R. Constable and C. R. Murthy. Finding computational content in classical proofs. In G. Huet and G. Plotkin, eds., Logical Frameworks, pp. 341-362. Cambridge University Press, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... typed calculi, see, e.g. 27, 28, 8] and applied to the compilation and optimization of typed languages, see, e.g. 19, 44] Grin s discovery initiated a series of studies on the computational content of classical proofs where CPS translations are a frequently employed tool, see, e.g. [13, 33, 38, 39, 34, 12, 4, 42, 24, 25, 35, 36, 6]. Inductive and coinductive types, see, e.g. 31, 29, 20, 15, 40] are syntactic representations for initial algebras (such as natural numbers and lists) resp. nal coalgebras (such as conatural numbers and streams) in typed calculi. Despite being pervasive in the type theoretical literature ....

R. Constable and C. R. Murthy. Finding computational content in classical proofs. In G. Huet and G. Plotkin, eds., Logical Frameworks, pp. 341-362. Cambridge University Press, 1990.

.... typed calculi, see, e.g. 27, 28, 8] and applied to the compilation and optimization of typed languages, see, e.g. 19, 44] Grin s discovery initiated a series of studies on the computational content of classical proofs where CPS translations are a frequently employed tool, see, e.g. [13, 33, 38, 39, 34, 12, 4, 42, 24, 25, 35, 36, 6]. Inductive and coinductive types, see, e.g. 31, 29, 20, 15, 40] are syntactic representations for initial algebras (such as natural numbers and lists) resp. nal coalgebras (such as conatural numbers and streams) in typed calculi. Despite being pervasive in the type theoretical literature ....

R. Constable and C. R. Murthy. Finding computational content in classical proofs. In G. Huet and G. Plotkin, eds., Logical Frameworks, pp. 341-362. Cambridge University Press, 1990.

....with Griffin s observation [12] that Felleisen s control operator C [9, 10] can be given the type of the stability scheme : A A. This initiated quite a bit of work aimed at extending the Curry Howard correspondence to classical logic, e.g. by Barbanera and Berardi [1] Constable and Murthy [5], Krivine [14] and Parigot [17] We now describe in more detail what the paper is about. In section 2 we fix our version of intuitionistic arithmetic for functionals, and recall how 1 classical arithmetic can be seen as a subsystem. Then our argument goes as follows. It is well known that from a ....

Robert L. Constable and Chetan Murthy. Finding computational content in classical proofs. In G. Huet and G. Plotkin, editors, Logical Frameworks, pages 341--362. Cambridge University Press, 1991.

....was the rst to stress the relation between sequential control and classical logic [12] His work is based on Felleisen s syntactic theory of sequential control, which provides an idealisation of Scheme call cc. Around the same time, Murthy studied the computational content of classical proofs [6, 14]. His work is based mainly on negative translations of classical logic, and CPS transforms. The work of GriOEn was extended by Barbanera and Berardi [2, 3] who noted that Felleinsen s reduction rules are similar to Prawitz s handling of double negation [20] They use a control operator akin to ....

R. Constable and C. Murthy. Finding computational content in classical proofs. In G. Huet and G. Plotkin, editors, Logical Frameworks, pages 341362. Cambridge University Press, 1991.

.... calculi is feasible and worthy. As we pointed out in the introduction, a system such as LK may be useful in practice. It can be used for interactive theorem proving and also provides a possible formalism to study the computational content of classical proofs, which is a problem addressed in [3]. At a more fundamental level, to design systems such as LK and to provide semantics for them may give us a better understanding of the nature of classical proofs. Related works on this topic include [3, 9, 12, 20] Nevertheless, LK is still an experimental system and problems remain. The two ....

....to study the computational content of classical proofs, which is a problem addressed in [3] At a more fundamental level, to design systems such as LK and to provide semantics for them may give us a better understanding of the nature of classical proofs. Related works on this topic include [3, 9, 12, 20]. Nevertheless, LK is still an experimental system and problems remain. The two main ones concern the de nition of suitable notions of conversion and reduction. The problem of de ning an appropriate conversion theory is related to the issue of completeness with respect to some class of models. ....

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R. Constable and C. Murthy. Finding computational content in classical proofs. In G. Huet and G. Plotkin, editors, Logical Frameworks, pages 341362. Cambridge University Press, 1991.

.... that the reduction rules for C were closely related to classical proof normalization as studied by Prawitz [81] Seldin [86,87] and Stalmarck [90] Griffin s discoveries were followed by a series of papers on classical logic, control operators and the Curry Howard isomorphism, see for example [3,4,18,24,42,62 66,69 72,82]. Most of these works introduce one typed classical calculus, i.e. a typed calculus enriched with control operators, and study its properties with respect to e.g. normalization, confluence and categorical semantics or its applications to e.g. classical theorem proving and witness extraction. ....

....[34] noted that Felleisen s extended translation maps simply typed C terms to Barthe, Hatcliff, S rensen simply typed terms. He further showed that this translation, when viewed as a translation on proofs, becomes the Kolmogorov embedding of classical logic into minimal logic [50] Murthy [18,62,63] and Griffin himself [35] later systematized these ideas by studying different logical embeddings, control operators, and CPS translations. More recently, CPS translations from classical typed calculi to typed calculi were studied by de Groote for [23] and exn [24] by Duba, Harper and ....

R. Constable and C.R. Murthy. Finding computational contents in classical proofs. In G. Huet and G. Plotkin, editors, Proceedings of the First Workshop on Logical Frameworks, pages 341--362. Cambridge University Press, 1990.

....works on classical logic, control operators and the Curry Howard isomorphism some initiated independently of his work. Most of these works study one typed calculus enriched with control operators; we call such calculi classical calculi. In the overwhelming majority of cases see for example [8, 9, 15, 22, 34, 48, 49, 50, 51, 52, 56, 57, 64] the calculus considered is essentially the simply typed or polymorphic calculus; other calculi considered include higher order calculus [31] ML [23] linear calculus [14] calculus with explicit substitutions [32, 68] or proof irrelevant logical pure type systems [79] In all cases, the ....

R. Constable and C.R. Murthy. Finding computational contents in classical proofs. In G. Huet and G. Plotkin, editors, Proceedings of the First Workshop on Logical Frameworks, pages 341--362. Cambridge University Press, 1990.

....V. Matiyasevich (Eds. Lecture Notes in Computer Science, Vol. 813, Springer Verlag (1994) pp. 142 152. In recent works, several authors have addressed the problem of extending the formulae astypes principle to classical logic, in order to express the computational content of classical proofs [2, 10, 12, 17, 18, 19]. This problem cannot have a unique solution because one knows that the technical content of the formulae as types principle, namely the Curry Howard isomorphism [3, 11, 14, 21] is strongly related to the constructive aspects of intuitionistic logic. Therefore, when dealing with classical logic, ....

R. Constable and C. Murthy. Finding computational content in classical proofs. In G. Huet and G. Plotkin, editors, Logical Frameworks, pages 341362. Cambridge University Press, 1991.

.... Gamma that not all functions are recursive (even though we cannot exhibit a counterexample ) A direct corollary is that there cannot be any recursive realization [12, 24] of G AC: For exactly this reason, and because the semantics of the system NuPrl is based on recursive realizability, the work [16, 4] restricts itself to a fragment of classical logic that does not include the Axiom of Choice. 2 This is to be contrasted with the induction schema over integers, whose negative interpretation is an instance of the induction schema itself. This often forces one to encode functions as relations, ....

R. Constable and C. Murthy. Finding Computational Content in Classical Proofs. In G.Huet and G. Plotkin, editors, Logical Frameworks, 341 - 362, (1991), Cambridge University Press.

....followed by several lines of work on classical logic, control operators, and the Curry Howard isomorphism some initiated independently of his work. It is not possible here to explain the aims and achievements of the individual lines of work; it must suffice simply to mention the work of Murthy [14, 59, 60, 61, 62], Barbanera and Berardi [2, 3, 4, 5, 6] Rezus [74, 75] Parigot [66, 67, 68, 69] de Groote [25, 27, 28, 29, 30] Krivine [57] Girard [42] Danos, Joinet, and Schellinx [18] Rehof and S rensen [73] Duba, Harper, and MacQueen[32] Harper and Lillibridge [46, 47] Coquand [15] Berardi, Bezem, ....

R. Constable and C.R. Murthy. Finding computational contents in classical proofs. In G. Huet and G. Plotkin, editors, Proceedings of the First Workshop on Logical Frameworks, pages 341--362. Cambridge University Press, 1990.

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