Girard, J.-Y. 1993. On the unity of logic. Annals of Pure and Applied Logic 59, 201--217.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... et al. 1999] The investigation of fragments of linear logic remains essential, as linear logic has no analogue of an explicitly scoped proof system, and so unlike intuitionistic logic and modal logic must be understood as a refinement of classical logic rather than an extension to it [Girard, 1993]. B, B D, B,D, F, B,D, B#D) F, C, C 3 B,C#D, B#D) F, C,F F, A, A,F 2 F, Figure 4: A goal directed proof in which multiple cases are considered. Each case is displayed in a separate ....

....an articulated SCLI. We represent assumptions as a pair P;G with P encoding the global program and G encoding local program statements; eventually local statements will be processed only in the current segment and then discarded. Compare the similar notation and treatment from [Girard, 1993]. Similarly, we represent goals as a pair D;Q, with Q encoding the restart goals and D encoding the local goals; ultimately, we will also describe inference rules which will discard D between segments. With this representation, principal formulas of logical rules are local formulas, in G or D; so ....

Girard, J.-Y. (1993). On the unity of logic. Annals of Pure and Applied Logic, 59:201--217.

....is purely formal introducing an articulated SCLI. We represent assumptions as a pair P;G with P encoding the global program and G encoding local clauses; eventually local clauses will be processed only in the current segment and then discarded. Compare the similar notation and treatment from [Girard, 1993]. Similarly, we represent goals as a pair D;Q, with Q encoding the restart goals and D encoding the local goals; ultimately, we will also describe inference rules which will discard D between segments. With this representation, principal formulas of logical rules are local formulas, in G or D; so ....

Girard, J.-Y. (1993). On the unity of logic. Annals of Pure and Applied Logic, 59:201--217.

....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 ....

Girard, J.-Y. (1993), On the unity of logic, Ann. Pure Appl. Logic 59, 201#217.

.... logic is not just a refinement of intuitionistic logic, such as ILL: there are expectations that CLL may tell us something fundamental about classical logic as well, indeed, that through linear logic a deep level of analysis may have been reached from which the unity of logic can be appreciated [13]. Therefore the relations between classical and intuitionistic components of CLL deserve careful investigation. A natural points of view to look at this issue is categorical logic. It has been known for years that monoidal closed categories provide a model for 2 intuitionistic linear logic, ....

J-Y. Girard. On the unity of logic. Ann. Pure App. Log. 59, pp.201-217, 1993.

....work on Display Logic, where it serves the same purpose as it does here, viz. to combine logics with different resource management regimes. In [Kracht 93, Wansing 92a] one finds recent applications in the context of modal logic. More recently, the same idea has been introduced in Linear Logic in [Girard 93] Definition 4.2 Multimodal Gentzen calculus: logical rules. Structure terms S : F (S,S) i . R i ] #,B) i # A # # A i B # # B #[A] # C #[ A i B,#) i ] # C [L i ] R i ] B,#) i # A # # B i A # # B #[A] # C #[ #,B i A) i ] # C [L i ] L. i ....

Girard, J.-Y. (1993), `On the unity of logic'. Annals of Pure and Applied Logic 59, 201-- 217.

....classical proofs. The following is a tentative (and incomplete) classification: ffl Algorithm extraction, control operators: Griffin [14] Murthy [22] Krivine [21] de Groote [9] Nakano [23] Hirokawa [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 ....

J.-Y. Girard. On the unity of logic. Ann. Pure Appl. Logic, 59:201--217, 1995.

....proof theory. This paper presents a new syntax for linear logic that resolves the Promotion problem. The new syntax follows naturally from the idea of using patterns in sequents to represent destructors. It is closely related to Girard s Logic of Unity, LU (though without the polarities) [Gir91]. Indeed, the syntax presented here is based on a suggestion from Jean Yves Girard, who pointed out to me that the problems I had noted with the standard syntax are resolved in the syntax of LU. The syntax also bears a passing resemblance to Moggi s calculus for monads [Mog89] The syntax has been ....

....law for monads, while our last equation has nothing to do with the associative law for comonads. However, the associative laws for comonads is important in verifying the coherence of the new syntax. 5 Logic of Unity The system described here is closely related to Girard s Logic of Unity (LU) [Gir91]. Indeed, it was inspired by it: the trick that avoids coherence problems was stolen from LU. To clarify the relation, this section present an appropriately simplified version of LU. Major differences from Girard s LU are that this version is restricted to the intuitionistic fragment, and there ....

J.-Y. Girard, On the unity of logic. Manuscript, 1991.

....is preserved, 2 each exponential in the image is necessary, as it is required by some instance of a contraction or weakening rule somewhere in the proof. We define a notion of lower decoration strategy , which in the case of the neutral fragment of ILU (Intuitionistic fragment of Unified Logic (Girard(1993)) is sub girardian in the sense that the result is a subdecoration of the uniform decoration obtained using Girard s well known translation of intuitionistic into linear logic. In fact, the linear proof we find is an optimal linear version of the original ILU derivation, with essentially the ....

....its linear interpretation it represents a formula that is not (yet) shrieked. The structural rule dL is the equivalent of the linear dereliction rule L. What we obtained is the (cut free) neutral fragment of intuitionistic implicational logic as it appears in Girard s system of Unified Logic (LU, Girard(1993)) We will refer to this fragment here as ILU. By construction its connection with the ( Delta) translation is flawless: Pi; Gamma ) A is derivable if and only if Pi ; Gamma ) A is derivable in linear logic. Moreover for a ILU derivation Girard s translation determines a ....

Girard, J.-Y. (1993). On the unity of logic. Annals of Pure and Applied Logic, 59:201--217.

.... the design of a noetherian and confluent normalization for LK 2 (that is, classical second order predicate logic presented as a sequent calculus) The method we present is powerful: since it allows us to recover as fragments formalisms as seemingly different as Girard s LC and Parigot s , FD ([10, 12, 29, 33]) delineates other viable systems as well, and gives means to extend the Krivine Leivant paradigm of programming with proofs ( 24, 25] to classical logic; it is painless: since we reduce strong normalization and confluence to the same properties for linear logic (for non additive proof nets, ....

....a correct converter, converting classical integers to linear integers. The reader should compare this to the LJ based approaches to this same conversion problem in [22, 23, 31] Another possibility would be the use of a calculus enabling the fusion of intuitionistic and classical reasoning (cf. [12]) as in [26] Quite surprisingly we even get a stronger result. Suppose one adds to classical logic the juxtaposition rule, then the same analysis shows that 0 will now be a juxtaposition of proofs 0i of ) N 0 (f(t) all containing an equational proof of S ] 0i 0 = f(t) Granted the ....

Girard, J.-Y. (1993) On the unity of logic. Annals of Pure and Applied Logic, 59:201--217.

....and our previous work. The extant and new encodings of the calculus can be obtained by encoding the calculus in linear logic and then translating linear logic into graphs. In turn, a further change of formal system illuminates the encoding of linear logic: a variant of Girard s unified logic [10] admits a simpler, more regular treatment than linear logic. This graph system may also serve as a basis for efficient implementations of the various programming languages inspired by linear logic in recent years (e.g. 11, 12, 3] In fact, work on the calculus [13] suggests an optimality ....

J.-Y. Girard, "On the unity of logic," tech. rep., June 1991. INRIA Report 1467.

....work on Display Logic, where it serves the same purpose as it does here, viz. to combine logics with different resource management regimes. In [Kracht 93, Wansing 92a] one finds recent applications in the context of modal logic. More recently, the same idea has been introduced in Linear Logic in [Girard 93] Definition 4.2. Multimodal Gentzen calculus: logical rules. Structure terms S : F j (S; S) i . R= i ] Gamma; B) i ) A Gamma ) A= i B Gamma ) B Delta[A] C Delta[ A= i B; Gamma) i ] C [L= i ] Rn i ] B; Gamma) i ) A Gamma ) Bn i A Gamma ) B Delta[A] C ....

Girard, J.-Y. (1993), `On the unity of logic'. Annals of Pure and Applied Logic 59, 201--217.

....and linear logic into one setting. In a linear sequent calculus it is natural to model context by dividing the terms to the left of the turnstile into a classical portion followed by a linear portion. Contextual categories are the proof theory of this fragment of Girard s unified logic [G93] and, indeed, we are convinced this is the basic building block of the categorical proof theory for this logic. The second connection, which we do not explore in this paper but rather leave to its sequel, is to the Action Calculi of Robin Milner. Strength is of course a pervasive notion in ....

....reader Instead we shall now consider coherence for contextual categories, via a study of proof circuits for these categories. 2 4 The context calculus It must be obvious by now that the structures we have been studying are very similar to Girard s approach to unifying classical and linear logic [G93]: context variables are classical and general variables are linear . This is represented by a morphism C ff G Gamma A; the position C before the ff is classical , while the position G after ff is linear . In the remainder of this paper we shall develop this idea, representing it by sequents ....

Girard, J.-Y. "On the unity of logic", Annals of Pure and Applied Logic 59 (1993) 201--217.

....be constrained to satisfy (8x; y; z 2 W ) R c xyz ) R c xzy. A ffi c B B ffi c A Gamma[ Delta 2 ; Delta 1 ) c ] A Gamma[ Delta 1 ; Delta 2 ) c ] A [P] 2 See [30, 49] for recent applications in a modal setting. More recently, the same idea has been introduced in Linear Logic in [18]. Multimodal communication. What we have done so far is simply put together the individual systems discussed before in isolation. This is enough to gain combined access to the inferential capacities of the component logics, and one avoids the unpleasant collapse into the least discriminating ....

Girard, J.-Y., (1993) `On the unity of logic'. Annals of Pure and Applied Logic, 59, 201--217.

....in the treatment of structural rules. Like its classical version LKT, it has been considered by Danos et al. for its good behaviour w.r.t. embedding into linear logic. The calculus LJT appears also as a fragment of ILU, the intuitionistic neutral fragment of unified logic described by Girard in [6]. The calculus ILU is itself a form of LJ constrained with a stoup, for which Girard pointed out that the formula [in the stoup] if there is one) is the analogue of the familiar head variable for typed calculi . Recently, Mints defined in [9] a notion of normal form for cut free proofs of LJ ....

J-Y. Girard: "On the Unity of Logic", Annals of Pure and Applied Logic, Vol 59, 1993, pp 201-217.

....linear logic into one setting. In a linear sequent calculus it is natural to model context by dividing the terms to the left of the turnstile into a classical portion followed by a linear portion. Context categories are the categorical proof theory of this fragment of Girard s unified logic [G93]. Many of the significant features of the semantics of unified logic may be studied in this fragment alone, and although we press on to a more symmetric system, we suggest some attention is warranted for this more modest fragment. Indeed, it does provide the basic building block of the categorical ....

....as their name suggests, correspond to cut rules in the sequent calculus we call the context calculus and which is the essence of Girard s unified logic. This suggests that there should be a corresponding nonsense cut rule missing from that system. This is indeed the case: to quote Girard [G93] there is no possibility of defining a cut between two occurrences of A with a classical maintenance in unified logic. 2 To extract the coherence conditions of a bicontext category it is expedient to develop in parallel the notion of bistrength in bicontextual bimodules or, more specifically, ....

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Girard, J.-Y. "On the unity of logic", Annals of Pure and Applied Logic 59 (1993) 201--217.

....a reading. The sharing of resources interpretation [20, 19] however, provides a reading of connectives consistent with the existence of Gamma typed functions using their arguments multiple times. Bunches are not similar to the zoned contexts used in some presentations of linear logic (e.g. [12, 5]) In particular, and , can be nested in a bunch, and ; just like , is internalized as a connective in the logic, while the stoup ; does not internalize in this way. Linear logic admits a decomposition of OE as OE Gammaffi ; BI admits no such decomposition. In common with R, BI ....

J.-Y. Girard. On the unity of logic. Ann. Pure Appl. Logic, 59:201--217, 1993.

.... and classical linear logic [1] Other models have been discussed by Chirimar, Gunter, and Riecke [3] Lincoln and Mitchell [13] Reddy [16] and Wadler [21, 22] The particular formulation of linear logic presented here is based on Girard s Logic of Unity, a refinement of linear logic [7]. This overcomes some technical problems with other presentations of linear logic, some of which are discussed by Benton, Bierman, de Paiva, and Hyland [2] and Wadler [23, 24] Much of the insight for this work comes from categorical models of linear logic [19, 15] The particular system ....

J.-Y. Girard, On the unity of logic. Manuscript, 1991.

....rule are such that their premise is the conclusion of an instance of the BC rule. Such a proof can be converted to a proof in L 00 by reversing the conversion mentioned for the first case. Girard has pointed out to us that this proposition should be provable directly within his LU proof system (Girard, 1991). A consequence of this proposition is that I 0 proofs involving Horn clauses or hereditary Harrop formulas are essentially the same as the L 00 proofs of their translations. This suggests how to design the concrete syntax of a linear logic programming language so that the interpretation of ....

Girard, J.-Y. (1991). On the unity of logic. Tech. rep. 26, Universit'e Paris VII.

....and approximately in [22] where Howard attributes the idea to Curry) Howard s treatment works well just for the implicational fragment; when other connectives are added, the treatment of implication needs modification. Herbelin s [20, 21] attributes the idea to Danos et al. [67] and to Girard [17]; also it appears in [18] where Girard suggests the creation of an improvement LI of the familiar intuitionistic sequent calculus LJ along these lines. In formal treatments (e.g. 28, 30] of logic programming [27] calculi with a formula in a distinguished place are well known; Pfenning [30] ....

Girard, J.-Y.: "On the unity of logic", Annals of Pureand Applied Logic 59 (1993), 201--217.

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Girard, J.-Y. 1993. On the unity of logic. Annals of Pure and Applied Logic 59, 201--217.

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J.-Y. Girard, On the unity of logic. Manuscript, 1991.

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GIRARD J.-Y.(1993), On the unity of logic, Annals of Pure and Applied Logic, 59, 201--217.

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