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Interaction Systems I: The theory of optimal reductions†

Published online by Cambridge University Press:  04 March 2009

Andrea Asperti  and
Cosimo Laneve
Andrea Asperti
Affiliation:
Dip. di Matematica, Universitá di Bologna, P.za di Porta S.Donato, 40126, Bologna, Italy. Email: asperti@cs.unibo.it
Cosimo Laneve
Affiliation:
INRIA Sophia Antipolis, 2004 Route des Lucioles BP 93, 06902, Valbonne, France. Email: cosimo@garfield.cma.fr
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Abstract

We introduce a new class of higher order rewriting systems, called Interaction Systems (IS's). IS's are derived from Lafont's (Intuitionistic) Interaction Nets (Lafont 1990) by dropping the linearity constraint. In particular, we borrow from Interaction Nets the syntactical bipartitions of operators into constructors and destructors and the principle of binary interaction. As a consequence, IS's are a subclass of Klop's Combinatory Reduction Systems (Klop 1980), where the Curry-Howard analogy still ‘makes sense’. Destructors and constructors, respectively, correspond to left and right logical introduction rules: interaction is cut and reduction is cut-elimination.

Interaction Systems have been primarily motivated by the necessity of extending the practice of optimal evaluators for λ-calculus (Lamping 1990; Gonthier et al. 1992a) to other computational constructs such as conditionals and recursion. In this paper we focus on the theoretical aspects of optimal reductions. In particular, we generalize the family relation in Lévy (1978; 1980), thus defining the amount of sharing an optimal evaluator is required to perform. We reinforce our notion of family by approaching it in two different ways (generalizing labelling and extraction in Levy (1980)) and proving their coincidence. The reader is referred to Asperti and Laneve (1993c) for the paradigmatic description of optimal evaluators of IS's.

Type
Research Article
Information
Mathematical Structures in Computer Science , Volume 4 , Issue 4 , December 1994 , pp. 457 - 504
Copyright
Copyright © Cambridge University Press 1994

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