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A rewriting calculus for cyclic higher-order term graphs

Published online by Cambridge University Press:  01 June 2007

PAOLO BALDAN ,
CLARA BERTOLISSI ,
HORATIU CIRSTEA  and
CLAUDE KIRCHNER
PAOLO BALDAN
Affiliation:
Dipartimento di Matematica, Pura e Applicata, Università di Padova, Italy Email: baldan@math.unipd.it
CLARA BERTOLISSI
Affiliation:
Université Henri Poincaré Email: Clara.Bertolissi@loria.fr LORIA, BP 239, 54506 Vandoeuvre-lès-Nancy CedexFrance
HORATIU CIRSTEA
Affiliation:
Université Nancy 2 Email: Horatiu.Cirstea@loria.fr LORIA, BP 239, 54506 Vandoeuvre-lès-Nancy CedexFrance
CLAUDE KIRCHNER
Affiliation:
INRIA Email: Claude.Kirchner@loria.fr LORIA, BP 239, 54506 Vandoeuvre-lès-Nancy CedexFrance
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Abstract

The Rewriting Calculus (ρ-calculus, for short) was introduced at the end of the 1990s and fully integrates term-rewriting and λ-calculus. The rewrite rules, acting as elaborated abstractions, their application and the structured results obtained are first class objects of the calculus. The evaluation mechanism, which is a generalisation of beta-reduction, relies strongly on term matching in various theories.

In this paper we propose an extension of the ρ-calculus, called ρg-calculus, that handles structures with cycles and sharing rather than simple terms. This is obtained by using recursion constraints in addition to the standard ρ-calculus matching constraints, which leads to a term-graph representation in an equational style. Like in the ρ-calculus, the transformations are performed by explicit application of rewrite rules as first-class entities. The possibility of expressing sharing and cycles allows one to represent and compute over regular infinite entities.

We show that the ρg-calculus, under suitable linearity conditions, is confluent. The proof of this result is quite elaborate, due to the non-termination of the system and the fact that ρg-calculus-terms are considered modulo an equational theory. We also show that the ρg-calculus is expressive enough to simulate first-order (equational) left-linear term-graph rewriting and α-calculus with explicit recursion (modelled using a letrec-like construct).

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Type
Paper
Information
Mathematical Structures in Computer Science , Volume 17 , Issue 3 , June 2007 , pp. 363 - 406
Copyright
Copyright © Cambridge University Press 2007

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