F. Barbanera and S. Berardi. Continuations and simple types: A strong normalization result. In ACM SIGPLAN Workshop on Continuations, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....(w b) x] if y; w 62 FV(b) y 6= w Deltax:x a 2 a if x 62 FV(a) Deltax:x ( Deltay:x a) 3 a if x; y 62 FV(a) 4. fi = fi [ 6 See [14, 28, 33] for applications of control operators in theorem proving, 27, 24] for applications of explicit substitutions in theorem proving, [3, 11, 15, 23, 28, 29, 30] for applications of control operators in proof theory and [12, 17, 32] for applications of explicit substitutions in proof theory. We let x; y; z; w; range over V and a; b; c; range over T . We consider terms modulo ff conversion (generalised over Delta) and assume the variable ....

F. Barbanera and S. Berardi. Continuations and simple types: A strong normalization result. In ACM SIGPLAN Workshop on Continuations, 1992.

....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 Felleisen s C to extract the computational content of classical proofs of Sigma 0 1 sentences. On the proof theoretic side, Parigot has introduced the ....

F. Barbanera and S. Berardi. Continuations and simple types: a strong normalization result. In Proceedings of the ACM SIGPLAN Workshop on Continuations. Report STAN-CS-92-1426, Stanford University, 1992.

....fruitful aspect of the Curry Howard Ismorphism [Gal92] p8. the proof theory literature, or to classical proof theory in general. This in fact applies to all the works on typing control operators that we have investigated [Gri90] Mur90] Mur91] Mur91b] Mur92a] Dub90] Har92] Wer92] [Bar92]. Considering the importance of this particular aspect of the isomorphism we feel that the relation between reduction on constructions and proof normalization should be investigated. It is debatable whether an extension of the isomorphism can be claimed if a close correspondence is not found. ....

....operator languages similar to ;oe Delta have been given by Griffin [Gri90] and Murthy [Mur92a] who use CPS translations to show termination for a particular reduction strategy 6 (and in the latter case for constructions only of a particular class of formulae. Finally, Barbanera and Berardi [Bar92] have a very technical proof of a SN theorem for a calculus with call by value control operator rules but with nothing similar to our rules (3) 4) While all this provides us with a substantial tool box of ideas for proving SN for ;oe Delta (1 4) we choose, in fact, to start all over and ....

Franco Barbanera & Stafano Beradi. Continuations and Simple Types: A Strong Normalization Result. In Proceedings of the ACM SIGPLAN Workshop on Continuations CW92. San Francisco, California, 1992.

....systems and classical proof normalization as found in the proof theory literature, or to classical proof theory in general. This in fact applies to all the works on typing control operators that we have investigated [Gri90] Mur90] Mur91] Mur91b] Mur92a] Dub90] Har92] Wer92] [Bar92]. Considering the importance of this particular aspect of the isomorphism we feel that the relation between reduction on constructions and proof normalization should be investigated. It is debatable whether an extension of the isomorphism can be claimed if a close correspondence is not found. ....

....operator languages similar to ;oe Delta have been given by Griffin [Gri90] and Murthy [Mur92a] who use CPS translations to show termination for a particular reduction strategy 8 (and in the latter case for constructions only of a particular class of formulae. Finally, Barbanera and Berardi [Bar92] have a very technical proof of a SN theorem for a calculus with call by value control operator rules but with nothing similar to our rules (3) 4) While all this provides us with a substantial tool box of ideas for proving SN for ;oe Delta (1 4) we choose, in fact, to start all over and ....

Franco Barbanera & Stafano Beradi. Continuations and Simple Types: A Strong Normalization Result. In Proceedings of the ACM SIGPLAN Workshop on Continuations CW92. San Francisco, California, 1992.

....p. 67, Proposition 8.3] In 1990, however, T. Griffin opened a new research area by introducing a classical formulae astypes notion of control based on Felleisen s C operator [9] Since then, various authors have defined different systems that enlighten the constructive content of classical logic [1, 2, 7, 13, 14, 15, 16, 17]. Despite its originality, Griffin s work has been criticized by some logicians. In [15] for instance, M. Parigot writes that the system he (Griffin) obtains is not satisfactory from the logical point of view: the reduction is in fact a reduction strategy and the type assigned to C doesn t fit in ....

....of negation. As for renaming, it can be simulated in Felleisen s calculus by using the notion of reduction that follows: M (C N ) c N (x: M x) where the type of M is of the form :A. This notion of reduction, which is absent from Felleisen s theory, is used by F. Barbanera and S. Berardi in [1]. In establishing our isomorphism, we did not use Felleisen s notion of reduction CR : M (C N ) c C (k: N (v: A (k (M v) This notion of reduction corresponds, at the level of the calculus, to the following reduction rule: M ( ff: N ) ff: N [M =ff] where N [M =ff] is obtained by ....

F. Barbanera and S. Berardi. Continuations and simple types: a strong normalization result. In Proceedings of the ACM SIGPLAN Workshop on Continuations. Report STAN-CS-92-1426, Stanford University, 1992.

....we have studied a framework for classical calculi and proved that the theory of classical calculus carry over to systems of dependent types. Much work remains to be done. At a practical level, the appropriateness of CPTSs as a foundation for classical theorem proving and program extraction [3, 4, 5, 6, 7] should be investigated. At a theoretic level, one needs to examine criteria for distinguishing between principal and minor rules further and study CPS translations in more depth. ....

F. Barbanera and S. Berardi. Continuations and simple types: A strong normalization result. In ACM SIGPLAN Workshop on Continuations, 1992.

....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, and Coquand [11] Ong [65] Underwood ....

F. Barbanera and S. Berardi. Continuations and simple types: A strong normalization result. In ACM SIGPLAN Workshop on Continuations, 1992.

....Proceedings of the Colloquium on Trees in Algebra and Programming CAAP 94, Lecture Notes in Computer Science, Vol. 787 Springer Verlag (1994) pp. 85 99. During the last three years, several authors have introduced various systems that clarify the computational content of classical proofs [2, 3, 5, 6, 8, 9, 10]. In this paper, we investigate one of these systems, namely Parigot s calculus [10] Our investigation tool is merely syntactic: we propose a translation of the calculus into the well known calculus. This interpretation, which obey a continuation passing style, works for any term. It is ....

F. Barbanera and S. Berardi. Continuations and simple types: a strong normalization result. In Proceedings of the ACM SIGPLAN Workshop on Continuations. Report STAN-CS-92-1426, Stanford University, 1992.

....witness was proved to be possible only using particular reduction strategies, like call by name and call by value, leaving open the problem of arbitrary reduction strategies. To face such a problem the authors studied the problem of strong normalization for the calculus C , a study that led in [2] to a proof of strong normalization for a system ( C Gamma ) similar to C , where the reduction rules present in C are somewhat restricted and where other reduction rules are added. In the present paper we show that this set of reduction rules, even if not the same as that of C , can be used ....

....for system PA C . The proof consists in a reduction preserving translation to the type system C Gamma , which it is already known to be strongly normalizable. System C Gamma is similar to PA C . In a sense it could be considered as a propositional version of it. It was introduced in [2], where a proof of strong normalization for it is given using a non trivial version of Girard s candidates of reducibility method. Indeed the system C Gamma we present here contains slight extensions with respect to its original version in [2] These extensions consist in the introduction of a ....

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Barbanera F., Berardi S. Continuations and simple types: a strong normalization result. Proceedings of the ACM SIGPLAN Workshop on Continuations. Report No STAN-CS-92-1426 Stanford University. San Francisco, June 1992.

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