R. Kennaway. A Conflict Between Call-By-Need Computation and Parallelism. In N. Dershowitz and N. Lindenstrauss, editors, Proc. of 4th Workshop on Conditional Term Rewriting Systems, CTRS'94, LNCS 968:247-261, Springer-Verlag, Berlin, 1994. 66  Home/Search   Document Details and Download   Summary   Related Articles   Check

....1 [KM91] Let R = F ; R) be a TRS. Let t; s 2 (F ; X ) and let p 2 Pos(t) Then, 1. t) t 2. t) t[ tj p ) p ) 3. t s ) t) s) 4. t s ) t) s) A term t 2 (F ; X ) is strongly root stable (or a strong head normal form) if (t) 6= Omega Gamma Proposition 2 [Ken94] Let R be a TRS. If t is strongly root stable, then t is root stable. Proof. If t is not root stable, then t oe(l) for some l 2 L(R) By Proposition 1, t) oe(l) Omega Gamma Hence (t) Omega Gamma a contradiction. 2 In general, the converse statement is not true [Ken94] Example ....

....2 [Ken94] Let R be a TRS. If t is strongly root stable, then t is root stable. Proof. If t is not root stable, then t oe(l) for some l 2 L(R) By Proposition 1, t) oe(l) Omega Gamma Hence (t) Omega Gamma a contradiction. 2 In general, the converse statement is not true [Ken94] Example 8 Consider the TRS of Example 7. The term f(b) is root stable, but, as shown in Example 7, it is not strongly root stable. A term t is rigid if (t) t and soft if (t) KM91, Klo92] We also say that the context C[ is rigid if ] is a rigid term. Proposition 3 [KM91, Klo92] ....

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R. Kennaway. A Conflict Between Call-By-Need Computation and Parallelism. In N. Dershowitz and N. Lindenstrauss, editors, Proc. of 4th Workshop on Conditional Term Rewriting Systems, CTRS'94, LNCS 968:247-261, Springer-Verlag, Berlin, 1994. 66

....rewriting systems too. Analogous questions arise fox other strategies. Fox example, how can one construct reductions that avoid head normal forms Besides strong sequentiality fox nor real forms studied in Huet Ldvy [HuLd91] strong sequentiality is studied w.r.t. head normal forms in Kennaway [Ken94]. Investigation of relative strong sequentiality and related to it strictness analysis (see e.g. N6k94] seems also as an interesting topic fox future research. In forthcoming papers, we extend the theory of relative normalization in two directions: we study minimal and optimal relative ....

Kennaway J. R. A conflict between call-by-need computation and parallelism. Workshop on conditional (and typed) term rewriting systems, Jerusalem, 1994.

....and f(A) has root needed redexes are equivalent. In a forthcoming joint paper with Ir ene Durand we present a unifying approach to decidable approximations of root stability, root neededness, neededness, and call by need, which simplifies and generalizes many works in this direction (e.g. [4, 6, 7, 8, 14]) Acknowledgements I thank Ir ene Durand, Vincent van Oostrom, and FerJan de Vries for helpful discussions, Nachum Dershowitz for suggesting the terminology context free (in Definition 6.3) and Femke van Raamsdonk for carefully reading the paper. The work reported in this paper is partially ....

J.R. Kennaway, A Conflict Between Call-by-Need Computation and Parallelism, Proceedings of the 4th Workshop on Conditional Term Rewriting Systems, Jerusalem, Lecture Notes in Computer Science 968, pp. 247--261, 1995.

....procedure. Since csr is complete w.r.t. root normalization of terms (Theorem 3.2) but not w.r.t. normalization of terms (Example 3.5) needed rewriting is not completely adequate for characterizing the cs computations. The notion of root needed computation has also been analyzed in the literature [5, 12]. We follow the formalization of Middeldorp [12] to develop our results. A redex in a term t is root needed, if it is contracted, itself or a descendant, in every rewrite sequence from t to a root stable form. Since root stable terms are an intermediate step to reach a normal form, every ....

....b at the occurrence 2 of g(a; b) is root needed. However, 2 is not a replacing occurrence. Thus, even in strongly replacing independent TRS, we cannot obtain I r R (t) O R (t) Theorem 4.6 shows that it is possible to perform root needed reduction in parallel. This makes Kennaway s claim in [5] unaccurate: in an orthogonal TRS, no term can have more than one strongly root needed redex. Kennaway argues that this justifies the existence of a conflict between call by need and parallelism. Our notion of strongly replacing independent TRSs reveals that this conflict depends on the election ....

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R. Kennaway. A Conflict Between Call-By-Need Computation and Parallelism. In Proc. of Workshop on Conditional and Typed Rewriting Systems, CTRS'94, LNCS 968:247-261, Springer-Verlag, Berlin, 1994.

....rewriting too. Analogous questions arise for other strategies. For example, how can one construct reductions that avoid head normal forms Besides strong sequentiality for normal forms studied by Huet L evy in [HuL e91] strong sequentiality is studied w.r.t. head normal forms in Kennaway [Ken94]. Investigation of relative strong sequentiality and related to it strictness analysis (see e.g. Nok94] seems also as an interesting topic for future research. Study of minimal and optimal relative normalization makes also sense for infinite reductions. The starting point here is the work ....

Kennaway J. R. A conflict between call-by-need computation and parallelism. Workshop on conditional (and typed) term rewriting systems, CTRS'94, Springer LNCS, vol. 968, N. Dershowitz, ed. Jerusalem, 1994, p. 247-261.

.... Delta do not normalize f(b) Theorem 4. 9 shows that normal forms of a left linear TRS R are root stable (whenever can R v ) As it is possible to normalize a term t by successively root normalizing maximal non root stable subterms of (reducts of) t, we can thought of root normalization [Ken94, Mid97] as a basis for defining normalizing computations, as every derivation leading to a normal form obtains a root stable term in some step of the derivation. In fact, whenever can R v , and according to Corollary 4.10, the normalization of a term t can be thought of as a preliminary ....

....for R is normalizing. Corollary 6.8 formalizes the use of root normalizing strategies for defining normalizing strategies. In the following sections, we investigate how to (effectively) define them. 6. 1 Root neededness and context sensitive rewriting The notion of root needed computation [Ken94, Mid97] provides a suitable formal framework for the definition of root normalizing, normalizing, and infinitary normalizing reduction sequences [Mid97] A redex in a term t is root needed if it is contracted (either itself or one of its descendants) in every rewrite sequence from t to a ....

R. Kennaway. A Conflict Between Call-By-Need Computation and Parallelism. In N. Dershowitz and N. Lindenstrauss, editors, Proc. of 4th Workshop on Conditional Term Rewriting Systems, CTRS'94, LNCS 968:247261, Springer-Verlag, Berlin, 1994.

....F . We say that redex Delta in C[ Delta] 2 T (F) is (S 1 ; S 2 ) root needed if there is no term t 2 RS (S2 ) ffi such that C[ Delta ffi ] S1 t. We abbreviate (S; S) root needed to S root needed. Our R s root needed redexes coincide with the strongly root needed redexes of Kennaway [11]. Lemma 35. Let R 1 and R 2 be TRSs over the same signature with approximations S 1 and S 2 . Every (S 1 ; R 2 ) root needed redex is (R 1 ; S 2 ) root needed. ut It should be noted that, for an approximation S of R, S root needed redexes need not be R root needed. Consider for instance the ....

....because the only term t such that f(a ffi ) R s t is f(a ffi ) itself and f(a ffi ) 2 RS (R s ) ffi because we have f(a ffi ) f(c) in the TRS (R s ) ffi = fa x; a ffi x; f(c) x; f ffi (c) xg. So not every strongly root needed redex is root needed, contradicting Theorem 16 in [11]. Definition 36. Let R and S be TRSs over the same signature F . The set of all terms C[ Delta ffi ] 2 M ffi R such that there is no term t 2 RSS ffi with C[ Delta ffi ] R t is denoted by (R; S) ROOT NEEDED. Here M ffi R = fC[ Delta ffi ] j C[ Delta] 2 T (F) and Delta is an R redexg ....

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J.R. Kennaway, A Conflict Between Call-by-Need Computation and Parallelism, Proc. 4th CTRS, LNCS 968 (1995) 247--261.

....of decidability results to these cases are only interesting for subclasses of rewrite systems (as e.g. described in [24] 6 Further applications We believe that the tree automata approach can be used successfully to other works on reduction strategies. For example the strong root stability of [12] is also expressible in WSkS for left linear rewrite systems. The use of automata should also be investigated in parallel reduction strategies, such as in [21] a run of the automaton not only gives an index position, but all index positions. Our approach could also be used for related notions of ....

Richard Kennaway, A conflict between call-by-need computation and parallelism, Workshop on Conditional Term Rewriting Systems (Jerusalem) (N. Dershowitz, ed.), 1994.

....of decidability results to these cases are only interesting for subclasses of rewrite systems (as e.g. described in [24] 6 Further applications We believe that the tree automata approach can be used successfully to other works on reduction strategies. For example the strong root stability of [12] is also expressible in WSkS for left linear rewrite systems. The use of automata should also be investigated in parallel reduction strategies, such as in [21] a run of the automaton not only gives an index position, but all index positions. Acknowledgments I thank Takahashi Nagoya, Masahito ....

R. Kennaway. A conflict between call-by-need computation and parallelism. In N. Dershowitz, editor, Workshop on Conditional Term Rewriting Systems, Jerusalem, 1994.

....branch which is explored. This requires some additional implementation machinery, which is out of the scope of this paper. 6 Further applications We believe that the tree automata approach can be used successfully to other works on reduction strategies. For example the strong root stability of [11] is also expressible in WSkS for left linear rewrite systems. The use of automata should also be investigated in parallel reduction strategies, such as in [19] a run of the automaton not only gives an index position, but all index positions. ....

R. Kennaway. A conflict between call-by-need computation and parallelism. In N. Dershowitz, editor, Workshop on Conditional Term Rewriting Systems, Jerusalem, 1994.

....parallel using the future annotation. In lenient languages, such as Id [25] and pH [26] by default all subexpressions can evaluate speculatively. With parallel implementations of lazy graph reduction [30, 18] speculative evaluation is used to overcome the inherent lack of parallelism of laziness [19, 36]. Although call by speculation is a powerful mechanism to achieve high degrees of parallelism, with current implementations it can be hard to understand the performance characteristics of a program without a reasonably deep understanding of the implementation. An important cause of this problem is ....

Richard Kennaway. A conflict between call-by-need computation and parallelism (extended abstract). In Proceedings Conditional Term Rewriting Systems-94, February 1994.

....rewriting systems too. Analogous questions arise for other strategies. For example, how can one construct reductions that avoid head normal forms Besides strong sequentiality for normal forms studied in Huet L evy [HuL e91] strong sequentiality is studied w.r.t. head normal forms in Kennaway [Ken94]. Investigation of relative strong sequentiality and related to it strictness analysis (see e.g. Nok94] seems also as an interesting topic for future research. In forthcoming papers, we extend the theory of relative normalization in two directions: we study minimal and optimal relative ....

Kennaway J. R. A conflict between call-by-need computation and parallelism. Workshop on conditional (and typed) term rewriting systems, Jerusalem, 1994.

....evaluation can have wide range of implementations, such as call by name, call by need (lazy) and call by speculation (lenient) 1 , and these implementations would have very different complexity models. The first two, call by name and call by need, actually offer no significant parallelism [23]. Call by speculation offers plenty of parallelism but does the same amount of work as applicativeorder semantics. In particular, a model that uses call byspeculation would give the same asymptotic work bounds as our model, although it might be possible to improve some depth bounds. Most ....

Richard Kennaway. A conflict between call-by-need computation and parallelism (extended abstract). In Proceedings Conditional Term Rewriting Systems-94, February 1994.

....in every redex prefix that has more than one. This eliminates a certain amount of non determinism from the system. Only a very restrictive class of systems have translations to lambda calculus which do not eliminate nondeterminism: in e#ect, the systems which do not have any to begin with. See [10] for a discussion of such systems. Assume we are given an orthogonal TRS. Choose, if possible, an operator symbol F , an integer i, and a constructor symbol C, such that 1. i is an extension site of F (w) 2. F (x, C(y) z) where the subterm C(y) is the ith argument) is a redexcomponent but ....

J.R. Kennaway. A conflict between call-by-need computation and parallelism. In N. Dershowitz and N. Lindenstrauss, editors, Proc. Workshop on Conditional and Typed Term Rewriting Systems (CTRS '94), pages 247--261, Jerusalem, 1994. Springer. LNCS 968.

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