Essays in Honor of Alan Robinson, pages 395-443. MIT Press, 1991.

....in formulations are explored and a review of abstract CR proofs is given (Tait Martin Lof, Newman, diamond property of special reductions, developments, parallel reduction of disjoint redex occurrences) 3.4. AN ABSTRACT REVIEW OF THE WORK ON CONFLUENCE 35 3.4. 3 Huet L evy Huet and L evy ( HL 91] work toward the Church Rosser Theorem and the Standardisation Theorem and explore reduction strategies in finite orthogonal TRSs. The results are restricted to OTRSs, where there are no critical pairs, the descendants of redexes are redexes themselves and where residuals of a set of disjoint ....

G. Huet, J.-J. L'evy, "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.

....of steps. This might suggest that the execution time of app2s (for suciently large inputs) should be about one half the execution time of app. However, executions of function app2s in several environments (e.g. in the lazy functional language Hugs [19] and the functional logic language Curry [17]) show that speedup is only around 10 . In order to reason about these counterintuitive results, we introduce several formal criteria to measure the eciency of a functional logic computation. We consider inductively sequential rewrite systems as programs. Inductive sequentiality ensures strong ....

....Therefore, to be formal, we consider only derivations in the technical results of this paper. Note that this is not a practical restriction in our context, since derivations are ecient, easy to compute, and used in the implementations of modern functional logic languages such as Curry [17] and Toy [27] The above property allows us to de ne the cost of a term as the cost of its derivation when the computed substitution is empty (since it is unique) De nition 5 (cost of a term) Let R be a program and t a term. Let C denote a cost criterion. We overload function C by partially ....

G. Huet and J.J. Levy. Computations in orthogonal rewriting systems, Part I + II. In J.L. Lassez and G.D. Plotkin, editors, Computational Logic { Essays in Honor of Alan Robinson, pages 395-443, 1992.

....subterms that are needed, in a precise technical sense, to obtain a result. Needed Narrowing. Needed narrowing is an optimal evaluation strategy w.r.t. both the length of derivations and the independence of computed solutions [10] It extends the Huet and L evy notion of a needed rewrite step [18] to functional logic programming. Following [10] a narrowing step t ; p;R;oe) t 0 is called needed iff, for every substitution such that oe , p is the position of a needed redex of (t) in the sense of [18] A narrowing derivation is called needed iff every step of the derivation is ....

....solutions [10] It extends the Huet and L evy notion of a needed rewrite step [18] to functional logic programming. Following [10] a narrowing step t ; p;R;oe) t 0 is called needed iff, for every substitution such that oe , p is the position of a needed redex of (t) in the sense of [18]. A narrowing derivation is called needed iff every step of the derivation is needed. An efficient implementation of needed narrowing exists for inductively sequential programs. The formal definition of inductive sequentiality is rather technical. In this paper, for the sake of completeness, we ....

G. Huet and J.J. L'evy. Computations in orthogonal rewriting systems, Part I + II. In J.L. Lassez and G.D. Plotkin, editors, Computational Logic -- Essays in Honor of Alan Robinson, pages 395--443, 1992.

....This theorem guarantees that there exists a standard reduction sequence to obtain normal form, thus we can concentrate on such reduction sequences in the computation of terms. Huet and L evy took an important step towards standardization theorem for orthogonal term rewriting systems in 1979 [6] and they demonstrated standardization theorem holds for orthogonal rewrite systems. Their result was extended to left linear TRSs by Boudol[2] Although, their proof involves in some unfamiliar notions like initial redex, external positions, or external redex positions, and is complicated. Just ....

G. Huet and J.-J. Levy. Computations in orthogonal rewriting systems, I. In J.-L. Lassez and G. Plotkin, editors, Computational Logic, Essays in Honor of Alan Robinson, chapter 11, pages 395-414. The MIT Press, 1991.

....oeg = oe 0 [ f(FVarG)g and oee = oe 0 [ f(FVarE)g. 3.2 Addresses and parallel reduction So, a consequence of mixing those two systems is the creation of a critical pair (non determinism) and thus non orthogonality. Fortunately, since this critical pair is at the root, the residual redex notion (Huet L evy 1991) can be extended in a straight forward way: We just observe that there is no residual redex of (FVarG) resp. FVarE) after applying (FVarE) resp. FVarG) We first establish that this is safe before we give the definition (Def. 3.7) Definition 3.1. A complete development of a preterm T is a ....

Huet, G. & L'evy, J.-J. (1991), Computations in orthogonal rewriting systems, II, in J.-L. Lassez & G. Plotkin, eds, `Computational Logic', The MIT press, chapter 12, pp. 415--443.

....can be assigned to them, and the approximants semantics, where terms are interpreted by the set of their approximants, and their interrelation. Approximants are defined as rooted finite sub trees of the (possibly infinite) normal form, based on the notion of Omega normal forms of Huet and Levy [16] (see also [18] The relation between the filter semantics and the approximation semantics has been studied extensively in the setting of the Lambda Calculus (LC) 6] see [8, 7, 1, 3] where it has been proved that they coincide [19, 3] But, perhaps surprisingly, this has never been studied ....

G. Huet and J.J. Levy. Computations in Orthogonal Rewriting Systems. In J.-L. Lassez and G. Plotkin, editors, Computational Logic. Essays in Honour of Alan Robinson. MIT Press, 1991.

.... programs) it can be eciently implemented by pattern matching and uni cation due to its local computation of a narrowing step (see, e.g. Antoy, 1996; Hanus, 1995; Loogen et al. 1993) On ground terms, needed narrowing falls back to the classical notion of needed reduction in the sense of Huet and L evy (1991). Evaluation by needed narrowing has also some similarities to pattern matching and lazy evaluation in functional languages like Haskell (Hudak et al. 1992) or Miranda (Peyton Jones, 1987) Note, however, that needed narrowing or needed reduction is a more powerful evaluation strategy than the ....

....now. Nevertheless, we strongly conjecture that LNT reductions are in fact normalizing which is witnessed by the following facts: LNT reductions are needed for reduction to a constructor normal form. LNT reductions fall back to ( rst order) needed reductions, which are known to be normalizing (Huet L evy, 1991), if the higher order features are not used. For terminating higher order inductively sequential rewrite systems, we show that LNT reductions compute a ground constructor normal form, if one exists. This implies a general completeness result for our strategy w.r.t. arbitrary reduction. Note that ....

[Article contains additional citation context not shown here]

Huet, G., & Levy, J.-J. (1991). Computations in orthogonal rewriting systems, I. Pages 395-414 of: Lassez, J.-L., & Plotkin, G. (eds), Computational Logic: Essays in Honor of Alan Robinson. Cambridge, MA: MIT Press.

....theorem [1] which shows that leftmost outermost reduction is a terminating strategy. These two fundamental theorems can be found in many different situations, such as in PCF (typed calculus augmented with several ffi rules, such as recursion or arithmetic) in orthogonal term rewriting systems [2], in orthogonal dags and in interaction networks. These different settings must share a common property to yield these two results. As proving these properties time and again is rather frustrating, there have been attempts to define an axiomatic version of the Church Rosser property [3] However, ....

....In our examples, the righmost outermost reductions loop. Intuitively, redexes may be created to the right, as in (y:a) Delta Delta) and a looping redex may be nested in a K redex. But left to right is no longer valid with ffi rules. Consider the case of first order term rewriting systems (TRS) [2], and, for instance, the following system fA A; B C; F (x; C) Dg. Then F (A; B) F (A; C) D reaches the normal form, whereas the leftmost reduction loops. Here terminating strategies are clearly rightmost, and we suspect that we may define a notion of normal reduction, representing the ....

[Article contains additional citation context not shown here]

G. Huet and J.-J. L'evy, "Computations in orthogonal rewriting systems," in Computational Logic; Essays in Honor of Alan Robinson (J.-L. Lassez and G. D. Plotkin, eds.), pp. 394--443, MIT Press, 1991.

....of intersection types that can be assigned to them, and the approximants semantics, where terms are interpreted by the set of their approximants. Approximants are defined as rooted finite sub trees of the (possibly infinite) normal form, based on the notion of Omega normal forms of Huet and Levy [15] (see also [17] The relation between the filter semantics and the approximation semantics has been studied extensively in the setting of the Lambda Calculus (LC) 6] see [8, 7, 1, 3] where it has been proved that they coincide [18, 3] But, perhaps surprisingly, this has never been studied ....

G. Huet and J.J. Levy. Computations in Orthogonal Rewriting Systems. In J.-L. Lassez and G. Plotkin, editors, Computational Logic. Essays in Honour of Alan Robinson. MIT Press, 1991.

.... via an ordinary calculus (a topic well explored in the so called explicit substitution framework [1] but the analysis of the concurrent semantics induced over the reduction steps by the axioms of rewriting logic, via a comparison with the permutation equivalence proposed by Jean Jacques L evy [40,51]. Roughly, the main point is that the rewriting steps are too concrete. For example, forgetting for the sake of readability the coercion operator and the subscript E , let us consider the terms (x: x) Delta z and (y: y) Delta z, belonging to the same equivalence class. The sequent [ x: x) ....

G. Huet and J.-J. L'evy. Computations in orthogonal rewriting systems, I. In J.-L. Lassez and G. Plotkin, editors, Computational Logic: Essays in honour of Alan Robinson, pages 395--414. MIT Press, 1991. 44

....redexes of t. A development of Phi is a reduction sequence such that after each initial segment ae, the next reduced redex is an element of Phinae. A complete development of Phi is a development ae such that Phinae = ut Next fact derives from the parallel moves lemma (see e.g. Lemma 2. 2 in [9]) Proposition16. All complete developments ae and ae 0 of a finite set of redexes Phi in a term t are finite, and end with the same term. ut Exploiting this result, we define the parallel reduction of a finite set of redexes as any complete development of them. Definition17 (finite parallel ....

G. Huet, J.J. L'evy, Computations in Orthogonal Rewriting Systems, chapter 11 of Computational Logic, eds. J.L. Lassez and G. Plotkin, MIT Press, 1991.

....the definition of a good narrowing strategy, i.e. a restriction on the narrowing steps issuing from t, without losing completeness. Needed narrowing [5] is currently the best known narrowing strategy due to its optimality properties. It extends the Huet and L evy s notion of a needed reduction [18]. The definition of needed narrowing [5] uses the notion of definitional tree [4] Roughly speaking, a definitional tree for a function symbol f is a tree whose leaves contain all (and only) the rules used to define f and whose inner nodes contain information to guide the (optimal) pattern ....

....methodology for lazy functional logic programs preserving the semantics of both values and answers computed by an efficient (currently the best) operational mechanism. For proving correctness, we extensively exploit the existing results from Huet and Levy s theory of needed reductions [18] and the wide literature about completeness of needed narrowing [5] rather than striving an ad hoc proof) We have shown that the transformation process keeps the inductively sequential structure of programs. We have also illustrated with several examples that the transformation process can be ....

G. Huet and J.J. L'evy. Computations in Orthogonal Rewriting Systems, Part I + II. In J.L. Lassez and G.D. Plotkin, editors, Computational Logic -- Essays in Honor of Alan Robinson, pages 395--443, 1992.

.... i.e. typical functional programs) it can be efficiently implemented by pattern matching and unification due to its local computation of a narrowing step (see, e.g. 9] On ground terms, needed narrowing falls back to the classical notion of needed reduction in the sense of Huet and L evy [13]. Evaluation by needed narrowing has also some similarities to pattern matching and lazy evaluation in functional languages like Haskell [12] or Miranda [29] Note, however, that needed narrowing or needed reduction is a more powerful evaluation strategy than the simpler left to right pattern ....

G'erard Huet and Jean-Jacques L'evy. Computations in orthogonal rewriting systems, I. In J.-L. Lassez and G. Plotkin, editors, Computational Logic: Essays in Honor of Alan Robinson, pages 395--414. MIT Press, Cambridge, MA, 1991.

No context found.

G. Huet, J.-J. L'evy, "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.

Gerard Huet and Jean-Jacques Levy. Computations in orthogonal rewriting systems, II. In Jean-Louis Lassez and Gordon Plotkin, editors, Computational Logic: Essays in Honor of Alan Robinson, chapter 12, pages 415-443. The MIT Press, Cambridge, MA, 1991.

No context found.

Gerard Huet and Jean-Jacques Levy. 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. The MIT Press, Cambridge, MA, 1991. This is a revision of [HL79].

No context found.

Gerard Huet and Jean-Jacques Levy. Computations in orthogonal rewriting systems, I. In Computational Logic: Essays in Honor of Alan Robinson, pages 395{ 414. MIT Press, 1991.

No context found.

G. Huet and J.-J. L#evy. Computations in orthogonal rewriting systems, II. In J.- L. Lassez and G. Plotkin, editors, Computational Logic: Essays in Honor of Alan Robinson,chapter 12, pages 415#443. MIT Press, Cambridge, MA, 1991.

No context found.

G. Huet and J.-J. L#evy. Computations in orthogonal rewriting systems, I and II. In J.-L. Lassez and G. Plotkin, editors, Computational Logic: Essays in Honor of Alan Robinson, pages 395#443. MIT Press, Cambridge, MA, 1991. This is a revision of #50#.

No context found.

G#erard Huet and Jean-Jacques L#evy. Computations in orthogonal rewriting systems, I and II. In J.-L. Lassez and G. Plotkin, editors, Computational Logic: Essays in Honor of Alan Robinson, pages 395#443. MIT Press, Cambridge, MA, 1991.

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