Comparing combinatory reduction systems and higher-order rewrite systems

Oostrom, Vincent van; Raamsdonk, Femke van

doi:10.1007/3-540-58233-9_13

Cited by 23 publications

(19 citation statements)

References 14 publications

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“…Our syntax is inspired by CRSs, but similar results can be obtained by using other higher-order systems, such as Nipkow's Higher-order Rewrite Systems [22], or Khasidashvili's Expression Reduction Systems [13]. The three formalisms are closely related [26]. CRSs have been used in previous work on interaction nets: Laneve [20] defined Interaction Systems as CRSs, and in [7] a translation function is given from interaction nets to CRSs.…”

Section: Introductionmentioning

confidence: 89%

A Higher-Order Calculus for Graph Transformation

Fernández

Mackie

Pinto

2007

Electronic Notes in Theoretical Computer Science

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This paper presents a formalism for defining higher-order systems based on the notion of graph transformation (by rewriting or interaction). The syntax is inspired by the Combinatory Reduction Systems of Klop. The rewrite rules can be used to define first-order systems, such as graph or term-graph rewriting systems, Lafont's interaction nets, the interaction systems of Asperti and Laneve, the non-deterministic nets of Alexiev, or a process calculus. They can also be used to specify higher-order systems such as hierarchical graphs and proof nets of Linear Logic, or to specify the operational semantics of graph-based languages.

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Section: Introductionmentioning

confidence: 89%

A Higher-Order Calculus for Graph Transformation

Fernández

Mackie

Pinto

2007

Electronic Notes in Theoretical Computer Science

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“…Hence it is a β-IDTS. In [31], van Oostrom and van Raamsdonk studied this translation in detail and proved:…”

Section: Application Of the General Schema To Hrssmentioning

confidence: 96%

Termination and Confluence of Higher-Order Rewrite Systems

Blanqui

2000

Rewriting Techniques and Applications

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In the last twenty years, several approaches to higher-order rewriting have been proposed, among which Klop's Combinatory Rewrite Systems (CRSs), Nipkow's Higher-order Rewrite Systems (HRSs) and Jouannaud and Okada's higher-order algebraic specification languages, of which only the last one considers typed terms. The later approach has been extended by Jouannaud, Okada and the present author into Inductive Data Type Systems (IDTSs). In this paper, we extend IDTSs with the CRS higher-order pattern-matching mechanism, resulting in simply-typed CRSs. Then, we show how the termination criterion developed for IDTSs with first-order pattern-matching, called the General Schema, can be extended so as to prove the strong normalization of IDTSs with higher-order pattern-matching. Next, we compare the unified approach with HRSs. We first prove that the extended General Schema can also be applied to HRSs. Second, we show how Nipkow's higher-order critical pair analysis technique for proving local confluence can be applied to IDTSs.

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“…In particular, it has been shown that crss and hrss have the same expressive power and therefore they can be considered equivalent [21]. Using this comparison, we can have an indirect representation of the ρ-calculus into hrss that is based on the translation from the crss to hrss and the translation from ρ-calculus to crss we have defined in this paper.…”

Section: Discussionmentioning

confidence: 99%

“…By the encoding of Section 3.1, Lemma 5 and the confluence results for orthogonal crss (see for example [20,21]). …”

Section: Corollary 1 (Confluence) the ρ-Calculus With Linear Algebramentioning

confidence: 99%

“…To this aim, we present an encoding of the ρ-calculus into crss, since for this kind of higher-order systems a well-developed meta-theory already exists. Studies on the comparison between different higher-order formalisms has already been conducted between Combinatory Reduction Systems and Higher-order Rewrite Systems in [21]. Moreover, an encoding of crss into the rewriting calculus has already been proposed by the authors in [3].…”

Section: Introductionmentioning

confidence: 99%

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The Rewriting Calculus as a Combinatory Reduction System

Bertolissi

Kirchner

Foundations of Software Science and Computational Structures

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Abstract. The last few years have seen the development of the rewriting calculus (also called rho-calculus or ρ-calculus) that uniformly integrates first-order term rewriting and λ-calculus. The combination of these two latter formalisms has been already handled either by enriching first-order rewriting with higher-order capabilities, like in the Combinatory Reduction Systems (crs), or by adding to λ-calculus algebraic features.In a previous work, the authors showed how the semantics of crs can be expressed in terms of the ρ-calculus. The converse issue is adressed here: rewriting calculus derivations are simulated by Combinatory Reduction Systems derivations. As a consequence of this result, important properties, like standardisation, are deduced for the rewriting calculus.

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Comparing combinatory reduction systems and higher-order rewrite systems

Cited by 23 publications

References 14 publications

A Higher-Order Calculus for Graph Transformation

A Higher-Order Calculus for Graph Transformation

Termination and Confluence of Higher-Order Rewrite Systems

The Rewriting Calculus as a Combinatory Reduction System

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