R. Di Cosmo and D. Kesner. A confluent reduction for the extensional typed \Gammacalculus with pairs, sums, recursion and terminal object. In A. Lingas, editor, Intern. Conf. on Automata, Languages and Programming (ICALP), LNCS. Springer-Verlag, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....new reduction system: weak confluence (section 4) and strong normalization (section 5) In section 8 we derive the confluence property in two different ways and finally in the conclusion we discuss some further applications of our proof techniques. An extended abstract of this work can be found in [DCK93]. 3 The Calculus It is now time to introduce the calculus we will deal with in this paper. There are two versions, one with bounded recursion, and the other with unbounded recursion, that differ just in the term formation rule and in the equality rule for recursive terms. We will now introduce ....

R. Di Cosmo and D. Kesner. A confluent reduction for the extensional typed \Gammacalculus with pairs, sums, recursion and terminal object. In A. Lingas, editor, Intern. Conf. on Automata, Languages and Programming (ICALP), LNCS. Springer-Verlag, 1993.

....Huet [Hue76] in the field of higher order unification, and by Mints [Min79] who was motivated by problems in category theory. But expansions seem however to have passed unnoticed for a long time, until these last two years, where many extensional calculi (without patterns) have been investigated [Jay92, JG92, Cub92, Dou93, Aka93, DCK93a]. Mainly inspired by [Aka93, DCK93a] we use expansion rules to provide a confluent reduction system for the extensional typed pattern calculus with layered, wildcard and product patterns equipped with fixpoints and product types. In particular, our system allows to rewrite any function P : A:M ....

....[Min79] who was motivated by problems in category theory. But expansions seem however to have passed unnoticed for a long time, until these last two years, where many extensional calculi (without patterns) have been investigated [Jay92, JG92, Cub92, Dou93, Aka93, DCK93a] Mainly inspired by [Aka93, DCK93a], we use expansion rules to provide a confluent reduction system for the extensional typed pattern calculus with layered, wildcard and product patterns equipped with fixpoints and product types. In particular, our system allows to rewrite any function P : A:M into another function P 0 : A:M 0 ....

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Roberto Di Cosmo and Delia Kesner. A confluent reduction for the extensional typed \Gammacalculus with pairs, sums, recursion and terminal object. In Andrzej Lingas, editor, Intern. Conf. on Automata, Languages and Programming (ICALP), number 700 in LNCS. Springer-Verlag, 1993.

....a modular property with respect to fixpoints. We considered anyway that it was worth presenting here in view of the fact that it does not seem to be very well known: in many works normalization with bounded fixpoints is proven again and again without noticing that it is a fairly general property [PV87, HM90, DCK93]. 4 This translation is different from the one suggested in [Dou93] and is crucial to be able to show the modularity of the confluence property. 5 Notice that we can reason in the same way if we directly take here a given fresh variable y for fix 0 . The traditional approach to confluence ....

Roberto Di Cosmo and Delia Kesner. A confluent reduction for the extensional typed \Gammacalculus with pairs, sums, recursion and terminal object. In Andrzej Lingas, editor, Intern. Conf. on Automata, Languages and Programming (ICALP), volume 700 of Lecture Notes in Computer Science, pages 645--656. Springer-Verlag, 1993.

....a modular property with respect to fixpoints. We considered anyway that it was worth presenting here in view of the fact that it does not seem to be very well known: in many works normalization with bounded fixpoints is proven again and again without noticing that it is a fairly general property [PV87, HM90, DCK93]. The traditional approach to confluence with bounded fixpoints would require here to go through any (usually old) already available proof of (local) confluence for the reduction relation without fixpoints, and find the spots where to fit the slight modifications necessary to accommodate the ....

Roberto Di Cosmo and Delia Kesner. A confluent reduction for the extensional typed \Gammacalculus with pairs, sums, recursion and terminal object. In Andrzej Lingas, editor, Intern. Conf. on Automata, Languages and Programming (ICALP), volume 700 of Lecture Notes in Computer Science, pages 645--656. Springer-Verlag, 1993.

.... URA 1347) Ecole Normale Sup erieure 45, Rue d Ulm 75230 Paris France E mail:dicosmo dmi.ens.fr INRIA Rocquencourt Domaine de Voluceau, BP 105 78153 Le Chesnay Cedex, France and CNRS and LRI B at 490, Universit e de Paris Sud 91405 Orsay Cedex, France E mail:kesner lri.lri.fr R. Di Cosmo and D. Kesner 2 N , give the same result MN . Similarly for pairs, the only operational way to use a pair is by projecting out the first or the second component, so we do not want to consider a term M of product type different from the term h 1 (M) 2 (M)i: the result of accessing any of these two terms ....

....of using expansion rules seems to have passed unnoticed for a long time, even if the so called j long normal forms were well known and used in the study of higher order unification problems (Hue76) only in z This is the Unit type in languages like ML. x The same idea is present in (Min77) R. Di Cosmo and D. Kesner 4 these last years there has been a renewed interest in expansion rules. In recent work (Jay92) still motivated by category theoretic investigation, Jay explores a simply typed calculus with just T and a natural number type N as base types, equipped with an induction combinator for terms of ....

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R. Di Cosmo and D. Kesner. A confluent reduction for the extensional typed \Gammacalculus with pairs, sums, recursion and terminal object. In A. Lingas, editor, Intern. Conf. on Automata, Languages and Programming (ICALP), LNCS. Springer-Verlag, 1993.

.... The traditional contractive way is well behaved in the simply typed calculus, but it breaks confluence in many other other calculi [CDC91, DCK94, Kes97b] On the other hand, the typed b j expansion can also be combined with many other reduction rules, such as expansive surjective pairing [DCK93, Aka93, Dou93, JG95], recursion [DCK93, Dou93] sums [DCK93, Dou93] algebraic rewriting systems [DCK94] and second order calculus [DCK95] All these combinations are well behaved as they preserve confluence and strong normalization. However, none of the mentioned works covers the combination of b j expansion ....

.... is well behaved in the simply typed calculus, but it breaks confluence in many other other calculi [CDC91, DCK94, Kes97b] On the other hand, the typed b j expansion can also be combined with many other reduction rules, such as expansive surjective pairing [DCK93, Aka93, Dou93, JG95] recursion [DCK93, Dou93], sums [DCK93, Dou93] algebraic rewriting systems [DCK94] and second order calculus [DCK95] All these combinations are well behaved as they preserve confluence and strong normalization. However, none of the mentioned works covers the combination of b j expansion with calculi having explicit ....

[Article contains additional citation context not shown here]

Roberto Di Cosmo and Delia Kesner. A confluent reduction for the extensional typed \Gammacalculus with pairs, sums, recursion and terminal object. In Andrzej Lingas, editor, Intern. Conf. on Automata, Languages and Programming (ICALP), number 700 in LNCS. Springer-Verlag, 1993.

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