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.

....than the degrees of their ancestor redexes, and the theorem follows from Lemma 3.6. 4 Static garbage collection In [6] a theory of needed reduction is developed for the l calculus, extending a similar theory of normalization by neededness for orthogonal Term Rewriting Systems (OTRSs) in [10]. The main result of the theory is that every term not in normal form has a needed redex one whose residual is contracted in every normalizing reduction of that term and that repeated contraction of needed redexes normalizes every term having a normal form. In the l calculus, the ....

....redexes normalizes every term having a normal form. In the l calculus, the leftmost outermost redex is needed in any term, while finding a needed redex in a term in an OTRS is a difficult task, and much work has been done to improve and strengthen the theory of finding needed redexes initiated in [10]. An alternative concept of neededness, that of essentiality, was independently developed in [11] for the l calculus and later extended to orthogonal ERSs (OERSs) see e.g. 9, 12] Essential are subterms that have descendants under any reduction of the given term, and in particular all ....

HUET G., L EVY J.-J.. Computations in Orthogonal Rewriting Systems. In: Computational Logic, Essays in Honor of Alan Robinson, J.-L. Lassezand G. Plotkin, eds. MIT Press, 1991.

.... of equations defines a rewrite system that reduces questions (subject terms) to provably correct answers (normal forms) The theoretical foundations of the work lie in the class of forward branching systems[Str89] an important subclass of strongly sequential systems as defined by Huet and L evy[HL91] and in turn indebted to Kahn s idea of a sequential predicate. Both classes are based on the construction of an index tree, which can be used to specify an efficient pattern matching automaton for identifying needed redexes in the subject term. Forwardbranching systems of equations have the ....

G'erard Huet and Jean-Jacques L'evy. Computations in orthogonal rewriting systems, ii. In Computational Logic: Essays in Honor of Alan Robinson, chapter 12, pages 415--443. The MIT Press, 1991.

....our result: Left linear development closed PRSs are confluent. Let s explain the terminology used. A rewrite system for which the rewrite rules do not depend on one another is called orthogonal. Formalising this notion can be quite involved depending on the rewrite formalism it is applied to ([Hue80, Klo80, HL, GLM, MN94, Oos94]) but the intuition to be captured is always the same: an application of a rule replaces some substructure by another one, and in orthogonal systems we moreover have that if two distinct substructures can be replaced then these substructures are independent. Some (non)examples are: 1. The rules F ....

....(FD, i.e. that any serialisation of a development step is finite) How the techniques employed above can be slightly strengthened to yield FD as a bonus can be found in [Oos94] but since we don t need it here we omit it. Moreover, taking development steps for the multi step derivations in [HL] there s no problem in setting up the theory of permutation equivalence. Again, since we don t need that part of the general abstract theory of independence here, we don t develop it. In general steps need of course not be independent, but if steps operate on disjoint subterms, they are. ....

[Article contains additional citation context not shown here]

G'erard Huet and Jean-Jacques L'evy. Computations in orthogonal rewriting systems. Chch. 11,12 in [LP91].

....d (possibly with variables) we say that d is the computed value and oe is the computed answer for t. 3 The Language Modern functional logic languages are based on needed narrowing and inductively sequential programs. Needed narrowing extends the Huet and L evy s notion of a needed reduction [9]. A precise definition of this class of programs and the needed narrowing strategy is based on the notion of a definitional tree [3] 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 the inner nodes contain ....

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.

....Ken89, vR96, vO94, KvOvR93, Kha90, Tak93, Wol93] Because one of the aims of standardization is nding normalizing rewriting strategies, much work on normalization is related. Huet and L evy devised the idea of needed redexes, those which must be contracted in any rewrite sequence to normal form [HL91a, HL91b]. To aid in nding needed redexes, they devised the notions of sequentiality and strong sequentiality. Klop and Middeldorp provide a quite readable discussion of strong sequentiality [KM91] Barendregt, Kennaway, Klop, and Sleep raised the idea of needed redexes to the calculus [BKKS87] ....

....that such functions are good and (3) presents a generic generator of subterm ordering functions. The methods developed here are general enough for the applications in the companion paper [MW00] There seems to be a close connection between the material here and the notion of strong sequentiality [HL91b], but we have not formally veri ed this. Subterm Ordering Function Let be a CRS and let u 2 Ter( A subterm occurrence ordering of u is a sequence = p] of members of Skel(u) When is clear from context, we may abbreviate subterm occurrence ordering by subterm ordering . A ....

[Article contains additional citation context not shown here]

Gerard Huet and Jean-Jacques Levy. Computations in orthogonal rewriting systems, II. In Lassez and Plotkin [LP91], pages 415-443.

....anti standard pairs with standard complete developments [Klo80] We use Klop s second method. Huet and L evy de ne standard reductions for orthogonal TRS s as outside in reductions, using a leftmost choice function to determine a unique standard reduction for a permutation equivalence class [HL91a]. Their proof of termination of the standardization algorithm depends on the disjointness of residuals through arbitrary rewrite sequences, a property of OTRS s but not of HORS s. Gonthier, L evy, 2 and Melli es proved a standardization theorem for abstract rewriting [GLM92] Melli es has done ....

....Ken89, vR96, vO94, KvOvR93, Kha90, Tak93, Wol93] Because one of the aims of standardization is nding normalizing rewriting strategies, much work on normalization is related. Huet and L evy devised the idea of needed redexes, those which must be contracted in any rewrite sequence to normal form [HL91a, HL91b]. To aid in nding needed redexes, they devised the notions of sequentiality and strong sequentiality. Klop and Middeldorp provide a quite readable discussion of strong sequentiality [KM91] Barendregt, Kennaway, Klop, and Sleep raised the idea of needed redexes to the calculus [BKKS87] ....

Gerard Huet and Jean-Jacques Levy. Computations in orthogonal rewriting systems, I. In Lassez and Plotkin [LP91], pages 395-414.

....derivation e ; # # true in R such that # # # # [V ] which completes the proof. 2 23 6.2 Completeness Firstly, we consider the notions of descendants and traces. Let A = t # u,l#r t # ) be a reduction step of some term t into t # at position u with rule l # r. The set of descendants [31] of a position v of t by A, denoted v A, is v A = 8 : if u = v, v if u ## v, u.p # .q r p # = x if v = u.p.q and l p = x, where x # X . The set of traces of a position v of t by A, denoted v A is v A = 8 : v if u = v, v if u ## v, u.p # .q r p # = x ....

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.

....Ken89, vR96, vO94, KvOvR93, Kha90, Tak93, Wol93] Because one of the aims of standardization is nding normalizing rewriting strategies, much work on normalization is related. Huet and L evy devised the idea of needed redexes, those which must be contracted in any rewriting sequence to normal form [HL91a, HL91b]. To aid in nding needed redexes, they devised the notions of sequentiality and strong sequentiality. Klop and Middeldorp provide a quite readable discussion of strong sequentiality [KM91] Barendregt, Kennaway, Klop, and Sleep raised the idea of needed redexes to the calculus [BKKS87] ....

Gerard Huet and Jean-Jacques Levy. Computations in orthogonal rewriting systems, II. In Lassez and Plotkin [LP91], pages 415-443.

....with standard complete developments [Klo80] We use Klop s second method. Huet and L evy de ne standard reductions for orthogonal term rewriting systems (TRS s) as outside in reductions, using a leftmost choice function to determine a unique standard reduction for a permutation equivalence class [HL91a]. Their proof of termination of the standardization algorithm depends on the disjointness of residuals through arbitrary rewriting sequences, a property of orthogonal TRS s but not of orthogonal HORS s. Gonthier, L evy, and Melli es proved a standardization theorem for abstract rewriting [GLM92] ....

....Ken89, vR96, vO94, KvOvR93, Kha90, Tak93, Wol93] Because one of the aims of standardization is nding normalizing rewriting strategies, much work on normalization is related. Huet and L evy devised the idea of needed redexes, those which must be contracted in any rewriting sequence to normal form [HL91a, HL91b]. To aid in nding needed redexes, they devised the notions of sequentiality and strong sequentiality. Klop and Middeldorp provide a quite readable discussion of strong sequentiality [KM91] Barendregt, Kennaway, Klop, and Sleep raised the idea of needed redexes to the calculus [BKKS87] ....

Gerard Huet and Jean-Jacques Levy. Computations in orthogonal rewriting systems, I. In Lassez and Plotkin [LP91], pages 395-414.

....normalizable term eventually yields its normal form, even if the term possesses infinite reductions as well. The reason is that such redexes are needed in every normalizable term t, i.e. they are contracted in every normalizing reduction starting from t. Based on this observation, Huet and L evy [HuL e91] defined a general normalizing strategy, called the needed strategy, for Orthogonal Term Rewriting Systems (OTRSs) They showed that any term t not in normal form has a needed redex, and that repeated contraction of needed redexes in t leads to its normal form whenever there is one. Barendregt et ....

....of Q can be considered as being an initial part of P , up to L , and Proposition 3.1. 1) 2) applies; recall that discrete neededness is invariant under L . UEA Norwich, UK Technical Report SYS C96 06 Z. Khasidashvili and J. Glauert 13 It is easy to see that standard reductions, in the sense of [Bar84, HuL e91, Klo80, GLM92], are self essential. The notion of self essential reduction is the best approximation to the notion of standard reduction for DRSs, and it captures the essence of standardization in many respects. For example, it is used below in the proof of the DN theorem (which implies the usual normalization ....

[Article contains additional citation context not shown here]

Huet G., L'evy J.-J. Computations in Orthogonal Rewriting Systems. In: Computational Logic, Essays in Honor of Alan Robinson, J.-L. Lassez and G. Plotkin, eds., pp. 394-443, MIT Press, 1991.

....strategy which enables one to construct reductions to normal form. It is well known that the leftmost outermost strategy is normalizing in the calculus [CuFe58] For Orthogonal Term Rewriting Systems (OTRSs) a general normalizing strategy, called the needed strategy, was found by Huet and L evy [HuL e91]. The strategy always contracts a needed redex one whose residual has to be contracted in any reduction to normal form. Huet and L evy showed that any term not in normal form has a needed redex, and that repeated contraction of needed redexes leads to its normal form whenever there is one. ....

....Q=P is external to W , and since W= Q=P ) WN = Q=P ) W u = Q=u) t N ) redexes in W= Q=P ) are created by P=Q. t Q e W u 2 o u Q=u o 0 3 W u = Q=u) u=Q W 2 s N Q=P = Q=u) N s 0 3 W N= Q=u) One can verify that the calculus [L ev78] orthogonal TRSs [HuL e91], orthogonal Higher Order Rewriting Systems (HORSs) OR94, Oos94] and hence other orthogonal higher order rewrite systems, such as CRSs [Klo80] ERSs [Kha92] and HRSs [Nip93] and Orthogonal Term Graph Rewriting Systems [KKSV93] form DRSs; the latter are equivalent to Maranget s orthogonal DAG ....

[Article contains additional citation context not shown here]

Huet G., L'evy J.-J. Computations in Orthogonal Rewriting Systems. In: Computational Logic, Essays in Honor of Alan Robinson, J.-L. Lassez and G. Plotkin, eds., pp. 394-443, MIT Press, 1991.

....systems, no subterm can be absorbed to the left of the contracted redex, so the leftmost outermost redexes are unabsorbed. In particular, the calculus and the combinatory logic are non left absorbing. Unabsorbed redexes can easily be found also in the wide class of strongly sequential OTRSs [3]. We develop a method for proving that the reductions constructed according to our perpetual strategy are indeed the longest, and for finding their lengths. Our method is similar to the Nederpelt s method [8] invented to reduce proofs of strong normalization to proofs of weak normalization ....

....We call a subterm s of a term t unabsorbed in a reduction P : t i e if the descendants of s do not appear inside redex arguments of terms in P , and call s absorbed in P otherwise. We call s unabsorbed in t if it is unabsorbed in any reduction starting from t, and absorbed in t otherwise. In [3], Huet and L evy introduced the notion of external redex of a term and proved that each term not in normal form possesses an external redex. It is easy to show that a redex u t is unabsorbed iff u is external in t. Thus we have the following lemma; a short direct proof of it can be found in ....

[Article contains additional citation context not shown here]

Huet G., L'evy J. -J. Computations in Orthogonal Rewriting Systems. In Computational Logic, Essays in Honor of Alan Robinson, ed. by J. -L. Lassez and G. Plotkin, MIT Press, 1991.

....reductions to normal form. It is well known that the leftmost outermost strategy is normalizing in the # calculus [10] Normalization by Needed Reduction For Orthogonal Term Rewriting Systems (OTRSs) a general normalizing strategy, called the needed strategy, was found by Huet and Levy in [18]. The needed strategy always contracts a needed redex a redex with at least one contracted residual in every reduction to normal form. Huet and Levy show that any term t not in normal form has a needed redex, and that repeated contraction of needed redexes in t leads to its normal form whenever ....

....to consider a set of terms S, and the induced set of reductions #S ending at a term 1 Note that descendants and residuals are sometimes treated as synonymous in the literature. 2 A multistep contracts a set of redexes simultaneously. 2 in S. For example, Huet and Levy neededness [18] is neededness w.r.t. the set NF of normal forms, Maranget neededness [35] is neededness w.r.t. all fair reductions, headneededness [7] is neededness w.r.t. the set of head normal forms, root neededness [40] is neededness w.r.t. the set of root stable forms, etc. We impose a natural condition on ....

[Article contains additional citation context not shown here]

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. MIT Press, 1991.

....which enables one to construct reductions to normal form. It is well known that the leftmost outermost strategy is normalizing in the calculus [CuFe58] For Orthogonal Term Rewriting Systems (OTRSs) a general normalizing strategy, called the needed strategy, was found by Huet and L evy in [HuL e91]. The needed strategy always contracts a needed redex a redex whose residual is to be contracted in any reduction to normal form. Huet and L evy showed that any term t not in normal form has a needed redex, and that repeated contraction of needed redexes in t leads to its normal form whenever ....

....if in no term redex patterns can overlap, i.e. if r i redex u contains an r j redex u 0 and i 6= j, then u 0 is in an argument of u, and the same holds if i = j and u 0 is a proper subterm of u. R is orthogonal (OERS) if it is left linear and non overlapping. As in the case of OTRSs [HuL e91] and the calculus [Bar84] for any co initial reductions P and Q, one can define in OERSs the notion of residual of P under Q, written P=Q, due to L evy [L ev78] PtQ abbreviates P Q=P . We write P Theta Q if P=Q = Theta is the L evy embedding relation) P and Q are called ....

[Article contains additional citation context not shown here]

Huet G., L'evy 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.

....it is no longer possible to duplicate it by reducing an outer redex. We can satisfy both these conditions if in each term s we contract a limit redex which is defined as follows: Choose in s an external redex u 1 (i.e. a redex whose descendants never appear inside the arguments of other redexes [19]) choose an erased argument s 1 of u 1 that is not in normal form; choose in s 1 an external redex u 2 , and so on as long as possible. The last redex chosen is a limit redex of s. This strategy, which we call a limit strategy, is not computable for orthogonal systems in general, but is ....

....choose in s 1 an external redex u 2 , and so on as long as possible. The last redex chosen is a limit redex of s. This strategy, which we call a limit strategy, is not computable for orthogonal systems in general, but is computable, for example, for the large class of strongly sequential OTRSs [19], where at least one external redex can be found effectively in any reducible term. The left normal OTRSs, where function symbols precede variables in left hand sides of rewrite rules, and in particular the Combinatory Logic [11] are strongly sequential as the leftmostoutermost redex is external ....

[Article contains additional citation context not shown here]

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

....but there is no adequate derivation starting in the request configuration hp(x) ffli. 5 Concluding Remarks In [7] we studied a generalization of logic programming where success and failure are replaced by predicates for adequacy and inadequacy. In that general setting we defined, inspired by [12], needed atoms as atoms that have to be selected eventually in order to reach an adequate state, and we showed that under certain conditions adequate SLD trees are obtained by repeatedly selecting needed atoms. The setting of the present paper is much more concrete. In particular, whereas needed ....

G. Huet and J.-J. L'evy. 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, 1991.

....It could then be argued that this axiomatisation is able to capture the concurrent behaviour of a system: Each equivalence class of sequents should intuitively describe the same set of causally unrelated computations. This is not so different in spirit from the well known permutation equivalence [5, 21], and there exists in fact a tight correspondence between the two notions [29] For a few initial considerations about the actual degree of concurrency expressed by the axioms, we refer to [8] Example 4 (equating sequents) Let us consider again the ars V . As shown in Example 3, the system ....

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.

.... 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.

....somewhat non standard concept of standardization. Firstly, our standard reductions are those in which later steps do not erase preceding ones. In DFSs we cannot define a left to right and or outside in concept of standard reduction familiar from the calculus and orthogonal rewriting systems [Bar84, HL91, Klo80], but we do not need such a standardization concept either. Secondly, we define a relativized standardization algorithm which standardizes any reduction Q in an ASDRS w.r.t. any other reduction P , co initial with Q. Definition 3.1 ( KG96] ffl Let P : t and u t. We call u P needed if there ....

....of mathematics. Obviously, relative) independence of redex sets is undecidable in general, as is neededness. However, we hope that decidable approximations for independence can be defined which will yield decidable concepts for large classes of rewrite systems, as is the case for the neededness [HL91]. For example, all the introduced concepts are decidable for Recursive Program Schemes (RPSs) both in first [Kha93] and higher order [Kha94] cases. Several kinds of RPS were extensively studied in the literature, mainly in the seventies [Cou90] Our (first order) RPSs correspond to Applicative ....

Huet G., L'evy 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, pp. 394-443.

....is deferred to the full paper. The representation of rewriting sequences as cells induces an obvious equivalence relation on them, relating two sequences if they are represented by the same cell. In the case of term rewriting, such equivalence coincides with the socalled permutation equivalence [5, 14], due to the axioms of cartesian 2 categories [20, 11] The precise characterization of this equivalence for the case of term graph rewriting is left as a topic for further research. A promising direction consists in relating the definition of term graph rewriting presented here with the algebraic ....

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 J. A. Robinson, pages 395--414. MIT Press, 1991.

....on terms which selects a perpetual redex in any 1 term, and selects any redex (if any) otherwise [Bar84] A redex (not an occurrence) is called perpetual iff its occurrence in any (admissible) context is perpetual. Finally, let us recall the concept of external redexes due to Huet and L evy [HuL e91]. These are redexes whose residuals or descendants can never occur in an argument of another redex. Any external redex is outermost, but not vice versa (e.g. the first a in f(a; a) in R = ff(x; g(y) y; a g(b)g, is outermost but not external) Definition 3.1 (1) Let 1(t) in an OCERS, and ....

Huet G., L'evy 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, pp. 394-443.

....Definition 3.2 Let t be a term in an OCRS R. We call a subterm s in t essential (written ES(s; t) if s has at least one descendant under any reduction starting from t and call it inessential (written IE(s; t) otherwise. The notion of essentiality is a generalization of the notion of neededness [6, 15] in a way that it works for all subterms, bound variables in particular. See [10] for the precise relationship between the notions) The following two lemmas are valid for all OCRSs; the proofs are similar to the case of orthogonal TRSs [10] Lemma 3.1 Let s 0 ; s k t be such that IE(s ....

Huet G., L'evy J.-J. Computations in Orthogonal Rewriting Systems. In Computational Logic, Essays in Honor of Alan Robinson, ed. by J.-L. Lassez and G. Plotkin, MIT Press, 1991.

....hypercollapsing towers, which are artificially made equivalent in the other approach, can be reduced to the same term, by a single parallel reduction. The paper is organized as follows. In Section 2 we summarize the basic definitions about infinite terms [GTWW77] orthogonal term rewriting [HL91] and parallel term rewriting [Cor93] In Section 3 we introduce (possibly cyclic) term graphs, and we make precise their relationship with (sets of) rational terms, via the unraveling function. Algebraic term graph rewriting is the topic of Section 4, where we recall the basics of the ....

....ft i g , 8w 2 : 9i : 8j i : t j (w) t(w) Moreover, every pair of terms has a greatest lower bound. All this amounts to say that CT Sigma (X) is an complete lower semilattice, or simply a CPO. 2. 2 Term Rewriting We recall here the basic definitions of (orthogonal) term rewriting [HL91] which apply to infinite terms as well. 5 Let X and Y be two sets of variables. A substitution (from X to Y ) is a function oe : X CT Sigma (Y ) used in postfix notation) Such a substitution oe can be extended in a unique way to a continuous (i.e. monotonic and lub preserving) function ....

[Article contains additional citation context not shown here]

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 J. A. Robinson, pages 395--414. MIT Press, 1991.

....horizontal category of cells. It is wellknown that 2 categories represent a faithful model for term rewriting systems [RS87, Pow89, CGM95] the arrows of the underlying category denote terms, and cells denote (equivalence classes of) rewrites (accordingly to the so called permutation equivalence [Bou85, HL91]) In the paper we show that a similar adequacy result holds for our tile model, if we consider double categories instead. Thus, the generality we claimed for out tile model is further confirmed by the richer structure of double categories with respect to 2 categories. Summarizing, the tile model ....

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

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.

No context found.

Huet G., L'evy 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.

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