Jean-Yves Girard. Normal functors, power series and the -calculus. Annals of Pure and Applied Logic, 37:129--177, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... m X ( Y may be seen as a function preserving some structure from X to Y , and conversely This paper provides a positive and natural answer to this question, inspired by a work of Girard who, before introducing qualitative domains, and already guided by his dilators intuitions, considered in [5] a quantitative semantics of calculus where the interpretation of a term takes into account the number of times a value is used in a computation. Actually, in that semantics, these numbers are sets, and morphisms are functors acting on families of sets, preserving directed limits, pullbacks and ....

Jean-Yves Girard. Normal functors, power series and the -calculus. Annals of Pure and Applied Logic, 37:129--177, 1988.

....coherent semantics is injective for the set of weakly polarized proof nets. Proof: Simply notice that a weakly polarized proof net is a ( LL proofnet. 55 4.1.6. Corollary. The coherent multiset based semantics is injective for the intuitionistic fragment ILU of Girard s uni ed logic ([Gir93]) Proof: The system ILU is actually the t fragment of LK (de ned in [DJS97] and Danos, Joinet and Schellinx proved that for this fragment Girard s translation A B = A ( B yields a denotational semantics for ILU . It then suces to notice that this translation uses only weakly polarized ....

Jean-Yves Girard. On the unity of logic. Annals of Pure and Applied Logic, 59:201-217, 1993.

.... an entire map from X to Y (K othe spaces) is by de nition a function from EX to E Y which is de nable by such a series (which is then necessarily unique) This interpretation of intuitionistic proofs as entire maps is of course completely in the spirit of Girard s quantitative semantics (see [Gir88, BE99a], and also [Has97] where a method is developed for computing the coecients of the power series associated to terms in Girard s quantitative semantics) The space X has a co algebraic structure, as it is standard in the semantics of linear logic (this structure is used for interpreting ....

....has a coecient di erent from 0 and 1. This is an e ect of non uniformity: observe that, in this example, the coecient 2 corresponds to a component of f which would not exist in a uniform setting. It must be pointed out that this e ect was already present in Girard s work on quantitative semantics [Gir88], since this model had no uniformity constraints, and Ryu Hasegawa showed in [Has97] how to compute e ectively the entire coecients associated to the interpretations of terms using generating functions. As a second example, we consider the map 2 : E ( 1) K f 7 f(f(0) which is a version ....

Jean-Yves Girard. Normal functors, power series and the -calculus. Annals of Pure and Applied Logic, 37:129-177, 1988.

....A full solution has still to be found. Much work is now possible such as an extension of our approach to second order quantifiers, the study of a geometry of interaction or of a game semantics for such proof nets, the continuation of this work towards the intuitionistic polarities as defined in [6], ....

Jean-Yves Girard. On the unity of logic. Annals of Pure and Applied Logic, 59:201-- 217, 1993.

....2.2 Typing Rules Our typing rules for the Mini ML fragment of the explicit language are completely standard. The problem of typing the modal fragment is well understood; we present here a variant of known systems [BdP92, PW95] inspired by zonal formulations of linear logic such as Girard s LU [Gir93]. In our formulation the typing judgment has two contexts. The first contains variables that may appear anywhere, since they represent code. The second contains variables that are only available in the current computation stage, including all ordinary Mini ML variables. Thus our judgement Delta; ....

Jean-Yves Girard. On the unity of logic. Annals of Pure and Applied Logic, 59:201--217, 1993.

....2.2 Typing Rules Our typing rules for the Mini ML fragment of the explicit language are completely standard. The problem of typing the modal fragment is well understood; we present here a variant of known systems [BdP92, PW95] inspired by zonal formulations of linear logic such as Girard s LU [Gir93]. Our typing judgment has two contexts, the first containing assumptions regarding all future worlds, and the second containing assumptions regarding the current world. Thus our judgement Delta; Gamma e E : A would correspond to 2 Delta; Gamma E : A in 2 e , where E is an ....

Jean-Yves Girard. On the unity of logic. Annals of Pure and Applied Logic, 59:201--217, 1993.

....a function of type B 2A can carry out all computation concerned with its argument while generating the residual code of type A. We begin by considering 2 e , a modal calculus based on a natural deduction formulation of intuitionistic modal S4. The presentation is new, but draws on ideas in [BdP92, PW95, Gir93]. We then construct a functional language Mini ML 2 e by augmenting 2 e with a fixpoint operator, natural numbers, and pairs and endow it with a natural call by value operational semantics along the lines of Mini ML [CDDK86] Mini ML 2 e can be somewhat awkward because it often requires a ....

....our main objective here. Our presentation simplifies that of the modal calculus 2 e from [BdP92, PW95] by eliminating the need for simultaneous substitution while preserving subject reduction. It is inspired by sequent calculi proposed by Andreoli [And92] for linear logic and by Girard [Gir93] for LU. Wadler [Wad93] has formulated a linear calculus with two contexts, which shares some features with our calculus. The methodolgy we followed is due to Martin Lof [ML85a, ML85b] although we have not seen the normative use of local soundness and completeness as witnessed by fi reduction ....

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Jean-Yves Girard. On the unity of logic. Annals of Pure and Applied Logic, 59:201--217, 1993.

....of the fi redex (x:B)t. We use the turnstile symbol as the mathematics level judgment that a sequent is provable: that is, Delta Gamma means that the two sided sequent Delta Gamma Gamma has a linear logic proof. The sequents of F are similar to those used in the LU proof system of Girard [10] except that we have followed the tradition of [1, 14] in writing the classical context (here, Psi and Upsilon) on the outside of the sequent and the linear context (here, Delta and Gamma) nearest the sequent arrow: in LU these conventions are reversed. Given the intended interpretation of ....

Jean-Yves Girard. On the unity of logic. Annals of Pure and Applied Logic, 59:201--217, 1993.

....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 the reduction system approach ....

....are studied under natural deduction style i.e. term of the form of abstraction( introduction) and application( elimination) Related Works: However at least CBN part of these frameworks should be considered as folklore. Girard already suggest the relation between calculus and ILU in [10]. Also DJS themselves gave a guess in the final remark of [3] and in there introduction of [4] about the relation between LKT, negative fragment of LC and Parigot s CND. DeGroote revealed the relation between CBN CPS and Parigot s calculus which is an computational interpretation of CND [5] ....

Jean-Yves Girard. On the unity of logic. Annals of Pure and Applied Logic, 59:201--217, 1993.

....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 soundness and completeness ....

....is clear. Because, by the simulation theorem, LKT (hence t ) inherits the denotation in linear logic s coherent space semantics. 1.3 Related Works Folklore: However, these frameworks should be considered as folklore. Girard has already suggested the relation between calculus and ILU in [8]. In addition, DJS themselves gave an explanation in their introduction of [3] about the relation between the LKT, the negative fragment of the LC, and Parigot s CND. Specifically, the second part of our work can be seen as a adaptation of the De Groote s work [4] His investigation is about the ....

Jean-Yves Girard. On the unity of logic. Annals of Pure and Applied Logic, 59:201--217, 1993.

....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 correspondence between classical ....

....to be new. As a natural consequence we could complete the reduction rules for v for the first time, which is partly appeared in [18] Related Works: However at least CBN part of these frame works should be considered as folklore. Girard already suggest the relation between calculus and ILUin [11]. Also DJS themselves gave a guess in the final remark of [4] and in there introduction of [5] about the relation between LKT, negative fragment of LC and Parigot s CND. Moreover there is a detailed work of Herbelin[14] about term calculus on LJTwhich is an intuitionistic fragment of LKT. He ....

Jean-Yves Girard. On the unity of logic. Annals of Pure and Applied Logic, 59:201--217, 1993.

....the first critical issue in an implementation of a proof of cut elimination. Felty s representation [Fel89] in Prolog, for example, uses lists of hypotheses which is advantageous for search but makes a formal meta theory prohibitively complex. Frameworks based on sequent calculi such as LU [Gir93] or Forum [Mil94] allow direct encodings, but they lack a notation for the proof terms that are required to describe cut elimination. In this section we develop a formulation of the sequent calculus for intuitionistic logic by transcribing the process of searching for a natural deduction into an ....

Jean-Yves Girard. On the unity of logic. Annals of Pure and Applied Logic, 59:201--217, 1993.

....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 soundness and completeness ....

....CBN and CBV CPS calculus are shown to be duals, since LKT and LKQ are duals in their denotations. 1.3 Related Works Folklore: However, at least the CBN portion of these frameworks should be considered as folklore. Girard has already suggested the relation between calculus and ILU in [10]. In addition, DJS themselves gave an explanation in their introduction of [4] about the relation between the LKT, the negative fragment of the LC, and Parigot s CND. With regards to isomorphism, DeGroote previously displayed the translation from calculus of Parigot[17] to CBN CPS calculus ....

Jean-Yves Girard. On the unity of logic. Annals of Pure and Applied Logic, 59:201--217, 1993.

....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 correspondence between classical ....

Jean-Yves Girard. On the unity of logic. Annals of Pure and Applied Logic, 59:201--217, 1993.

....system CLV introduced in the next section. 4. 2 CLV : A Two Sided Formulation of Classical Linear Logic with Four Zones In this section we present a formulation of a sequent calculus for linear logic, called CLV very close to Andreoli s dyadic system Sigma 2 [And92] and similar to Girard s LU [Gir93]. This formulation was specifically devised to allow a simple proof of cut elimination. A sequent has the form Psi; Gamma = Delta; Theta which may be interpreted as Psi; Gamma = Delta; Theta in ordinary linear sequent calculus. Thus the outer zones in the sequents represent non linear ....

Jean-Yves Girard. On the unity of logic. Annals of Pure and Applied Logic, 59:201--217, 1993.

....in derivations, thus naturally capturing occurrence conditions. Critical is the elimination of structural rules, both for the informal proofs and their formalizations, which leads us quite naturally to Kleene s sequent system G 3 [Kle52] for intuitionistic and classical logic and a variant of LU [Gir93] for linear logic. These formulations can easily be seen to be equivalent to more traditional sequent calculi. The reader interested in structural cut elimination for intuitionistic, classical, or linear logic, but not in its formalization, should be able to follow this paper by ignoring the ....

....Roorda [Roo91] gives a different proof of cut elimination by generalizing the cut rule to multiple occurrences of modal formulas. The main challenge is to isolate the non linear reasoning and the associated structural rules. Our solution is close to Andreoli s Sigma 2 [And92] and Girard s LU [Gir93] in that we divide a sequent into linear and non linear zones, and that we have several forms of cut. The structural aspects of non linear reasoning are treated in the manner of the earlier sections. This leaves a version of dereliction as the only structural rule, and it can be handled directly ....

Jean-Yves Girard. On the unity of logic. Annals of Pure and Applied Logic, 59:201--217, 1993.

....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 correspondence between classical ....

....As a natural consequence we could complete the reduction rules for v for the first time, which is partly appeared in OS s work[18] Related Works: However at least CBN part of these frameworks should be considered as folklore. Girard already suggest the relation between calculus and ILU in [10]. Also DJS themselves gave a guess in the final remark of [3] and in there introduction of [4] about the relation between LKT, negative fragment of LC and Parigot s CND. Moreover there is a detailed work of Herbelin[13] about term calculus on LJT which is an intuitionistic fragment of LKT. He ....

Jean-Yves Girard. On the unity of logic. Annals of Pure and Applied Logic, 59:201--217, 1993.

....of existing systems were explored in numerous experiments. This hiatus has recently come to a close with implementations of frameworks based on inductive definitions such as FS0 [Fef88, MSB93] and ALF [Mag95] partial inductive definitions [Hal91, Eri93, Eri94] and substructural logics [SH91, Gir93, Mil94, Cer96] There is a different approach to logical frameworks based on equational rather than deductive reasoning. While one can be interpreted in the other without much difficulty, the meta languages based on equational reasoning take a rather different form and we will not discuss them ....

Jean-Yves Girard. On the unity of logic. Annals of Pure and Applied Logic, 59:201--217, 1993.

....in CLL, a system with explicit weakening, contraction, and dereliction for modal formulas. This is achieved by dividing a sequent into linear and non linear zones whose constituents are treated differently in the sequent rules. Examples of such calculi are Andreoli s Sigma 2 [And92] Girard s LU [Gir93], and Hodas Miller s L [HM94, Hod94] We take a two sided version of classical linear logic quite close to Sigma 2 with three rules of Cut as in LU and endow the resulting calculus LV with proof terms. We then prove admissibility of the cut rules in LV by three nested structural inductions. Cut ....

.... j 1 A Phi B j (A NB ) 0 j 9x: A j (8x: A ) 3 LV: Another Sequent Calculus for Linear Logic In this section we present a formulation of a sequent calculus for linear logic, called LV very close to Andreoli s dyadic system Sigma 2 [And92] and similar to Girard s LU [Gir93]. It may also be considered a complete classical analogue of Hodas Miller s L [HM94, Hod94] a formulation of a fragment of intuitionistic linear logic. LV will be amenable to a structural proof of cut elimination following ideas from an analysis of intuitionistic and classical sequent calculi ....

[Article contains additional citation context not shown here]

Jean-Yves Girard. On the unity of logic. Annals of Pure and Applied Logic, 59:201--217, 1993.

....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 soundness and completeness ....

....clear. Because, by the DJS s simulation theorem, LKT (hence t ) inherits the denotation in linear logic s coherent space semantics. 1.3 Related Works Folklore: However, these frameworks should be considered as folklore. Girard has already suggested the relation between calculus and ILU in [7]. In addition, DJS themselves gave an explanation in their introduction of [2] about the relation between the LKT, the negative fragment of the LC, and Parigot s CND. Especially, the second part of our work can be seen as a refinement of the De Groote s work [3] His investigation is about the ....

Jean-Yves Girard. On the unity of logic. Annals of Pure and Applied Logic, 59:201--217, 1993.

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