Toyama, Y., Commutativity of term rewriting systems. In Fuchi, K. and Kott, L. (eds.), Programming of future generation computers II, pp. 393--407. North-Holland, Amsterdam, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....combinatory reduction systems (CRSs) Khasidashvili s expression reduction systems (ERSs) and Nipkow s higher order pattern rewriting systems (PRSs) 1 Introduction This paper is concerned with a method to prove confluence of rewriting systems. It s an extension of some confluence results in [CR36, Hue80, Toy88, Klo80, Kha92, Raa93, Tak, MN94, Oos94, ORb] and we refer the reader to these papers and to the handbook chapters [DJ, Klo] for motivation and for standard definitions as well. Here we will mainly be concerned with proving our result: Left linear development closed PRSs are confluent. Let s explain the terminology used. A rewrite system ....

....by assumption, contributing to m, but not to m 0 . fi As remarked above, the diamond property for ffi Gamma implies its confluence, which in turn implies confluence of , so we have our main result: Corollary13. Development closed PRSs are confluent. 5. 2 Toyama The main result of Toyama s [Toy88] is an improvement of Huet s result by weakening the condition on a critical pair (s;t) in case of overlay (i.e. root overlap) to the existence of a term r such that s k Gamma r j t. Analogous to the above extension of Huet s result, Toyama s result can be extended to only requiring s ffi ....

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Yoshihito Toyama. Commutativity of term rewriting systems. In F. Fuchi and L. Kott, editors, Programming of Future Generation Computer, volume II, pages 393--407. North-Holland, 1988. This article was processed using the L a T E X macro package with LLNCS style

....the Finite Developments theorem (FD) In Section 4 we show that FD can be used to show confluence for non orthogonal untyped lambda calculi with eta contraction, Omega rule and eta expansion respectively. In Section 5 we extend Huet s parallel closed result ( Hue80] and Toyama s extension of it ([Toy88]) using a method which generalises to higher order term rewriting systems. Finally, we show that the Knuth Gross strategy is normalising for weakly orthogonal term rewriting systems. 1 2. Rewriting In this section we explain the analogy rewriting = substitution rules, which forms the basis ....

....as well. cf. MN94] for an extension to first order like HRSs) Let us remark that the result does extend the confluence by weak orthogonality result of Section 4, but only in the case of patterm rewriting systems, since the proof here uses the term structure in an essential way. 8 Toyama In [Toy88], Toyama improved upon Huet s result somewhat by weakening the condition on a critical pair (s;t) in case of overlay (i.e. root overlap) to the existence of a term r such that s k Gamma r j t. Analogous to the above extension of Huet s result, Toyama s result can be extended to only requiring s ....

Yoshihito Toyama. Commutativity of term rewriting systems. In F. Fuchi and L. Kott, editors, Programming of Future Generation Computer, volume II, pages 393--407. North-Holland, 1988. 11

....Q for every critical pair P; Q . Any left linear and parallel closed TRS is CR ( 3] Note. For any critical pair P; Q (called an outside critical pair) P j Q can be relaxed to P j L Q for some term L and the relaxed condition ensures that left linear TRS R is CR. Y. Toyama [10]) De nition 5. A TRS R is upside parallel closed if for every critical pair P; Q u of R, P Q or Q W j P for some W where jwj juj for all w 2 W For any critical pair P; Q , the above condition can be relaxed to P ( j [ L( W j [ Q for some term L and W ....

Y.Toyama, On commutativity of term rewriting systems, Trans. IECE Japan, J66-D, pp. 1370-1375 (1983).

....These counterexamples must necessarily be non left linear. 1 In fact, TRSs which are left linear and non overlapping, i.e. orthogonal , are confluent (cf. e.g. Ros73] This fundamentally important positive result has been considerably generalized by Huet ( Hue80] and further by Toyama ([Toy88]) by allowing critical pairs, but imposing certain strong joinability properties on them (cf. Theorems 3.2, 3.5 and 3.7 below) The first result, Theorem 3.2, however, has the severe drawback that it additionally requires right linearity, a rather unnatural condition as pointed out in [Hue80] ....

....(oel) p oer 0 ] p oel oer is the corresponding critical peak . If p = we speak of an outside critical peak (or critical overlay)and outside critical pair , respectively. Otherwise, i.e. if p , we speak of an inside critical peak and inside critical pair , respectively (following [Toy88]) If the two rules are renamed versions of the same rule we do not consider the case p = which gives only rise to improper divergences) The set of all critical pairs between rules of R is denoted by CP(R) If CP(R) R is said to be non overlapping. It is called orthogonal if it is ....

[Article contains additional citation context not shown here]

Y. Toyama. Commutativity of term rewriting systems. In K. Fuchi and L. Kott, editors, Programming of Future Generation Computer, volume II, pages 393--407. North-Holland, 1988.

....deals with net rewriting, which is introduced here. We recapitulate the Finite Developments theorem (FD) and the confluence by developments result for orthogonal rewriting systems known from literature. In Section 3 we extend Huet s parallel closed result ( Hue80] and Toyama s extension of it ([Toy88]) using a method which applies to higher order term rewriting systems as well. As an easy consequence, we show that the Knuth Gross strategy (full substitution) is normalising for weakly orthogonal TRSs. 2. Confluence by Developments We recapitulate the main ingredients of the confluence by ....

....our earlier confluence results for weakly orthogonal (only having trivial critical pairs) pattern HORSs ( ORb, Oos94a] in fact the rewriting system of the example above is weakly orthogonal) but the proof here uses the term structure in an essential way. Toyama The main result of Toyama s [Toy88] is an improvement of Huet s result by weakening the condition on a critical pair (s;t) in case of overlay (i.e. root overlap) to the existence of a term r such that s k Gamma r j t. Analogous to the above extension of Huet s result, Toyama s result can be extended to only requiring s ffi ....

Yoshihito Toyama. Commutativity of term rewriting systems. In F. Fuchi and L. Kott, editors, Programming of Future Generation Computer, volume II, pages 393--407. North-Holland, 1988.

.... pair t s (where t = u[roe] u[loe] g d = s, for some rules l r and g d) we have t k s (t reduces in one parallel step to s) The condition t k s can be relaxed to t k r k s for some r when the critical pair is generated from two rules overlapping at the roots; see [89]. What if s k t for every critical pair t s What if for every t s we have s = t (Here = is the reflexive closure of . What if for every critical pair t s, either s = t or t = s In the last case, especially, a confluence proof would be interesting; one would then ....

Y. Toyama. Commutativity of term rewriting systems. In K. Fuchi and L. Kott, eds., Programming of Future Generation Computers II, pp. 393--407, North-Holland, 1988.

....note that is parallel closed, because for each critical pair P; Q resulting from an overlapping at root position Q; P is also a critical pair. 2 Also, Toyama proved a mild version which states that overlay LL TRSs are confluent if for every critical pair P; Q , P jj Q. See [Toy88]. Reviewing the structure of the proof of lemma 4.1, one can observe that there is another more simple question to answer: Are LL TRSs for whose critical pairs P; Q , Q 1 P holds, confluent Obviously a positive answer to the main open problem implies a positive answer to the previous ....

Y. Toyama. Commutativity of Term Rewriting Systems. In K. Fuchi and L. Kott, editors, Programming of Future Generation Computers II, pages 393--407. North-Holland, 1988.

....e.g. 3] These counterexamples must necessarily be non leftlinear. 2 In fact, TRSs which are left linear and non overlapping, i.e. orthogonal , are confluent (cf. e.g. 12] This fundamentally important positive result has been considerably generalized by Huet ( 3] and further by Toyama ([13]) by allowing critical pairs, but imposing certain strong joinability properties on them (cf. Theorems 4, 6 and 8 below) Theorems 6 and 8 of Huet Toyama are particularly interesting, since they do not require right linearity. They are proved by showing strong confluence of parallel reduction, ....

....(oel) p oer 0 ] p oel oer is the corresponding critical peak . If p = we speak of an outside critical peak (or critical overlay)and outside critical pair , respectively. Otherwise, i.e. if p , we speak of an inside critical peak and inside critical pair , respectively (following [13]) If the two rules are renamed versions of the same rule we do not consider the case p = which gives only rise to improper divergences) A critical pair hs; ti is said to be joinable if s # t. The set of all critical pairs between rules of R is denoted by CP(R) If CP(R) R is said to be ....

[Article contains additional citation context not shown here]

Y. Toyama. Commutativity of term rewriting systems. In K. Fuchi and L. Kott, eds., Programming of Future Generation Computer, vol. II, pp. 393--407. NorthHolland, 1988.

.... termination, both UN and CR are undecidable in general [13,9] Without termination limitation, there are many sufficient conditions for CR, most of which assume that a TRS is non overlapping (i.e. it has no critical Preprint submitted to Elsevier Science 19 January pairs) or its extensions [29,12,30,27,28,24]. However, the non overlap assumption alone is not sufficient for concluding CR. In [12] the following two counter examples are presented: R 1 = 8 : d(x; x) 0 d(x; f (x) 1 2 f(2) 9 = R 2 = 8 : d(x; x) 0 f(x) ....

Y. Toyama. On commutativity of term rewriting systems. In Programming of Future Generation Computers II. Proceedings of the Second France-Japanese Symposium, pages 393--407. North-Holland, 1988.

....function. Persistency of confluence will be shown here. This can be obtained by a straightforward modification of the proof of modularity of confluence, whose details are given in this report. 1 Introduction Confluence is a property that is widely studied in the theory of term rewriting [5] 2] [9]. One of the approaches to guarantee confluence of a given TRS is to infer it from those of its subsystems. Y. Toyama showed that confluence of the disjoint sum of TRSs can be inferred from those of its components [8] Following standard terminology, we say a property P is modular if P is ....

Toyama, Y., Commutativity of term rewriting systems. In Fuchi, K. and Kott, L. (eds.), Programming of future generation computers II, pp. 393--407. North-Holland, Amsterdam, 1988.

....sorts. It will be shown here that the persistency of confluence is preserved for this extension. Moreover, a new result on the modularity of confluence will be presented making use of this. 1 Introduction Confluence is a property that is widely studied in the theory of term rewriting [4] 7] [10]. One of the approaches to guarantee confluence of a given TRS is to infer it from those of its subsystems. It is shown that confluence of the disjoint union of TRSs can be inferred from those of its components [9] Following standard terminology, we say a property P is modular if P is inferred ....

Y. Toyama. Commutativity of term rewriting systems. In K. Fuchi and L. Kott, editors, Programming of future generation computers II, pages 393--407. North-Holland, Amsterdam, 1988.

No context found.

Toyama, Y. (1988). Commutativity of Term Rewriting Systems. In: "Programming of Future Generation Computer II" (eds. K. Fuchi and L. Kott), pp. 393--407, North-Holland.

No context found.

Yoshihito Toyama (1988). Commutativity of Term Rewriting Systems. In: K. Fuchi, L. Kott (eds.). Programming of Future Generation Computers II. Elsevier Science Publishers.

No context found.

Yoshihito Toyama (1988). Commutativity of Term Rewriting Systems. In: K. Fuchi, L. Kott (eds.). Programming of Future Generation Computers II. Elsevier Science Publishers.

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