Jean-Yves Girard. A new constructive logic: classical logic. Mathematical Structures in Computer Science, 1(3):255-296, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the characterization of boxes given by proposition 2.4.7 is still valid for R 0 and R 0 0 . We can then deduce R 0 = R 0 0 , i.e. R R 0 . We now de ne the notion of weakly polarized formula , which is related to the one of polarized formula , widely studied in the last 10 years (see [Gir91], DJS97] QTdF96] TdF97] Lau99] LQTdF00] 4.1.3. Definition. Weakly polarized formulas) A propositional formula P (resp. N) of LL is weakly positive (resp. weakly negative) when it is built in the following way (where X is an atomic formula) P : X j P P j P N j N P N : X j ....

....polarized when it is weakly positive or weakly negative. A proof net of MELL is weakly polarized when the types of its conclusions are all subformulas of weakly polarized formulas. 4.1.4. Remark. The di erence between the weakly polarized formulas and the (strongly) polarized ones (coming from [Gir91]) is that every atomic formula is both weakly positive and weakly negative: we do not suppose anything on the atoms for weakly polarized formulas. In particular, a weakly polarized formula A is not equivalent to A nor to A (contrary to polarized formulas, see [DJS97] or [TdF00b] 4.1.5. ....

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Jean-Yves Girard. A new constructive logic: classical logic. Mathematical Structures in Computer Science, 1(3):255-296, 1991.

....of a certain number of proofs. The cognoscenti might also notice that the two normal proof nets (of PN 2 ) of gure 3 are equivalent up to reversion : by reverting one of them one obtains the other one. This operation of reversion has been recently exploited in several ways (see [Gir91a], DJS97] LQTdF01] The proofnet of gure 3 containing a multibox can then also be though as a canonical representative of the equivalence class induced by reversion. 4.4. Remark. The step [r( does not modify the notions of lift and residue of a box, de ned in 2.13. Let us stress the fact ....

....the strong normalization theorem for PN 2 (see [TdF00a] and extend it to the calculus with multiboxes, which might be non trivial. 8. 2 Polarization, focalization, and hypersequentialization The two crucial properties of focalization and reversion have been widely studied and exploited ( AP91] [Gir91a], DJS97] LQTdF01] Thanks to the notion of positive tree (coming from [Lau99] the problems of con uence and strong normalization have been entirely solved in the polarized and focalized fragment of LL (see [LQTdF01] But polarized and focalized proof nets have a very particular shape, ....

Jean-Yves Girard. A new constructive logic: classical logic. Mathematical Structures in Computer Science, 1(3):255-296, 1991.

....direct isomorphism between LKQ and a familiar CBV # calculus with a let construct equipped with a classical extension. As for the relation between constructive classical logic and CPS translation, Murthy s pioneering work is also noteworthy[8] He shows that one can interpret Girard s LC[5] (of which the negative fragment is LKT) by means of CPS calculi with intuitionistic extract method. 2 2 Background In this section, we recall necessary definitions and notations for our presentation. Basically, we follow the notion of indexed logical system according to Parigot[14] It was ....

Jean-Yves Girard. A new constructive logic: Classical logic. Mathematical Structures in Computer Science, 1:255--296, 1991.

....to yield a unique cut free object in some important cases. So what The problem is that the objects (the proof nets) used are de nitely not canonical. Some better ( more canonical) solutions have been proposed in [9] and [15] Recently, a new fragment of LL appeared to have a great interest: in [6] and [3] the polarized fragment of LL is shown to be enough to translate faithfully classical logic. A study of proof nets for such a fragment was undertaken in [13] and the the notion of [9] drastically simpli ed. In [14] a proof of strong normalization and con uence of the cut elimination ....

Jean-Yves Girard. A new constructive logic: classical logic. Mathematical Structures in Computer Science, 1(3):255-296, 1991.

.... for MELL [2] is not completely satisfactory as it uses non deterministic strategies for modeling terms (and is therefore incomplete) In this paper we build a game model of the polarized fragment LLP of LL and introduce a polarization of formulas games in the spirit of Girard s polarities [7]. As LLP is expressive enough to encode LC, the calculus, 15, 16, 17] we therefore get a model for various classical deduction systems. Moreover these encodings are de nable for both call by name and call by value evaluations giving a uniform setting for interpreting them. Negative ....

.... : the strategy f g composed with any strategy gives the strategy f g (because strategies aren t empty) 2 In fact this result may be extended to a focalized calculus for MALL (HC [9] for example) by replacing the constraint of a negative context in the # rule by a focalization constraint (stoup [7] or constraint [6] for example) even if provable sequents in these systems may contain several positive formulas. 3 Exponentials 3.1 Games De nition 14 (Multiple game) A game A is a multiple game if it is well opened and: m1) if s 2 PA , A m and m A n then s mn 2 PA where s mn is ....

Jean-Yves Girard. A new constructive logic: classical logic. Mathematical Structures in Computer Science, 1(3):255-296, 1991.

....for LKT can also be seen as an immediate successor of these works, although we do not use explicitly of these results. 3 As for the relation between constructive classical logic and CPS translation, Murthy s pioneering work is also noteworthy[18] He shows that one can interpret Girard s LC[10] (of which the negative fragment is LKT) by means of CPS calculi with intuitionistic extract method. However, he could not give an answer to the question whether this term extraction method is appropriate or not. This is because he doesn t consider the relation between the computation and ....

Jean-Yves Girard. A new constructive logic: Classical logic. Mathematical Structures in Computer Science, 1:255--296, 1991.

....Myriam Quatrini IML CNRS Marseille quatrini iml.univ mrs.fr Lorenzo Tortora de Falco Roma III tortora logique.jussieu.fr Abstract We give the precise correspondence between polarized linear logic and polarized classical logic. The properties of focalization and reversion of linear proofs [AP91, Gir91a, DJS97] are at the heart of our analysis: we show that the tq protocol of normalization (de ned in [DJS97] for the classical systems LK pol and LK ; pol perfectly ts normalization of polarized proof nets. In section 6, some more semantical considerations allow to recover LC as a re nement of ....

....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 the cut elimination ....

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Jean-Yves Girard. A new constructive logic: classical logic. Mathematical Structures in Computer Science, 1(3):255-296, 1991.

....Proof Nets for LC (Extended Abstract) Olivier Laurent Institut de Math ematiques de Luminy CNRS Marseille, France olaurent iml.univ mrs.fr Abstract. We define a notion of polarization in linear logic (LL) coming from the polarities of Jean Yves Girard s classical sequent calculus LC [4]. This allows us to define a translation between the two systems. Then we study the application of this polarization constraint to proofnets for full linear logic described in [7] This yields an important simplification of the correctness criterion for polarized proof nets. In this way we ....

....takes an important place in proof theory. Much work is spent to deal with commutation of rules for cut elimination in sequent calculi. The introduction of proof nets (see [7] for instance) solves commutation problems and allows us to define a clear notion of reduction and complexity. In [4], Jean Yves Girard defines the sequent calculus LC using polarities. LC is a refinement of LK with a deterministic cut elimination. J. Y. Girard leaves open the following problem about the syntax: Find a better syntax (which would be to LC what typed calculus is to LJ) for normalization [ ....

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Jean-Yves Girard. A new constructive logic : classical logic. Mathematical Structures in Computer Science, 1(3):255--296, 1991.

....for classical logic: nding some computational interpretation of classical proofs, similar to the CurryHoward correspondence for intuitionistic logic. We will be interested in two kinds of solutions that have been proposed. The sequent calculus approach has two main instances: Girard s LC [5] is a deterministic sequent calculus for classical logic based on a polarization of formulas, with a semantics of proofs in coherent spaces; the LK tq system of Danos Joinet Schellinx [2] gives an extensive description of the deterministic reduction strategies that may be applied to LK. Both LC ....

Jean-Yves Girard. A new constructive logic: classical logic. Mathematical Structures in Computer Science, 1(3):255-296, 1991.

....constructive classical logic, i.e. classical logic equipped with Strongly Normalizing (SN) and Church Rosser(CR) cut elimination procedure. The investigation has started in Girard s two important works. They are the Linear Logic (LL) 7] and the new constructive logic: classical logic LC[8]. In fact, these works gave impetus to our work reported in this paper. This approach continues to LKT and LKQ through Danos, Joinet and Schellinx(DJS) 2] LKT and LKQ are generalized into LK j in [3] The constructive classical logic LK j is a variant of LK. The CR property of ....

.... context which is a set of indexed formulas. Similarly, 1 is a context which is a set of indexed formulas. Comma means taking union as sets. Thus, the set 0 0 [0 1 is denoted by 0 0 ; 0 1 and fA x g [ 0 by A x ; 0 . 5 denotes at most one unindexed formula. We say 5 is in the stoup [8]. Otherwise said, 5 insist on being attractive and main in axiom rule or logical rule(in case it become a cut formula) Please note that implication is the only logical rule in our presentation. We only handle multiplicative rules in every logical system. That is, contexts (and contexts) in ....

Jean-Yves Girard. A new constructive logic: Classical logic. Mathematical Structures in Computer Science, 1:255-- 296, 1991.

....ffi 1 ffi A 1 ffi A Les coupures croisees de la logique classique de [Gen34] sont en fait des coupures dont les deux pre misses principales sont des formules qui viennent d etre contractees. Ceci est une cause majeure de son non determinisme, comme en te moigne l exemple 2 de l annexe B de [Gir91]. Ceci ne peut se produire en logique lineaire, puisque pour e tre contractee une formule doit e tre de la forme A, et que par suite sa negation est A qui ne peut pas e tre contractee. Pour e tudier un tel phenomene il faut disposer d une modalite autoduale, autorisant la contraction tant en ....

....ffi A A rien ffi 1 ffi Il n y a pas a l heure actuelle de syntaxe e tendant le calcul ordonne a cette modalite. Il faut pour cela e tudier les cas de base d elimination des coupures et les diagrammes commutatifs qui en resultent comme cela a e te fait pour le systeme LC de [Gir91] dans [Qua95] On peut s etonner que j etudie un e ventuel modele avant de savoir precisement ce qu il modelise. Disons que c est un moyen de guider les choix syntaxiques en e vitant la degenerescence du systeme, et que c est un peu l analogue pour les modeles de Heyting, de ce que la theorie des ....

Jean-Yves Girard. A new constructive logic: classical logic. Mathematical Structures in Computer Science, 1(3):255--296, November 1991.

....There is a long line of proof theoretical approaches to understanding deconstructive classical logic. That is, classical logic that has Strongly Normalizing (SN) and confluent (Church Rosser or CR) cut elimination procedure. This thread began with Girard s linear logic(LL) 8] followed by LC [9] and the logic of unity (LU) 10] It reaches to LKT and LKQ [3] and more general LK tq [4] through Danos, Joinet and Schellinx (DJS) These works are all based on logic in Gentzen style sequent calculus [7] We unify the proof theoretical approach (i.e. SN and CR cut elimination procedure) and ....

....may be a fixed arbitrary atomic formula OE. We define intuitionistic negation :A as A OE. In LKT, 5 h denotes at most one indexed formula. In LKQ, 5 denotes at most one unindexed formula. The place on the left of semi colon where 5 h is located in LKT is called stoup according to Girard[9]. We also call this specially placed index as head index and always denote by h. We also call the place on the right of semi colon as stoup in LKQ. In our version of CND, there is no unindexed formula in lhs. This is contrast to Parigot s original system, where the rhs of CND has exactly one ....

Jean-Yves Girard. A new constructive logic: Classical logic. Mathematical Structures in Computer Science, 1:255--296, 1991.

....relative au connecteur autodual precede dans la categorie des espaces coherents et dans celle des hypercoherences. Mots cle : Semantique denotationnelle. Logique, theorie de la demonstration, logique lineaire. A self dual modality for before 3 1 Presentation This note strongly relies on [Gir91], where the question was raised. This section is just a concise reminder. The structural rules of classical logic are responsible for the non determinism of classical logic, and linear logic which carefully handles these rules is especially adequate for a constructive treatment of classical logic, ....

....theorem of Gentzen as a rule called MIX [Gen34] This rule is a generalised cut between several occurences of A and several occurences of :A, which is in fact a cut between two formulae coming both from a contraction. This is a major cause of non determinism: an example can be found in [Gir91], Appendix B, Example 2, p 294. In linear logic, such a cut may not happen, since contraction applies on A formulae, while their negation for applying a cut is A which can not come from a contraction. Let us now quote the precise paragraph of [Gir91] p 257 which motivates this note: The ....

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Jean-Yves Girard. A new constructive logic: classical logic. Mathematical Structures in Computer Science, 1(3):255--296, November 1991.

....in this paper. Proof theory: There is also a long line of proof theoretical approaches to understanding deconstructive Gentzen style classical logic, i.e. classical logic equipped with SN and CR cut elimination procedure. This thread began with Girard s linear logic(LL) 6] followed by LC [7] and the logic of unity (LU) 8] It reaches to LKT and LKQ [2] and LK j [3] through Danos, Joinet and Schellinx(DJS) LKT and LKQ are dual variations of Gentzen s original system LK. Confluency (i.e. the CR property) is recovered by adding some restrictions on logical rules to the LK, though ....

....a context which is a set of indexed formulas. Similarly, 1 is a context which is a set of indexed formulas. Comma means taking union as sets. Thus, the set 0 0 [ 0 1 is denoted by 0 0 ; 0 1 and fA x g [ 0 by A x ; 0 . 5 denotes at most one unindexed formula. We say 5 is in the stoup [7]. Otherwise said, 5 insist on being attractive and main in axiom rule or logical rule(in case it become a cut formula) Please note that implication is the only logical rule in our presentation. We only handle multiplicative rules in every logical system. That is, contexts (and contexts) in ....

Jean-Yves Girard. A new constructive logic: Classical logic. Mathematical Structures in Computer Science, 1:255--296, 1991.

....with no accessible path to them) requires execution, and is thus undecidable. 7 Extensions This work leaves open a number of interesting questions. Foremost is the extension of this formalism to the whole of linear logic, including additives and quantifiers, and from there to classical logic [Gir91]. We should establish a clear relationship between the coherence semantics of linear logic [Gir87] and the geometry of interaction. This should give us some guidance for the remaining two problems: finding a type system that abstracts the behavior on the left wire well enough to extend to ....

Jean-Yves Girard. A new constructive logic: Classical logic. Technical report, June 1991. INRIA Report 1443.

....necessary for a good understanding of linear logic was quite extensive, and we found it convenient to break this paper into two parts. The first part gives an exposition of background material (with the exception of the Girard translation of classical logic into intuitionistic logic, which is new [9]) The second part is devoted to linear logic and proof nets. In our presentation of background material, we have tried to motivate the introduction of various concepts by showing that they are indispensable to achieve certain natural goals. For pedagogical reasons, it seems that it is best to ....

....twists due to Girard [8] We conclude with a fairly extensive discussion of the reduction of classical logic to intuitionistic Research Report No. 8 May 2 Jean Gallier logic. Besides the standard translations due to Godel, Gentzen, and Kolmogorov, we present an improved translation due to Girard [9] (based on the notion of polarity of a formula) 2 Natural Deduction and Simply Typed Calculus We first consider a syntactic variant of the natural deduction system for implicational propositions due to Gentzen [3] and Prawitz [14] In the natural deduction system of Gentzen and Prawitz, a ....

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Jean-Yves Girard. A new constructive logic: classical logic. Logic and Computation (April 1991). To appear.

....There is a long line of proof theoretical approaches to understanding deconstructive classical logic. That is, classical logic that has Strongly Normalizing (SN) and confluent (ChurchRosser or CR) cut elimination procedure. This thread began with Girard s linear logic(LL) 9] followed by LC [10] and the logic of unity (LU) 11] It reaches to LKT LKQ [4] and more general LK tq [5] through Danos, Joinet and Schellinx (DJS) These works are all based on logic in Gentzen style sequent calculus [8] Continuation Passing Style(CPS) Since Griffin s influential work [13] on the Curry Howard ....

....which is also a set of indexed formula. 5 denotes at most one unindexed formula in LK. In LKT, 5 h denotes at most one indexed formula, while 5 denotes at most one unindexed formula. The place on the left of semi colon where 5 h is located in LKT is called stoup according to Girard[10]. We also call this specially placed index as head index and always denote by h. We also call the place on the right of semi colon as stoup in LKQ. The rhs of the sequent of LJ is an unindexed formula which may be a fixed arbitrary atomic formula OE. We define intuitionistic negation :A as A ....

Jean-Yves Girard. A new constructive logic: Classical logic. Mathematical Structures in Computer Science, 1:255--296, 1991.

....on their types. Proof theory: There is also a long line of proof theoretical approaches to understanding deconstructive Gentzen style classical logic, i.e. classical logic equipped with SN and CR cutelimination procedure. This thread began with Girard s linear logic(LL) 8] followed by LC [9] and the logic of unity (LU) 10] It reaches to LKT and LKQ [3] and LK j [4] through Danos, Joinet and Schellinx(DJS) LKT and LKQ are dual variations of Gentzen s original system LK. Confluency (i.e. the CR property) is recovered by adding some restrictions on logical rules to the LK, though ....

....a context which is a set of indexed formulas. Similarly, 1 is a context which is a set of indexed formulas. Comma means taking union as sets. Thus, the set 0 0 [ 0 1 is denoted by 0 0 ; 0 1 and fA x g [ 0 by A x ; 0 . 5 denotes at most one unindexed formula. We say 5 is in the stoup [9]. C denotes exactly one unindexed formula. If maps formulas to formulas, then if 1 = A 1 x1 ; An xn , we write 1 for the set (A 1 x 1 ) A n xn ) For example, 1 q for ( A q 1 ) x1 ; A q n ) xn , where q maps an LKQ formula to an LJ formula. We ....

Jean-Yves Girard. A new constructive logic: Classical logic. Mathematical Structures in Computer Science, 1:255--296, 1991.

....LKQ term calculus (CBV calculus on classical proofs) in Gentzen style sequent as the subsystem of Proof Net of Girard[9] 1 Introduction There is a long line of proof theoretical approach to understand constructive classical proofs. This thread began with Girard s linear logic[9] followed by LC [10] logic of unity (LU) 11] It reaches to LKT LKQ [5] and LK tq [4] by Danos, Joinet and Schellinx(DJS) As usual proof theoretical argument goes, these works are based on logic in Gentzen style sequent calculus[8] On the other hand, since Griffin s influential work[13] on Curry Howard ....

Jean-Yves Girard. A new constructive logic: Classical logic. Mathematical Structures in Computer Science, 1:255--296, 1991.

....There is a long line of proof theoretical approaches to understanding deconstructive classical logic. That is, classical logic that has Strongly Normalizing (SN) and confluent (ChurchRosser or CR) cut elimination procedure. This thread began with Girard s linear logic(LL) 8] followed by LC [9] and the logic of unity (LU) 10] It reaches to LKT LKQ [3] and more general LK tq [4] through Danos, Joinet and Schellinx (DJS) These works are all based on logic in Gentzen style sequent calculus [7] Continuation Passing Style(CPS) Since Griffin s influential work [12] on the Curry Howard ....

....as sets. Thus the set 0 0 [0 1 is denoted by 0 0 ; 0 1 . fA x g[0 by A x ; 0. 5 denotes at most one unindexed formula in LK and LKQ. In LKT, 5 h denotes at most one indexed formula. The place on the left of semi colon where 5 h is located in LKT is called stoup according to Girard[9]. We also call this specially placed index as head index and always denote by h. We also call the place on the right of semi colon as stoup in LKQ. The rhs of the sequent of LJ is an unindexed formula which may be a fixed arbitrary atomic formula OE. We define intuitionistic negation :A as A ....

Jean-Yves Girard. A new constructive logic: Classical logic. Mathematical Structures in Computer Science, 1:255--296, 1991.

....show in this paper. Proof theory: There is also a long line of proof theoretical approaches to understanding deconstructive Gentzen style classical logic, i.e. classical logic equipped with SN and CR cut elimination procedure. This thread began with Girard s linear logic(LL) 5] followed by LC[6] and the logic of unity (LU) 7] It reaches to LKT and LKQ [1] and LK j [2] through Danos, Joinet and Schellinx(DJS) LKT and LKQ are dual variations of Gentzen s original system LK. Confluency (i.e. the CR property) is recovered by adding some restrictions on logical rules to the LK, though ....

....a context which is a set of indexed formulas. Similarly, 1 is a context which is a set of indexed formulas. Comma means taking union as sets. Thus, the set 0 0 [ 0 1 is denoted by 0 0 ; 0 1 and fA x g [ 0 by A x ; 0 . 5 denotes at most one unindexed formula. We say 5 is in the stoup [6]. Otherwise said, 5 insist on being attractive and main in axiom rule or logical rule(in case it become a cut formula) Please note that implication is the only logical rule in our presentation. We only handle multiplicative rules in every logical system. That is, contexts (and contexts) in ....

Jean-Yves Girard. A new constructive logic: Classical logic. Mathematical Structures in Computer Science, 1:255--296, 1991.

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