M. Parigot. Free deduction: an analysis of computations in classical logic. Proc. Logic Progr. and Autom. Reasoning, St Petersbourg. L.N.C.S. 592, p. 361-380 (1991).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....We conclude: 1) the natural notion of computation for intuitionistic logic is that of asymmetric reductions with garbage collection as defined in the natural deduction or sequent calculus representation. By contrast, in classical natural deduction, as in Parigot s system of free deduction [14], a reduction step may be described not only as a substitution of a subderivation d 1 for an assumption of d 2 but also as a substitution of a subderivation d 2 for a conclusion of d 1 . This ambiguity is the root of the non determinism and non confluence of classical normalization, when ....

....of classical natural deduction, then strong normalization holds. Moreover, if some syntactic device of systematic disambiguation is introduced (e.g. a labelling of the formulas) then the normalization process becomes strongly normalizing and confluent (with respect to the given labelling, see [7, 14]) Such a disambiguation is regarded as a constructivization of classical logic. A way to disambiguate classical logic is through translations into linear logic. Such a use of linear logic is exemplified in Girard [10] and it has been developed extensively by V. Danos, J B. Joinet and H. ....

M. Parigot. Free Deduction: an analysis of computation in classical logic. In Voronkov, A., editor, Russian Conference on Logic Programming, pp. 361380. Sprinver Verlag. LNAI 592, 1991.

....with a deterministic normalization procedure. This means that cutelimination enjoys some particular properties: it has a denotational semantics, and it enjoys strong normalization and (usually) con uence. Let us quote, among the works following this approach, Parigot s FD and calculus [Par91, Par92], Girard s LC [Gir91a] LK tq and its subsystems [DJS97] Although all this work is not entirely based on it, linear logic (LL) introduced in [Gir87] seems to play a pre eminent role, because it can be used as a looking glass. On the one hand LL suggests a reasonable way to make choices in 1 ....

Michel Parigot. Free deduction: an analysis of computation in classical logic. In Russian Conference on Logic Programming, volume 592 of Lecture Notes in Articial Intelligence, pages 361-380. Springer, 1991.

.... and Coq [14] In [27] a computational interpretation of classical natural deduction is discussed, in which lambda terms may be extracted from proofs; this investigation is continued in [28] For other papers dealing with computational interpretations of classical logic see [19] 15] 25] and [26]. These systems typically supply a computational interpretation to all classical proofs; in contrast, our system uses classical logic for the part of the proof that does not affect the computation at all. As for logics dealing with fixed points, we can mention the logic for computable functions ....

M. Parigot. Free deduction: an analysis of computations in classical logic. In Proc. Russian Conference on Logic Programming, pages 361--380, St. Petersburg, Russia, 1991.

.... and predicate logic, the correspondence has been extended to more complex, but essentially intuitionistic, systems by several authors; see, for example, the references in [14] More recently, the correspondence has been extended to classical propositional 8 and predicate logic by Parigot [123, 124, 101], to propositional intuitionistic linear logic (see [166] and to a bunched logic, combining linear and intuitionistic predicate logics [116, 147] by Ishtiaq and Pym [86] A good view of the propositions as types correspondence for minimal intuitionistic logic is given by the cube [14] in ....

M. Parigot. Free deduction: an analysis of computations in classical logic. In 1th and 2th Russian Conference on Logic Programming, LNAI 592, pages 361-380, St. Petersburg, Russia, July 1991.

....P embedding into linear logic is a decoration. Introduction Apr es que le contenu calculatoire de la logique classique fut mis en evidence par Felleisen [2] de nombreuses tentatives de pr esentation constructive de la logique classique ont et e r ealis ees. Dans les syst emes LC ( 3] et FD ([4]) des choix et des restrictions ont et e op er es pour se d ebarrasser des situations conflictuelles, responsables du non d eterminisme de l elimination des coupures de LK (le calcul des s equents classique de Gentzen) L approche de Danos, Joinet et Schellinx [1] consiste a rajouter a LK ....

M. Parigot, 1991. Free Deduction: an analysis of computation in classical logic. In Voronkov, A., editor, Russian Conference on Logic Programming, Springer Verlag. LNAI 592: 361-380. . 4

....is now a cut elimination step in one of the constructive slices superimposed by the classical proof. 1 Introduction Much work has been done in the past 6 7 years to extract the computational content from classical proofs. On the proof theoretical side, let s quote for example [Girard 91] LC) Parigot 91] FD) DJS 95] LK tq and its restrictions) and [BarBer 95] the symmetric calculus) With the notable exception of the symmetric calculus of [BarBer 95] which enjoys only strong normalization, all the systems previously mentioned enjoy the usual good computational properties (strong ....

....leads to the collapse of the whole system: we have to identify all the proofs of a given sequent (see [Girard 91] and [DJS 95] The solution proposed by the works previously mentioned is either to avoid such difficult rules (this is possible keeping completeness w.r.t. classical provability, see [Parigot 91] and [BarBer 95] or to restrict the application of such difficult rules to the premises which lead to key cases the elimination of which is unproblematic (see [Girard 91] and [DJS 95] A trace of this restriction is the fact that the translations from LK (with polarized or coloured formulas) to ....

Parigot M., Free Deduction: an analysis of computation in classical logic. In: Voronkov, A, editor, Russian Conference on Logic Programming, pp. 361-380. Springer Verlag. LNAI 592. 1991.

....functions, but, in some cases, it can produce more efficient algorithms than those provided by intuitionistic proofs. In order to gain this, many different constructive proofs systems for classical logic have recently been proposed. Among others we can mention Girard s LC [3] Parigot s FD [7] and the system LK t;q developed in [2] Most of these systems have a relevant aspect in common, namely that of sort of breaking the inner symmetry of classical logic. In fact such a symmetry, clear in Gentzen s sequent system for classical logic, can be seen as one of the main causes of the ....

Parigot, M. (1991) Free Deduction: an analysis of computation in classical logic. In Voronkov, A., ed., Russian Conference on Logic Programming, 361380. Springer Verlag. LNAI 592.

.... the design of a noetherian and confluent normalization for LK 2 (that is, classical second order predicate logic presented as a sequent calculus) The method we present is powerful: since it allows us to recover as fragments formalisms as seemingly different as Girard s LC and Parigot s , FD ([10, 12, 29, 33]) delineates other viable systems as well, and gives means to extend the Krivine Leivant paradigm of programming with proofs ( 24, 25] to classical logic; it is painless: since we reduce strong normalization and confluence to the same properties for linear logic (for non additive proof nets, ....

....observe that the resulting LL normalization theorem is essentially different from Girard s strong normalization theorem for proof nets. Several such proof systems for classical logic have recently been proposed, notably LC (Logique Classique, 10] by Jean Yves Girard and FD (Free Deduction, [29]) by Michel Parigot. At first sight these do not seem to have much more in common than the bare fact of being both complete for provability. Girard s LC is distinguished by coming equipped with a denotational semantics, while Parigot developed an extension of the calculus, the so called ....

[Article contains additional citation context not shown here]

Parigot, M. (1991) Free Deduction: an analysis of computation in classical logic. In: Voronkov, A., editor, Russian Conference on Logic Programming, pp. 361--380. Springer Verlag. LNAI 592.

.... and predicate logic, the correspondence has been extended to more complex, but essentially intuitionistic, systems by several authors; see, for example, the references in [14] More recently, the correspondence has been extended to classical propositional and predicate logic by Parigot [123, 124, 101], to propositional intuitionistic linear logic (see [166] and to a bunched logic, combining linear and intuitionistic predicate logics [116, 147] by Ishtiaq and Pym [86] A good view of the propositions as types correspondence for minimal intuitionistic logic is given by the cube [14] in which ....

M. Parigot. Free deduction: an analysis of computations in classical logic. In 1th and 2th Russian Conference on Logic Programming, LNAI 592, pages 361--380, St. Petersburg, Russia, July 1991.

....connectives and simply forgetting about colours and orientations. The present paper contains in some detail a proof of strong normalization for all style LK tq , hence a fortiori for any of its (complete) fragments. 4 The implication am corresponds e.g. to the arrow in Free Deduction ([8]) We will assume that the reader has some acquaintance with linear logic, especially with proof nets and their reduction. 2 Some terminology: main active interspaces The following conventions are used in distinguishing between the occurrences of formulas in a given logical rule, e.g. L : ....

Parigot, M. (1991). Free Deduction: an analysis of computation in classical logic. In Voronkov, A., editor, Russian Conference on Logic Programming, pages 361--380. Springer Verlag. Lecture Notes in Artificial Intelligence 592.

No context found.

M. Parigot. Free deduction: an analysis of computations in classical logic. Proc. Logic Progr. and Autom. Reasoning, St Petersbourg. L.N.C.S. 592, p. 361-380 (1991).

No context found.

PARIGOT M. (1991), Free Deduction: an analysis of computation in classical logic, in A. Voronkov editor, Russian Conference on Logic Programming, Springer Verlag, LNAI 592, 361--380.

No context found.

M. Parigot. Free deduction: an analysis of computations in classical logic. Proc. of Log. Prog. and Automatic Reasoning. St Petersbourg. L.N.C.S. 592 p. 361-380 (1991).

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