C. Laneve and U. Montanari, Axiomatizing permutation equivalence in the - calculus, in: H. Kirchner and G. Levi, eds., Proc. Third Int. Conf. on Algebraic and Logic Programming, Volterra, Italy (LNCS 632, Springer-Verlag, 1992) 350-- 363.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....between lambda terms. The natural inclusion of the lambda calculus within rewriting logic using explicit substitution was pointed out in [115] An illuminating investigation of parallel computations in the lambda calculus using rewriting logic has been carried out by Laneve and Montanari [87, 88], who have considered the even more general case of orthogonal, left normal combinatory reduction systems as formalized by Aczel [1] that contain the lambda calculus as a special case. They show that such systems exactly correspond to rewrite theories R whose equational part E consists of ....

C. Laneve and U. Montanari. Axiomatizing permutation equivalence in the - calculus. In H. Kirchner and G. Levi, editors, Proc. Third Int. Conf. on Algebraic and Logic Programming, Volterra, Italy, September

.... models of computation have been naturally expressed in rewriting logic: 1) equational programming, which is the special case of rewrite theories whose set of rules is empty and whose equations are Church Rosser, possibly modulo some axioms A; 2) lambda calculi and combinatory re7 duction systems [218,192,193,295,292]; 3) labeled transition systems [218] 4) grammars and string rewriting systems [218] 5) Petri nets, including place transition nets, contextual nets, algebraic nets, colored nets, and timed Petri nets [218,223,293,297,268,289] 6) Gamma and the Chemical Abstract Machine [218] 7) CCS and ....

Cosimo Laneve and Ugo Montanari. Axiomatizing permutation equivalence in the -calculus. In H. Kirchner and G. Levi, editors, Algebraic and Logic Programming, Third International Conference, Volterra, Italy, September 2--4,

....sometimes we will denote the corresponding reduction by t Delta s. Sequential term rewriting, as just defined, can be generalized to parallel term rewriting by allowing for the simultaneous application of two or more redexes to a term. The definitions below summarize those in [6] see also [23, 7]) and are valid for orthogonal trs s only: as for subterm replacement, all definitions and results lift smoothly to terms. Definition 12 (residuals) Let Delta = w; d) and Delta 0 = w 0 ; d 0 : l 0 r 0 ) be two redexes in a term t. The set of residuals of Delta by Delta 0 , ....

C. Laneve and U. Montanari. Axiomatizing permutation equivalence in the - calculus. Mathematical Structures in Computer Science, 6:219--249, 1996.

....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 approach to graph transformation for which a rich ....

C. Laneve and U. Montanari. Axiomatizing permutation equivalence in the - calculus. Mathematical Structures in Computer Science, 6:219--249, 1996.

....and, at the same time, places rules in all possible contexts. Vertical composition acts instead as sequential composition. Furthermore, the generated rewriting sequences are subject to an equivalence that coincides with the so called permutation equivalence [7] due to the axioms of 2 categories [21, 42]. A similar construction can be followed for term graph rewriting as well, as it is shown in [10] The idea is to add to the gs monoidal category generated by Sigma cells representing the rules of a term graph rewriting system, and to consider the (gs monoidal) 2 category freely generated by ....

C. Laneve and U. Montanari. Axiomatizing permutation equivalence in the -calculus. In H. Kirchner and G. Levi, editors, Algebraic and Logic Programming, volume 632 of LNCS, pages 350--363. Springer Verlag, 1992.

....between lambda terms. The natural inclusion of the lambda calculus within rewriting logic using explicit substitution was pointed out in [75] An illuminating investigation of parallel computations in the lambda calculus using rewriting logic has been carried out by Laneve and Montanari [52, 53]. Before addressing the case of the lambda calculus, Laneve and Montanari [53] first clarify the exact relationship between the equivalence of rewrites obtained by the equations identifying proof terms in the free model TR (X) of a rewrite theory R, and Boudol s notion of permutation equivalence ....

C. Laneve and U. Montanari. Axiomatizing permutation equivalence in the - calculus. In H. Kirchner and G. Levi, editors, Proc. Third Int. Conf. on Algebraic and Logic Programming, Volterra, Italy, September 1992, volume 632 of LNCS, pages 350--363. Springer-Verlag, 1992.

....specific, the first axiom schemata could be considered analogous to the conditions required for permutation equivalence on term rewriting [2] since the rewrites occur on disjoint parts of a process. For general results on the relationships between rewriting logic and permutation equivalence, see [17,6,12]. Proof sketch We proceed like in the proof of Theorem 4.1: we first find a normal form for both equivalent proved computations and concurrent proof terms, and we show then that c and ff c are inverse to each other over the classes of elements in normal form. Roughly, we are looking for proved ....

C. Laneve and U. Montanari. Axiomatizing permutation equivalence in the -calculus. Mathematical Structures in Computer Science, 6:219--249, 1996.

....and extra axioms on stable DRSs may be needed to ensure it. We expect that a similar extraction algorithm will be applicable, but proofs will become more complicated. Event Structure semantics for orthogonal rewrite systems with duplicating residual relation are studied, among others, in [LaMo92, Lan93], but the results there are limited because of the problems with erasure illustrated by the example in Section 6. Second, it is natural to extend DPESs with an axiomatized erasure relation or an axiomatized permutationequivalence relation to enable DPESs to give an adequate semantics to ....

Laneve C., Montanari U. Axiomatizing permutation equivalence in the -calculus. In proc. of the 3 rd International Conference on Algebraic and Logic Programming, ALP'92, Springer LNCS, vol. 632, 1996, p.350363.

....have chosen any other logic. It has only the expository advantage of building upon an example already introduced in this paper. In fact, our equational treatment of quantification, inspired by ideas of Laneve and Montanari on the definition of the lambda calculus as a theory in rewriting logic [57, 58], is very general and encompasses not only existential and universal quantification, but also lambda abstraction and other such binding mechanisms. The main idea is to internalize as operations in the theory the notions of free variables and substitution that are usually defined at the metalevel. ....

.... calculus with explicit substitutions defined by Abadi, Cardelli, Curien, and L evy in [1] and to the work of Talcott on binding structures [106, 107] We begin by presenting the example of the lambda abstraction binding mechanism in the lambda calculus, as defined by Laneve and Montanari in [57] (see also [58] where this technique is generalized to combinatory reduction systems) Since in this case the syntax is much simpler, the main ideas can become more explicit and clearer to the reader. We assume a parameterized functional module SET[X] that provides finite sets over a parameter ....

C. Laneve and U. Montanari, Axiomatizing permutation equivalence in the -calculus, in: H. Kirchner and G. Levi (eds.), Proc. Third Int. Conf. on Algebraic and Logic Programming, Volterra, Italy, September 1992, LNCS 632, Springer-Verlag, 1992, pages 350--363.

....scope of the assumption. In the chosen case studies, instead, the basic rules interact via local information, explicitly carried over by the evolution steps, without resorting to any condition on the structure of the deduction. 3 logic; it is a simplified version of the translation described in [49]. We close the section showing how the rewriting logic paradigm can be applied to data structures different from terms. Action structures were originally proposed by Robin Milner [64] as a foundational model for different kinds of process algebras, especially for mobile and higher order systems. ....

....axiom defining ff conversion is x:t = y: t[ y = x ] for y 62 fn(t) The remaining axioms allow reducing every ground term to one built up without using substitution. As for the one encoding ff conversion, they are just schemata, i.e. they represent a denumerable set of axioms. We refer to [49] for a conditional, finitary presentation inducing the same equivalence relation. c(x) M = x ] M c(x) M = y ] c(x) for x 6= y (M Delta N ) O = x ] M [ O = x ] Delta (N [ O = x ] x: t 1 ) t 2 = y ] x: t 1 [ t 2 = y ] for x 6= y and x 62 fn(t 2 ) We denote by = ....

[Article contains additional citation context not shown here]

C. Laneve and U. Montanari. Axiomatizing permutation equivalence in the - calculus. In H. Kirchner and G. Levi, editors, Algebraic and Logic Programming, volume 632 of Lect. Notes in Comp. Science, pages 350--363. Springer Verlag, 1992.

....a well accepted model for expressing the concurrency of an abstract formalism. On the contrary, this is not true in general for the derivation spaces associated to Meseguer s model (based on cartesian structure plus interchange axiom) 4 The same result could be inferred also by the analysis of [LM92, Lan94]. In the first paper it is shown that the equivalence induced on rewrites in Meseguer s space of computations coincides with permutation equivalence. In the second, it is shown that in the case of calculus (and more generally, for any trs) permutation equivalence cannot be the basis for a prime ....

C. Laneve, U. Montanari, Axiomatizing Permutation Equivalence in the - calculus, in Proc. 3 rd ALP, LNCS 632, 1992, pp. 350-363.

....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 entails the sequents d Omega (d d 1 ) and (d Omega d) id u Omega d 1 ) ....

C. Laneve and U. Montanari. Axiomatizing permutation equivalence in the - calculus. In H. Kirchner and G. Levi, editors, Algebraic and Logic Programming, volume 632 of LNCS, pages 350--363. Springer Verlag, 1992.

.... of the best known formalisms for distributed systems, namely Petri nets [73] and we recall the description of its concurrent behaviour proposed in [54] Afterwards, we present an encoding of the untyped calculus [3] into rewriting logic; it is a simplified version of the translation described in [49]. We close the section showing how the rewriting logic paradigm can be applied to data structures different from terms. Action structures were originally proposed by Robin Milner [64] as a foundational model for different kinds of process algebras, especially for mobile and 1 More explicitly, we ....

....axiom defining ff conversion is x:t = y: t[ y = x ] for y 62 fn(t) The remaining axioms allow reducing every ground term to one built up without using substitution. As for the one encoding ff conversion, they are just schemata, i.e. they represent a denumerable set of axioms. We refer to [49] for a conditional, finitary presentation inducing the same equivalence relation. c(x) M = x ] M c(x) M = y ] c(x) for x 6= y 15 (M Delta N ) O = x ] M [ O = x ] Delta (N [ O = x ] x: t 1 ) t 2 = y ] x: t 1 [ t 2 = y ] for x 6= y and x 62 fn(t 2 ) We denote ....

[Article contains additional citation context not shown here]

C. Laneve and U. Montanari. Axiomatizing permutation equivalence in the - calculus. In H. Kirchner and G. Levi, editors, Algebraic and Logic Programming, volume 632 of LNCS, pages 350--363. Springer Verlag, 1992.

....behaviour of a system: Each equivalence class of sequents should intuitively describe the same set of causally unrelated events. This is not so different in spirit from the well known permutation equivalence [20, 5] and there exists in fact a tight correspondence between the two notions [28]. For a few initial considerations about the actual degree of concurrency expressed by the axioms, we refer to [9] Example 4 (equating sequents) Let us consider again the ars V. As shown in Example 3, the system entails the sequents d Omega (d d 1 ) and (d Omega d) id u Omega d 1 ) ....

C. Laneve and U. Montanari. Axiomatizing permutation equivalence in the - calculus. Mathematical Structures in Computer Science, 6:219--249, 1996.

.... The generalization of permutation equivalence to ambiguous term rewriting systems need some preliminary notion of compatibility among derivations (see [1] Another approach overcoming this drawback is to define an axiomatization over derivations capturing permutation equivalence [8] Actually, in [5], it is shown that the axiomatization which results from Meseguer s technique in general does not match with permutation equivalence due to the interplay of the rewriting rules with the axioms quotienting the term algebra. Therefore, case by case, a package of further axioms must be provided. ....

C. Laneve and U. Montanari. Axiomatizing permutation equivalence in the - calculus. In 3 th Int. Conf. on Algebraic and Logic Programming, volume 632 of Lecture Notes in Computer Science, pages 350 -- 363. Springer-Verlag, 1992.

....a fi reduction with the one involving an ff equivalent abstraction. We already proved that, in the calculus case, Meseguer s axiomatization can be completed w.r.t. permutation equivalence by adding some simple axioms modeling the interplay between the operation of substitution and fi reduction [Laneve and Montanari 1992]. Here we generalize this result to generic rewriting systems with mechanisms of binding and substitution. More precisely, we study Klop s orthogonal Combinatory Reduction Systems (oCRS s, in the following) The restriction of orthogonality (no critical pair is admitted) is adopted for ....

C. Laneve and U. Montanari (1992) Axiomatizing permutation equivalence in the -calculus. In 3 th Int. Conf. on Algebraic and Logic Programming, volume 632 of Lecture Notes in Computer Science, pages 350 -- 363. Springer-Verlag.

....Then for each n 2 IlN and s 2 T Sigma (x 1 : xn ) there is a one to one correspondence between the families of L evy equivalent derivations entailed by R originating from s, and the cells in F c (Th(R) with source s Sigma : n 1. The same result could be inferred also by the analysis of [LM92, Mes90], relating respectively permutation equivalence and coherence axioms with the concurrent term rewriting proposed in [Mes92] A further discussion, describing a different equivalence [Ste94] and involving also concurrency issues, is proposed in [CGM95a] 4 Double Categories and Process Description ....

C. Laneve, U. Montanari, Axiomatizing Permutation Equivalence in the - calculus, in Proc. 3 rd ALP Conference, LNCS 632, 1992, pp. 350-363.

No context found.

C. Laneve and U. Montanari, Axiomatizing permutation equivalence in the - calculus, in: H. Kirchner and G. Levi, eds., Proc. Third Int. Conf. on Algebraic and Logic Programming, Volterra, Italy (LNCS 632, Springer-Verlag, 1992) 350-- 363.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC