H. Herbelin. A -calculus Structure Isomorphic to Sequent Calculus Structure. In Computer Science Logic, volume 933 of LNCS, pages 67--75. Springer Verlag, 1995.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....syntax directed proof search) whose proofs can be translated in a 1 1 way with the normal natural deductions for intuitionistic logic. MJ has the advantages of a sequent calculus system, whilst reflecting the structure of normal natural deductions. The calculus originates with Herbelin ( Her95] [Her96]) and has also been investigated and developed by Dyckhoff and Pinto ( DP96] DP98a] Herbelin calls his calculus LJT, but here we follow Dyckhoff Pinto in calling it MJ, as a calculus intermediate between natural deduction (NJ) and sequent calculus (LJ) This nomenclature also avoids a clash ....

....and completeness of MJ (Corollary 1.1) but also (via Lemmas 1.1 and 1.2) that proofs of MJ correspond in a 1 1 to the normal natural deductions for intuitionistic logic. We give the translations here, and state the lemmas and theorems, all of which can be found in [DP96] DP98a] Her95] [Her96]. Sequent Calculus Natural Deduction: M N (x; Ms) 0 (var(x) Ms) x:M) x: M) pair(M 1 ; M 2 ) pr( M 1 ) M 2 ) inl(M) i( M) inr(M) j( M) u:M) u: M) pairq(T; M) prq(T; M) CHAPTER 1. INTRODUCTION AND BACKGROUND 7 Gamma P Gamma [ ....

H. Herbelin. A -calculus Structure Isomorphic to Sequent Calculus Structure. Unpublished, 1996.

.... normalising, i.e. they employ a particular reduction strategy (for example an inner most reduction strategy or the elimination of the cut with the highest rank) Besides these weakly normalising methods a few strongly normalising cut elimination procedures have been developed; for example in [4 7, 13, 14]. However, all those methods impose some form of restriction on the reduction rules to ensure strong normalisation. A common restriction is to not allow a cut rule to pass over another cut rule (exceptions are [6, 13] However this limits, in the intuitionistic case, the correspondence between ....

....some form of restriction on the reduction rules to ensure strong normalisation. A common restriction is to not allow a cut rule to pass over another cut rule (exceptions are [6, 13] However this limits, in the intuitionistic case, the correspondence between cut elimination and beta reduction [8, 14]. Therefore in this paper we develop a strongly normalising cut elimination procedure adapting the standard cut elimination steps for logical cuts and allowing commuting cuts to pass over other cuts. A cut rule is said to be a logical cut when both cut formulae are introduced by axioms or ....

H. Herbelin. A -calculus Structure Isomorphic to Sequent Calculus Structure. In Computer Science Logic, volume 933 of LNCS, pages 67-75. Springer Verlag, 1995.

....formalism between # reduction and cut elimination, and not just the same. One can then do basically two things. Either define an explicit substitution calculus interpreting cut elimination, in such a way to have a perfect Curry Howard correspondence between them, as is done by Hugo Herbelin in [20]: there terms encode proofs, types encode propositions and reduction encodes cut elimination in intuitionistic sequent calculus. However, this comes at a price, since, exactly as in [33] his calculus does not allow to simulate every possible one step # reduction, because substitutions cannot go ....

H. Herbelin. A #-calculus structure isomorphic to sequent calculus structure. In Proceedings of Annual Conference of the European Association for Computer Science Logic (CSL), LNCS, 1994.

....calculus and Joshi and Kulick s partial proof manipulation system [8] which is used to represent linguistic information. Another relevant system is Herbelin s lambda calculus isomorphic to a variant of sequent calculus, where proofs of certain sequents are interpreted by applicative contexts [6]. These results suggest some interesting connections between context calculus and proof systems. Acknowledgements The authors thank Pierre Louis Curien, Laurent Dami, Yasuhiko Minamide, Didier R emy, Masahiko Sato and anonymous referees for their careful reading of a draft of this paper and ....

H. Herbelin. A -calculus Structure Isomorphic to Sequent Calculus Structure. In Proc. Annual Conference of the European Association for Computer Science Logic (CSL'94), 1994.

....LJT, in question. However, LJT is a very specialised L system and, hence, only admits a restricted, albeit interesting in its own right, description of the computational meaning of the cut rule. We should mention that Herbelin uses a calculus of proof terms with variable names but he suggests in [26] how to adapt the correspondence to a calculus a la de Bruijn. This does not alter the specialised nature of the calculus 5 We have omitted all definitions of standard rewriting notions such as SR, WIN, etc. due to space constraints. We refer instead to [3, 23, 37] 3 A completely different ....

H. Herbelin. A -calculus structure isomorphic to sequent calculus structure, DRAFT. Available from the author's homepage: http://capella.ibp.fr/~herbelin/publis.html.

....proofs for intuitionistic propositional logic under the name of clausal calculus. He also uses sequent left rules to build nested patterns but only some reduction rules are suggested in the spirit of cut elimination, while the explicit substitution paradigm is not incorporated at all. Herbelin [Her94] proposes a calculus which is also inspired by the Curry Howard isomorphism. His calculus does not have abstraction on patterns but only on variables, and reductions of terms correspond exactly to cut elimination rules. However, fi reduction cannot be simulated by this calculus because ....

H. Herbelin. A -calculus structure isomorphic to sequent calculus structure. In CSL, LNCS 845, 1994.

....[16] Schwichtenberg and Berger [4] Coquand [6] etc. ffl Formal systems and calculi: Girard [11, 12] Parigot [24] Berardi and Barbanera [2] Danos, Joinet and Schellinx [8] etc. ffl Proofs and semantics of cut elimination: Girard [11] Hofmann [17] Coquand [5] Pfenning [25] Herbelin [15], etc. Of these Parigot s ideas have provided much inspiration and, at an early stage of this work, the basis for believing that a categorical characterization of classical proofs was up for grabs . Girard s work on lc [11] presents, among other things, a denotational model of a version of ....

H. Herbelin. A -calculus structure isomorphic to sequent calculus structure. preprint, 1995.

.... this alternative definition corresponds to a version of ILU in which one demands that stoup formulas are always active (or in the conclusion of the derivation) cf. Danos et al. 1994) 2) The proximity of the sequent calculus ILU to natural deduction for intuitionistic logic is put to use in Herbelin(1994), where a calculus with explicit substitution is introduced, that is isomorphic to (a small variation on) ILU. 6 Decorated derivations and normalization When normalizing derivations in sequent calculus (i.e. eliminating the cuts) we come across counterparts of the structural rules of weakening ....

Herbelin, H. (1994). A -calculus structure isomorphic to sequent calculus structure. Manuscript. To appear in the proceedings of CSL'94.

....an object logical connective and the Isabelle meta level connectives. 2 A Specific Example To look at a concrete example, we take the implicational fragments of Gentzen s NJ [Gen34, Pra65] and a permutation free sequent calculus MJ, due to Dyckhoff and Pinto[DP95] based on work by Herbelin in [Her94] 2.1 The Systems NJ and MJ The syntactic terms of these systems are A and N for NJ, M and Ms for MJ, defined as follows: A : ap(A; N) j var(V) N : V:N j an(A) M : V ; Ms) j V:M Ms : j M : Ms where V is the set of variables. The appropriate rules for the two calculi are as ....

H. Herbelin. A -calculus structure isomorphic to sequent calculus structure, 1994. To appear in CSL '94.

....with proof terms 1 is, however, a non trivial exercise. If not actually impossible, it requires far more working around ALF s intended methods than working within them. 1 Introduction To illustrate the problems encountered in ALF, the following term calculus, MJ (based on work by Herbelin in [Her94], developed by Dyckhoff and Pinto in [DP95] is used: M : V ; Ms) j V:M Ms : j M : Ms where Vis the set of variables. These syntactic categories and the sequent rules: Gamma; x : P Gamma Gamma Gamma P Ms : R Gamma; x : P ) x ; Ms) R Select Gamma; x : P )M : Q Gamma ) ....

H. Herbelin. A -calculus structure isomorphic to sequent calculus structure, 1994. To appear in CSL '94.

....proofs for intuitionistic propositional logic under the name of clausal calculus. He also uses sequent left rules to build nested patterns but only some reduction rules are suggested in the spirit of cut elimination, while the explicit substitution paradigm is not incorporated at all. Herbelin [22] proposes a calculus which is also inspired by the Curry Howard isomorphism. His calculus does not have abstraction on patterns but only on variables, and reductions of terms correspond exactly to cut elimination rules. However, fi reduction cannot be simulated by this calculus because ....

H. Herbelin. A -calculus structure isomorphic to sequent calculus structure. In CSL, LNCS 845, 1994.

....may be a good formal language to describe the whole compilation process. This confirms the versatility of oe, which has been used recently to study advanced topics in the calculus, such as higher order unification [12] or issues in logic, such as the interpretation of sequent calculus [15]. 2 Preliminaries 2.1 The lambda calculus with explicit substitutions The traditional weak calculus is an attempt to model the execution of machine code within the calculus; it conforms with the basic intuition that functions, once compiled, are code and cannot change (otherwise, there would ....

H. Herbelin, "A -Calculus Structure Isomorphic to Sequent Calculus Structure", CSL'94.

....proofs for intuitionistic propositional logic under the name of clausal calculus. He also uses sequent left rules to build nested patterns but only some reduction rules are suggested in the spirit of cut elimination, while the explicit substitution paradigm is not incorporated at all. Herbelin [22] proposes a calculus which is also inspired by the Curry Howard isomorphism. His calculus does not have abstraction on patterns but only on variables, and reductions of terms correspond exactly to cut elimination rules. However, fi reduction cannot be simulated by this calculus because ....

H. Herbelin. A -calculus structure isomorphic to sequent calculus structure. In CSL, LNCS 845, 1994.

....we may use the same methods for encoding natural deduction systems. 2 A Specific Example To look at a concrete example, we take the implicational fragments of Gentzen s NJ [Gen34, Pra65] and a permutation free sequent calculus MJ , due to Dyckhoff and Pinto[DP95] based on work by Herbelin in [Her94]. 2.1 The Systems NJ and MJ The syntactic terms of these systems are A and N for NJ, M and Ms for MJ, defined as follows: A : ap(A; N) j var(V) N : V:N j an(A) M : V ; Ms) j V:M Ms : j M : Ms where V is the set of variables. The appropriate rules for the two calculi are as ....

H. Herbelin. A -calculus structure isomorphic to sequent calculus structure, 1994. To appear in CSL '94.

....replaced does not occur in the term. The rule for explicit substitution fflL can thus be used to model the oeL rule of the classical sequent calculus directly. In [27] a similar analysis is provided for a proof system for SLD resolution over propositional implicational Horn clauses. Herbelin [10] also uses explicit substitutions, for a similar reason, in his version of a translation of intuitionistic sequent calculus (LJ) into a modified calculus. His concern, however, is to restrict LJ so as obtain a bijective correspondence between terms and LJ derivations. Now we extend strong ....

H. Herbelin. A -calculus structure isomorphic to sequent calculus structure. In: Proc. Computer Science Logic '94, Kazimierz, Poland , Lecture Notes in Computer Science 933, Springer, 1995.

....may be a good formal language to describe the whole compilation process. This con rms the versatility of , which has been used recently to study advanced topics in the calculus, such as higher order uni cation [12] or issues in logic, such as the interpretation of sequent calculus [15]. 2 Preliminaries 2.1 The lambda calculus with explicit substitutions The traditional weak calculus is an attempt to model the execution of machine code within the calculus; it conforms with the basic intuition that functions, once compiled, are code and cannot change (otherwise, there would ....

H. Herbelin, \A -Calculus Structure Isomorphic to Sequent Calculus Structure", CSL'94.

....formalism between # reduction and cut elimination, and not just the same. One can then do basically two things. Either define an explicit substitution calculus interpreting cut elimination, in such a way to have a perfect Curry Howard correspondence between them, like is done by Hugo Herbelin in [17]: there terms encode proofs, types encode propositions and reduction encodes cut elimination in intuitionistic sequent calculus. However, this come at a price, since, exactly as in [28] his calculus does not allow to simulate every possible one step # reduction, because substitutions cannot go ....

H. Herbelin. A #-calculus structure isomorphic to sequent calculus structure. In Proceedings of Annual Conference of the European Association for Computer Science Logic (CSL), LNCS, 1994.

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H. Herbelin. A -calculus Structure Isomorphic to Sequent Calculus Structure. In Computer Science Logic, volume 933 of LNCS, pages 67--75. Springer Verlag, 1995.

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