A. Middeldorp. Call by Need Computations to Root-Stable Form. In Proceedings of the 24th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, pages 94--105. ACM, New York, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....infinite strongly converging sequence t = t 1 t 2 Delta Delta Delta has a limit s (written t s which is necessarily an infinite term. If t s or s, we write t s. A rewrite sequence is called infinitary normalizing if it strongly converges to a (possibly infinite) normal form [25]. An infinite rewrite sequence that is not infinitary normalizing is called perpetual. 2.1 Context sensitive rewriting Given a signature F , a mapping : F P(N) is a replacement map (or F map) if for all f 2 F ; f) f1; ar(f)g. The replacement map determines the argument positions ....

....a using the leftmost outermost rewriting strategy yields the strongly converging sequence: a f(a,a) f(f(a,a) a) Delta Delta Delta which does not converge to a normal form. In order to ensure correctness of nf S w.r.t. nf, we need to use an infinitary normalizing strategy S(see [16, 20, 25] for a discussion about the definition of such strategies. 3.2 Semantics of context sensitive computations The rewriting semantics cs nf for a TRS R computes the set of normal forms of each term t: cs nf (t) fs 2 NF We also consider the infinitary version: cs nf (t) fs 2 ....

A. Middeldorp. Call by Need Computations to Root-Stable Form. In Conference Record of the 24th Annual ACM Symposium on Principles of Programming Languages, POPL'97, pages 94-105, ACM Press, 1997.

....this setting, the canonical replacement map (denoted by R ) is specially important, as it specifies the See, e.g. OV02] for a very recent survey on the topic where this requirement is part of the definition of strategy (Definition 8.1. 1) similar requirements can be found in [AM96, Klo92, Mid97] In contrast, BEGK 87] admits strategies which are not forced to reduce terms containing redexes. Here, a value is a term which contains no defined symbol, i.e. symbols occurring at the outermost position of the left hand sides of any rule of the TRS. most restrictive replacement map ....

....but have no normal form. Since each redex in a term which does not have a normal form is needed, neededness is not useful for discriminating the redexes which should be contracted to normalize (such) a term. Instead, we use Middeldorp s to define root normalizing and root needed computations [Mid97] A redex in a term is root needed if the redex (itself or some of its descendants) is reduced in every root normalizing derivation issued from this term. Every term which is not root stable contains a root needed redex and reduction of root needed redexes is root normalizing [Mid97] Since we ....

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A. Middeldorp. Call by Need Computations to Root-Stable Form. In Conference Record of the 24th Annual ACM Symposium on Principles of Programming Languages, POPL'97, pages 94-105, ACM Press, 1997.

....for correspondence: Institute of Information Sciences and Electronics, University of Tsukuba, Tsukuba 305, Japan. Email: ami score.is.tsukuba.ac.jp. In this paper we also study decidable call by need computations to root stable form, the underlying theory of which is expounded in Middeldorp [14]. This is more complicated because root needed redexes, which are redexes that are contracted in every rewrite sequence to rootstable form, are not uniform. More precisely, we have to employ weak second order monadic logic in order to get the strongest decidability results. Nevertheless, we ....

....on optimal normalizing reduction strategies for orthogonal TRSs. Theorem 2.1 Let R be an orthogonal TRS. 1) Every reducible term contains a needed redex. 2) Repeated contraction of needed redexes results in a normal form, whenever the term under consideration has a normal form. Middeldorp [14] generalized the above result to rewrite sequences to root stable form. A redex Delta in a term t is root needed if in every rewrite sequence from t to root stable form a descendant of Delta is contracted. In [14] it is argued that root neededness is the proper generalization of neededness when ....

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A. Middeldorp, Call by Need Computations to Root-Stable Form, Proc. 24th POPL, 1997. To appear.

.... of paralleloutermost reduction may also be used as essential property (this is exactly what is needed in the construction) And it seems plausible ( Mid97b] that hyper normalization of parallel outermost reduction for weakly orthogonal TRSs can be proved by using ideas of [SR93] OR94] [Mid97a]. Yet, this remains to be checked in detail. Concerning normalization of outermost fair reduction we are not aware of any further positive results (besides the one mentioned above) For instance, it seems to be open whether outermost fair reduction is also normalizing for weakly orthogonal TRSs. ....

Aart Middeldorp. Call by need computations to root-stable form. In Proc. 24th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, pages 94--105, Paris, 1997.

....They correspond to in nitary normalizing strategies if we restrict our attention to computing (in nite) values rather than arbitrary (in nite) normal forms. It is possible to provide an e ective notion of in nitary normalizing strategy by using Middeldorp s theory of root needed computations [Mid97] and their decidable approximations [Luc98] Remark 3.4 We obtain a ground semantics for the de ned symbols f 2 F as follows: f( CRew 1 (f( for all 2 T (C ) ar(f) Similarly, it is possible to describe a ground semantics under a given strategy F by using CRew 1 F . 4 Narrowing as ....

A. Middeldorp. Call by Need Computations to Root-Stable Form. In Conference Record of the 24th ACM Symposium on Principles of Programming Languages, pages 94-105, 1997.

....in place replacement. 2. Only somewhat needed steps are executed. We use the quali er somewhat because di erent notions of need have been proposed for di erent classes of rewrite systems. We execute a particular kind of steps that for reductions in orthogonal systems is known as rootneeded [30]. Thus, reductions that are a priori useless are never performed. We call this principle useful step. 3. Don t know non deterministic reductions are executed in parallel. Both narrowing computations (in most rewrite systems) and reductions (in interesting rewrite systems) are non deterministic. ....

A. Middeldorp. Call by need computations to root-stable form. In Proc. 24th ACM Symposium on Principles of Programming Languages (Paris), pages 94-105, 1997.

.... is not difficult to see (considering Theorems 4 and 1) that, for every orthogonal TRS R, GammaSeval R needed redexes are needed in the sense of [KKSV95] Unfortunately, S neededness does not always naturally coincide with other well known notions of neededness such as, e.g. root neededness [Mid97]. Example 8. Consider the TRS R: a b c b f(x,b) g(x) 11 and t = f(a,c) which is not root stable (hence, f(a,c) 62 W hnf ) Since derivation f(a,c) f(a,b) g(a) does not reduce redex a in t, a is not root needed. However, f(ffl,c) f(ffl,b) g(ffl) which means that a is hnf needed. ....

A. Middeldorp. Call by Need Computations to Root-Stable Form. In Proc. of POPL'97, pages 94-105, ACM Press, 1997.

....or another step is performed on t, provided that these operations are compatible with the necessary step. Lemma 26 shows that a step computed by INS must be performed, eventually, to reach a certain class of constructor rooted terms. These computations are more fundamental than needed computations [11]. Theorem 27 shows that any derivation that narrows a term at the root must perform a step computed by INS on t. Its proof further shows that INS lays the foundations for a sequence of steps that must be performed to compute a root stable form. Although INS does not compute a minimal (most ....

A. Middeldorp. Call by need computations to root-stable form. In Proc. 24th ACM Symposium on Principles of Programming Languages, pages 94-105, Paris, 1997.

....# b allows a trivial step, but can be normalised to a. The elimination is caused by a critical step in the former, and by an erasing step in the latter case. These are the only problems. A term allowing a trivial head step cannot be normalising in an (almost) orthogonal TRS by the results of [2, 3]. Lemma A term s allowing a trivial head step # by rule # : l # r is not normalising in a weakly orthogonal term TRS, i.e. a left linear TRS such that s = t for every critical pair (s, t) Proof We construct a prefix C, such that l# # C # t for any s # t, where l# = l [#x: # : ....

A. Middeldorp. Call by need computations to root-stable form. POPL97, pp. 94--105, 1997.

....0 . By induction this yields a reduction t LNT c, since I c (t) LNT case c of c : c X is the only way to transform I c (t) to fg. It remains to show that LNT reductions are needed to compute a constructorheaded term. For a normalization result for the rst order case, we refer to (Middeldorp, 1997). Theorem 6.13 If t reduces to c, then t has a LNT redex at a position p and t must be reduced at p eventually. Otherwise, t is not reducible to c. Proof In the computation of I c (t)#, we have goal systems of the form case pn s of : Xn ; case p1 X 2 of c : c X 1 ....

Middeldorp, A. (1997). Call by need computations to root-stable form. Pages 94-105 of: Proc. 24th ACM Symposium on Principles of Programming Languages.

....is root stable if it cannot be rewritten to a redex. The set of infinite terms is denoted by T 1 ( Sigma; X ) see [KKSV95] for a formal definition of infinite term) An infinite rewrite sequence is an infinite sequence t 1 ; t 2 ; of terms such that t n t n 1 for all n 2 IN. Following [Mid97], in this paper we are only interested in infinite rewrite sequences of length (unlike [KKSV95] We write t s if either t s or s is the (possibly infinite) limit of an infinite rewrite sequence starting from t (according to the standard notion of Cauchy convergence, see [KKSV95] for ....

A. Middeldorp. Call by Need Computations to Root-Stable Form. In Proc. of POPL'97, pages 94-105. ACM Press, 1997.

.... considered as the (infinite) value of from(0) According to this situation, some research has been done concerning infinitary rewriting, i.e. rewriting that also considers infinite reduction sequences, probably involving infinite terms, and even term rewriting systems built from infinite terms [DKP91,KKSV95,Mid97]. The redundancy of an argument of a function f in a TRS R depends on the semantic properties of R that we are interested in observing. In [AEL00] we consider different (reduction) semantics including the standard normalization semantics nf (typical of pure rewriting [Jou94] and the evaluation ....

A. Middeldorp. Call by Need Computations to Root-Stable Form. In Conference Record of the 24th Annual ACM Symposium on Principles of Programming Languages, POPL'97, pages 94-105, ACM Press, 1997.

....as strong sequentiality. Keywords: infinitary normalization, normalization, sequentiality, strategies, term rewriting. 1 Introduction Root needed reduction provides a suitable formal framework for the definition of root normalizing, normalizing, and infinitary normalizing reduction sequences [Mid97]. A root stable term (also called a head normal form) is a term which cannot be reduced to a redex. A redex in a term is root needed if the redex (itself or one of its descendants) is reduced in each rewriting sequence leading to a root stable term. Root neededness is undecidable and it must be ....

....programs, in the context of lazy languages infinitary normalizing strategies should be better considered since infinite data structures (and their finite approximations) can be returned as the result of certain computations in lazy systems. Example 1. Consider the CS R to generate prime numbers [Mid97] (where the rules for defining the boolean operator x y for checking whether y divides x were not included) primes sieve(from(2) from(x) x:from(x 1) sieve(x:y) x:sieve(filter(x,y) filter(x,y:z) if(x y,filter(x,z) y:filter(x,z) if(true,x,y) x if(false,x,y) y Assuming that ....

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A. Middeldorp. Call by Need Computations to Root-Stable Form. In Conference Record of the 24th Annual ACM Symposium on Principles of Programming Languages, POPL'97, pages 94-105, ACM Press, 1997.

....needed reductions: Definition 6.12 A term t has a needed redex p if I(t)# is of Invariant 2 with p = p 1 Delta Delta Delta p n . It remains to show that needed reductions are indeed needed to compute a constructor headed term. For a normalization result for the first order case, we refer to [21]. Theorem 6.13 If t reduces to true, then t has a needed redex at a position p and t must be reduced at p eventually. Otherwise, t is not reducible to true. Proof The first claim, that t has a needed redex at p, follows from Lemma 6.10. We show that t must be reduced at p (or is not reducible ....

Aart Middeldorp. Call by need computations to root-stable form. In Proc. 24th ACM Symposium on Principles of Programming Languages, pages 94--105, January 1997.

.... than other neededness notions, since root needed normalization provides a unified framework to analyze reductions leading to a finite normal form (i.e. it includes Huet and L evy s theory) and also reductions leading to an infinite normal form for which Huet and L evy s theory does not apply [12]. He also generalizes Sekar and Ramakrishnan s work on necessary sets of redexes [14] from constructor based TRSs to root necessary sets of redexes in general TRSs. Thus the notion of root necessary reduction, which applies in parallel strategies, is also available. In this work, we prove that, ....

....V ) T = t t2T t , is the minimum replacement map which makes T compatible. Given a TRS R = Sigma; R) the canonical replacement map com R of R, is com R = L(R) Note that com R makes compatible each lhs of R. Example 3. 1 Let us consider the TRS R which is mainly borrowed from [12]. primes sieve(from(s(s(0) if(true; x; y) x from(x) x : from(s(x) if(false; x; y) y sel(0; x : y) x 0 x x sel(s(x) y : z) sel(x; z) s(x) y s(x y) sieve(x : y) x : sieve(filter(x; y) filter(x; y : z) if(xjy; filter(x;z) y : filter(x; z) Assume that j ....

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A. Middeldorp. Call by Need Computations to Root-Stable Form. In Conference Record of the 24th ACM Symposium on Principles of Programming Languages, pages 94-105, 1997.

....decides whether a left linear TRS belongs to CBN NF g . Section 4 also contains an example illustrating the various constructions. The complexity of the construction is analyzed in the next section. In Section 6 we consider call by need computations to root stable form. As argued in Middeldorp [15], rootstable forms and root neededness are the proper generalizations of normal forms and neededness when it comes to infinitary normalization. In this case we again obtain a double exponential upperbound, which is a significant improvement over the non elementary upperbound of the complexity of ....

A. Middeldorp. Call by need computations to root-stable form. In Proc. 24th POPL, pages 94--105, 1997.

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A. Middeldorp. Call by Need Computations to Root-Stable Form. In Proceedings of the 24th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, pages 94--105. ACM, New York, 1997.

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A. Middeldorp. Call by need computations to root-stable form. In Proceedings of the 24th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, pages 94--105. ACM Press, New York, 1997.

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A. Middeldorp. Call by need computations to root-stable form. In Proceedings of the 24th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL'97, pages 94--105. 1997.

No context found.

A. Middeldorp. Call by Need Computations to Root-Stable Form. In Conference Record of the 24th Annual ACM Symposium on Principles of Programming Languages, POPL'97, pages 94-105, ACM Press, 1997.

No context found.

A. Middeldorp. Call by Need Computations to Root-Stable Form. In Proceedings of the 24th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, pages 94--105. ACM, New York, 1997.

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A. Middeldorp. Call by need computations to root-stable form. In Proc. 24th ACM Symposium on Principles of Programming Languages (Paris), pages 94--105, 1997.

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Middeldorp, A., Call by need computations to root-stable form, In Conference Record of the 24th Annual ACM symposium on Principles of Programming Languages, POPL'97, 94-105, 2001.

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A. Middeldorp, Call by need computations to root-stable form, In Conference Record of the 24th Annual ACM symposium on Principles of Programming Languages, POPL'97, 94-105, 2001.

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A. Middeldorp. Call by Need Computations to Root-Stable Form. In Proceedings of the 24th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, pages 94--105. ACM, New York, 1997.

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