Satish R. Thatte. On the correspondence between two classes of reduction systems. Information Processing Letters, 20(1985):83--85, February 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... Equivalent Term Rewriting Systems are Strongly Sequential: a direct proof Ir ene DURAND Bruno SALINIER Laboratoire Bordelais de Recherche en Informatique Universit e de Bordeaux I 351, Cours de la Lib eration 33405 Talence, France email: idurand labri.u bordeaux.fr Abstract In [8], Thatte demonstrated the possibility of simulating an orthogonal TRS with a left linear constructor system obtained from the original system via a simple transformation. The class of strongly sequential systems (SS) was defined in [3] In [2] we have defined the class of constructor ....

.... systems admit an efficient one step reduction strategy; strong sequentiality is decidable [3] but is conjectured to be NP complete [6] This work has been sup ported by the ESPRIT Basic Research Working Group COMPUGRAPH II ( Computing by Graph Transformation ) and the PRC MathInfo In [8], Thatte demonstrated the possibility of simulating an orthogonal TRS with a left linear constructor system obtained from the original system via a simple transformation. In [2] we have defined the class of constructor equivalent systems (CE) for which Thatte s transformation preserves strong ....

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Satish Thatte. On the correspondence between two classes of reduction systems. Information Processing Letters 20, pages 83--85, 1985. 13

....are applicable to functional programming languages as well. ffl For languages based on orthogonal systems such as O Donnell s equational language [OD85] programs that violate the constructor discipline can be transformed into constructor based systems using Thatte s transformation procedure [That85] 3 . We would like to remark that the above transformation procedures are often not necessary since most programs in practice (e.g. almost all the programs in [OD85] appear to be already path sequential and constructor based. 3 A caveat here is that Thatte s transformation does not always ....

Satish Thatte, On the correspondence between two classes of Reduction systems, Information Processing Letters 20 (2), pp. 83-85.

....will be dealing with orthogonal programs that follow the constructor discipline 2 . Note that this entails no loss of generality since equational programs almost exclusively follow this discipline [OD85] and those that do not can be syntactically transformed so as to follow it (details appear in [That85]) Finally, we would like to clarify how our work in this paper differs from our previous work on parallel evaluation of path sequential systems presented in [SPR89, SR92] As pointed out by one of the reviewers of this paper, it appears on the surface that these two works are related since they ....

Satish Thatte, On the correspondence between two classes of Reduction systems, Information Processing Letters 20 (2), pp. 83-85, 1985.

....the resulting normal forms, first we add most general rules, to make the TRS simply complete. Note that in a simply complete TRS, function symbols that occur at the head of a left hand side do not occur in normal forms. The transformation is an adaptation of a transformation introduced by Thatte [44]. Assume a left linear TRS R over Sigma = V ; F ; ar) The following program adds most general rules, to obtain a simply complete left linear TRS. Procedure Add Most General Rules applied to (R; F ; ar) If R is simply complete, then output (R; F ; ar) Else, select a function symbol f 2 F ....

S.R. Thatte. On the correspondence between two classes of reduction systems. Information Processing Letters, 20(2):83--85, 1985.

.... Equivalent Term Rewriting Systems are Strongly Sequential: a direct proof Ir ene DURAND Bruno SALINIER Laboratoire Bordelais de Recherche en Informatique Universit e de Bordeaux I 351, Cours de la Lib eration 33405 Talence, France email: idurand labri.u bordeaux.fr Abstract In [8], Thatte demonstrated the possibility of simulating an orthogonal TRS with a left linear constructor system obtained from the original system via a simple transformation. The class of strongly sequential systems (SS) was defined in [3] In [2] we have defined the class of constructor equivalent ....

.... systems admit an efficient one step reduction strategy; strong sequentiality is decidable [3] but is conjectured to be NP complete [6] This work has been sup ported by the ESPRIT Basic Research Working Group COMPUGRAPH II ( Computing by Graph Transformation ) and the PRC MathInfo In [8], Thatte demonstrated the possibility of simulating an orthogonal TRS with a left linear constructor system obtained from the original system via a simple transformation. In [2] we have defined the class of constructor equivalent systems (CE) for which Thatte s transformation preserves strong ....

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Satish Thatte. On the correspondence between two classes of reduction systems. Information Processing Letters 20, pages 83--85, 1985.

....of F 2 F F replaced by C F and t 00 [T ] the term with all occurrences so replaced. Sigma # is the smallest system which satis es the following two assertions: 1) If L R 2 Sigma then t 0 [L] R 2 Sigma # . 2) If M is a functional subterm, t 0 [M ] t 00 [M ] 2 Sigma # . In [11], Thatte shows how to simulate any orthogonal TRS Sigma with the corresponding system Sigma # . 4.2 De nition of Constructor Equivalent Systems In [2] it was showed that Thatte s transformation does not always preserve neither orthogonality nor strong sequentiality 1 . The class of strong ....

Satish Thatte. On the correspondence between two classes of reduction systems. Information Processing Letters, 20:8385, 1985. This article was processed using the L A T E X macro package with LLNCS style

....in FB, outermost evaluation can be preserved while still doing innermost stabilization (computing strong head normal forms) leading to an efficient strategy for sequences of reductions. Furthermore, Durand [1] has proved that the forward branching property can be decided in quadratic time. Thatte [8] demonstrated the possibility of simulating an orthogonal TRS with a left linear constructor system obtained from the original system via a simple syntactic transformation. Unfortunately, it does not preserve strong sequentiality. The constructor equivalent systems, for which strong sequentiality ....

Satish Thatte. On the correspondence between two classes of reduction systems. Inform. Processing Letters, 20:83--85, 1985. This article was processed using the L a T E X macro package with LLNCS style

....evaluation can be preserved while still doing innermost stabilization (computing strong head normal forms) leading to an eOEcient strategy for sequences of reductions (Strandh, 1988) Furthermore, Durand (1994) has proved that the forward branching property can be decided in quadratic time. Thatte (1985) demonstrated the possibility of simulating an orthogonal TRS with a left linear constructor system obtained from the original system via a simple syntactic transformation. Unfortunately, this transformation does not always preserve orthogonality nor strong sequentiality. The class of constructor ....

Thatte, S. (1985). On the correspondence between two classes of reduction systems. Information Processing Letters, 20:8385.

....normal forms, first we add so called most general rules to the left linear TRS. In the resulting left linear TRS, which is called simply complete, function symbols that occur at the head of a left hand side do not occur in normal forms. This transformation is an adaptation of one introduced by Thatte [1985]. For the sake of compilation, it is imperative to fix a deterministic rewriting strategy. We apply innermost rewriting, which reduces a redex that is as close as possible to the leaves of the parse tree of a term; if there are several such redexes, then we reduce the rightmost one. The innermost ....

Thatte, S. R. 1985. On the correspondence between two classes of reduction systems. Inf. Process. Lett. 20, 2, 83--85.

.... [Str88] Furthermore, Durand has proved that the forward branching property can be decided in quadratic time [Dur94] Thatte demonstrated the possibility of simulating an orthogonal TRS with a left linear constructor system obtained from the original system via a simple syntactic transformation [Tha85]. Unfortunately, this transformation does not always preserve orthogonality nor strong sequentiality. The class of constructor equivalent systems (CE ) for which strong sequentiality is preserved by Thatte s transformation, is a subclass of FB [DS93, DS94] In this paper, we present a new ....

Satish Thatte. On the correspondence between two classes of reduction systems. Information Processing Letters, 20:8385, 1985.

....is sound, complete, and termination preserving [14] Hence, no information on normal forms for the lazy rewrite relation ;L with respect to the original TRS is lost. The introduction of the function symbol f d in the two transformation principles is reminiscent of a transformation by Thatte [51] to make TRSs constructor based; see [52, 11, 54, 29] for subsequent improvements on the correctness of this transformation. Salinier and Strandh [43] use a similar transformation technique to simulate a forward branching TRS by a constructor based strongly sequential TRS; they also prove ....

S.R. Thatte. On the correspondence between two classes of reduction systems. Information Processing Letters 20(2):83--85, 1985.

....that they are correct, most notably for the notion of simulation from Kamperman and Walters. 1 Introduction Quite a number of papers deal with particular examples of transformations of rewrite systems, usually with the aim to obtain a rewrite system which satisfies some desirable property, e.g. [17, 12, 2, 18, 16, 20, 19, 23, 9, 10, 8]. In most of these papers, correctness of the transformation is stated, meaning that the original and the transformed rewrite system are in some sense equivalent . This claim is based on the observation either that desirable properties such as confluence and termination are preserved by the ....

....rewrite system with priorities can be transformed into a rewrite system where the priorities are captured in the syntax; this method has been applied in the implementation of the CAML system. Laville s transformation leaves the structure of reduction trees in tact, so clearly it is correct. Thatte [17] showed how a left linear non overlapping rewrite system R can be transformed into a left linear non overlapping rewrite system S that is constructor based, where R ranges over a signature Sigma and S ranges over an extended signature Sigma. Thatte showed that the original and the transformed ....

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S.R. Thatte. On the correspondence between two classes of reduction systems. Information Processing Letters, 20(2):83--85, 1985.

....of automata, such that if automaton A simulates automaton B, then A can be seen as an implementation of B. ffl In papers dealing with transformations for Reduction Systems and their correctness, it is usually proved that the normal form relations of the original and its transform are related [Tha85, Tha88, Ver95a, Ver95b]. This is because the normal form relation of a Reduction System is its computational content; it relates input and result. Kamperman and Walters initiated a new approach towards correctness proofs of transformations for Reduction Systems [KW96, Kam96] In order to make correctness of their ....

....we call simulation. We study its preservation properties and prove that the existence of a termination preserving simulation implies correctness as defined by Fokkink and Van de Pol. We apply our results to a transformation proposed by Thatte. This transformation was introduced in 1985 in [Tha85] for regular Term Rewriting Systems. In [Tha88] Thatte claimed that it preserved confluence and (up to some translation on terms) the normal form relation for nonoverlapping systems, but it was shown by Verma in [Ver95b] that this claim was false. Verma, for his part, tried to fix Thatte s proof ....

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Satish R. Thatte. On the correspondence between two classes of reduction systems. Information Processing Letters, 20(1985):83--85, February 1985.

....of compilation steps for functional programming languages which stay inside the domain of rewrite systems. Quite a number of papers deal with particular examples of transformations of rewrite systems, usually with the aim to obtain a rewrite system which satisfies some desirable property, e.g. [17, 10, 18, 16, 19, 22, 7]. In most of these papers, correctness of the transformation is stated, meaning that the original and the transformed rewrite system are in some sense equivalent . This claim is based on the observation either that desirable properties such as confluence and termination are preserved by the ....

....In [4, 14] a transformation of an equational specification of abstract data types is called a correct implementation if the initial algebras of the original and the transformed specification are isomorphic. This notion is considerably stronger than our notion of a correct transformation. Thatte [17, 18] defined a transformation of certain types of rewrite systems, over a signature Sigma , into rewrite systems that are constructor based, over an extended signature Sigma . The relation between the original and the transformed rewrite system, with the latter restricted to the part that is ....

S.R. Thatte. On the correspondence between two classes of reduction systems. Information Processing Letters, 20(2):83--85, 1985.

....for Scientific Research (NWO) under contract SION 612 33 008. y Research supported in part by NSF grant CCR 9303011. 1 of Fokkink and Van de Pol [FP96] to show that simulation preserves the normal form relation. We conclude by investigating a transformation proposed by S.R. Thatte [Tha85], that transforms arbitrary rewrite systems into constructor based systems. Contrary to what is stated in [Ver95b] this transformation does not preserve confluence for weakly persistent systems. On the other hand, it does preserve semi completeness. For terminating rewrite systems a simpler ....

....Kamperman and Walters (see [Kam96] propose another correctness criterion. Roughly speaking, they call a simulation sound if each reduction in B simulates a reduction in A, and complete if each reduction in A is simulated by one in B. These properties also play a part in preservation proofs in [Tha85] and [Ver95b] Definition 3.6 A simulation hOE; i of A by B preserves reduction graphs if 1. a A a 0 implies OE(a) B OE(a 0 ) 2. if b iB b 0 for b 2 dom( then there exists c 2 dom( such that (b) i A (c) and b 0 iB c. Condition 1 generalizes the notion of completeness of ....

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S.R. Thatte. On the correspondence between two classes of reduction systems. Information Processing Letters, 20(2):83--85, 15 February 1985.

....of automata, such that if automaton A simulates automaton B, then A can be seen as an implementation of B. ffl In papers dealing with transformations for Reduction Systems and their correctness, it is usually proved that the normal form relations of the original and its transform are related [Tha85, Tha88, Ver95a, Ver95b]. This is not surprising, because the normal form relation of a Reduction System is its computational content; it relates input and result. Kamperman and Walters initiated a new approach towards correctness proofs of transformations for Reduction Systems [KW95, KW96] In order to make correctness ....

....transformation as defined by Fokkink and Van de Pol. In sections 5 and 6 we study the applicability of our notion of simulation and the notion of correct transformation of Fokkink and Van de Pol by means of a notorious transformation of Thatte. This transformation was introduced in 1985 in [Tha85] for the class of regular Term Rewriting Systems. In [Tha88] Thatte claimed that his transformation preserved confluence and (up to some translation on terms) the normal form relation for the class of nonoverlapping systems, but it was shown by Verma in [Ver95b] that this claim was false. 2 1. ....

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Satish R. Thatte. On the correspondence between two classes of reduction systems. Information Processing Letters, 20(1985):83--85, February 1985.

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Satish R. Thatte. On the correspondence between two classes of reduction systems. Information Processing Letters, 20(1985):83--85, February 1985.

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S. Thatte, On the correspondence between two classes of Reduction systems, Information Processing Letters 20 (2), 1985.

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