Valeria de Paiva. A Dialectica-like model of linear logic. In Pitt et al. [23], pages 341--356.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to reduce the logic to classical logic. In these sections I describe and comment on several approaches to avoiding this problem. The final two sections are about connections between game semantics and two other semantical systems for linear logic, namely de Paiva s Dialectica like semantics [9] and Girard s coherence spaces [12] Throughout this paper, we shall consider only propositional linear logic. Much 1 Full version of an invited paper presented at the 3rd Workshop on Logic, Language, Information and Computation (WoLLIC 96) May 8 10, 1996, Salvador (Bahia) Brazil, organised ....

....of game semantics to mediate applications of linear (or a#ne) logic to actual computing. 11 Dialectica like semantics In this section and the next, we briefly point out connections between game semantics and two other approaches to modeling linear logic. The Dialectica like model of de Paiva [9] is based on quite a general category, but we concentrate here on the case where that category is the category of sets. With this specialization, de Paiva s model interprets formulas as pairs of sets equipped with a binary relation between them, i.e. A ,A ,A)w hereA#A A . These are the ....

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Valeria de Paiva. A Dialectica-like model of linear logic. In Pitt et al. [23], pages 341--356.

.... A [26, 88, 18, 19] Other versions of game semantics are given by Abramsky and Jagadeesan [2] and by Lafont and Streicher [65] Event spaces, which come about from Pratt s work in semantics of concurrency, also provide models for certain linear logic proofs [82] Models investigated by de Paiva [39] are motivated by important proof theoretic transformations. A mathematical model for the linear logic provability relation is given by phase spaces, discussed by Girard [45] and by Avron [15, 16] Kripke style models are investigated by Allwein and Dunn [7] A mathematical structure underlying ....

V.C.V. de Paiva. A dialectica-like model of linear logic. In Category Theory and Computer Science, pages 341--356. Springer LNCS 389, September 1989.

.... spaces are the case V = Set of the construction described by Po Hsiang Chu in the appendix of Barr s book on autonomous (i.e. self dual closed) categories [Bar79] Chu s construction takes a closed monoidal category V with pullbacks and completes it to a self dual category Chu(V; k) De Paiva [dP89a, dP89b] and Brown and Gurr [BG90, BGdP91] apply the Chu construction to respectively a version of Godel s Dialectica and Petri nets. Lafont and Streicher study Chu spaces over K, which they call games [LS91] the term Chu construction had been around previously, and the name Chu space for V = Set was ....

V. de Paiva. A dialectica-like model of linear logic. In Proc. Conf. on Category Theory and Computer Science, LNCS 389, pages 341--356, Manchester, September 1989. Springer-Verlag.

....in ILL #2 . For evidence on the naturality of the proposal to split into two operators we mention that one finds in V. Pratt s Chu spaces [15] two maximal solutions for a storage operator and Girard s operator emerges as the set theoretic intersection of the two (V. Pratt [16] De Paiva s [14] construction of the operator also goes through a preliminary construction of two auxiliary comonads S and T related by a natural transformation : ST TS which allows for the construction of a composite comonad , modeling the storage operator of Linear Logic (it should be pointed out, however, ....

....operators # and 2, one controling contraction, the other controling weakening. From our standpoint, however, and as in Jacobs s report [13] the critical issue is the interaction of the two, modeling the crucial property # 2 = 2 #, which is reminiscent of the situation in both de Paiva s model [14] and the Chu space semantics. It would be interersting if someone could show how this interaction condition can be modeled in the setting of either [21] or [3] We have kept things simple enough in that we avoided considering in the intuitionistic setting the order duals and 3 of # and 2 ....

V. C. V. de Paiva, "A Dialectica-like Model of Linear Logic", Lecture Notes in Computer Science, vol 389, Springer-Verlag.

....been studied by Dimitrovici et al. in [20, 19, 21, 18] These results have then been applied by Hummert in his dissertation [42] to define a notion of net module. The most novel approach is the one taken by Carolyn Brown and Doug Gurr. It is based on the Dialectica Categories of Valeria de Paiva [15, 16] which are a category theoretic model of Godels Dialectica interpretation of higher order arithmetic [37] The constructs available in these categories give rise to very interesting constructions on nets that have a connection with linear logic. Unfortunately the approach currently only works ....

de Paiva, V. C. Dialectica-like model of linear logic. Category Theory and Computer Science. Lecture Notes in Computer Science 389. Springer Verlag, Berlin, 1989, pp. 341--356.

....then proposed the Chu construction as a means of producing constructive models of linear logic [Bar91] The subsequent history of Chu spaces has been one of successive weakenings of the enriching category V . Barr and Chu took V to be any symmetric monoidal closed category with pullbacks, de Paiva [dP89] and Brown and Gurr [BG90] restricted to order enrichment, and finally Lafont and Streicher banished enrichment altogether by taking V = Set [LS91] and calling the resulting objects games after von Neumann and Morgenstern. Chu spaces are indeed games, and moreover of the asynchronous kind, ideally ....

V. de Paiva. A dialectica-like model of linear logic. In Proc. Conf. on Category Theory and Computer Science, LNCS 389, pages 341--356, Manchester, September 1989. Springer-Verlag.

....category to a weakly distributive category by passing to the Kleisli category of the exception monad; for example, although Sets is not weakly distributive, PointedSets is weakly distributive, with the usual sum and product. 1 The system FILL (full intuitionistic linear logic) of de Paiva [dP89] amounts to having just the second of these (families of) maps. From the autonomous category viewpoint, these are the more natural maps, as they correspond to evaluations. The symmetry of the autonomous viewpoint then suggests the first (family of) maps. One point must be made about the ....

de Paiva V.C.V. A Dialectica-like model of linear logic. in D.H.Pitt et al., eds., Category Theory and Computer Science, Lecture Notes in Computer Science 389, Springer-Verlag, Berlin, Heidelberg, New York, 1989.

....ffi GammaA is then (f 0 ; f ) as an application of commutativity. Note this is not the same thing as evidence against A GammaffiB, i.e. for A Omega B , namely a pair (a; x) consisting of evidence for A and evidence against B. With Godel s Dialectica interpretation and the work of de Paiva [dP89a, dP89b] in mind, we call this variant of the simply typed calculus the dialectic calculus. The two language features distinguishing it from the simply typed calculus, taken to have the usual exponentiation operator and also Theta for convenience, are a second implication , and the notion ....

V. de Paiva. A dialectica-like model of linear logic. In Proc. Conf. on Category Theory and Computer Science, volume 389 of Lecture Notes in Computer Science, pages 341--356, Manchester, September 1989. Springer-Verlag.

....of the category Chu(Set; 2) of Chu spaces over the twoelement set 2 = f0; 1g. Chu spaces have an equally evident duality obtained by transposition, which has the side effect of reversing the direction of the Chu transforms. As noticed by Barr [Bar91] and further developed by several authors [dP89, BG90, LS91, BGdP91, Pra93b], Chu spaces provide a straightforward interpretation of full linear logic; the fact that Set (as well as Pos and other even larger cartesian closed categories) is comonadic (cotripleable) in Chu gives a straightforward interpretation of Girard s bang operation A as the functor of that comonad. ....

V. de Paiva. A dialectica-like model of linear logic. In Proc. Conf. on Category Theory and Computer Science, LNCS 389, pages 341--356, Manchester, September 1989. Springer-Verlag.

....machine or signals over an electric wire: they are created and immediately consumed. An exponential formula corresponds to a datum stored in memory. With this interpretation (and with a bottom up reading) the first three rules above can be 1 de Paiva considers non involutive negation in [51, 52]. thought of as the actions of duplication, discarding and reading respectively. Promotion corresponds to storing a datum which is produced only from stored data (the before Delta play the role of for formulas on the right side, which is justified by the duality of and ) It can be ....

....categorial spirit , as early as 1975. Categorial notions (like adjoints, monads, Kleisli categories) give useful insights for the design of logics and interpretations, as well as programming languages and abstract machines. Another semantics in this class are dePaiva s Dialectica categories [51, 52]. A recent new development which has important applications to models of concurrency based on LL are the game semantics [34, 103, 2] A LL proof is interpreted as a history independent winning strategy in a two opponent zero sum game. Something which has been long missing for LL is the ....

V. de Paiva. A Dialectica-like model of linear logic. In D. Pitt et al., editors, Category Theory and Computer Science, number 389 in LNCS, pages 313--340, Manchester, Sept. 1989.

....for the appropriate spaces when referring to the units. autonomous, that is to say, symmetric monoidal closed (with or without products and coproducts, depending on whether or not the additives are wanted) There is an intermediate notion, full intuitionistic linear logic due to de Paiva [dP89], in which the morphism A Gamma A need not be an isomorphism. And as mentioned above, there is the notion of weakly distributive category [CS91, BCST] where negation and internal hom are not required. One classically important class of autonomous categories are the compact categories ....

de Paiva, V.C.V. "A Dialectica-like model of linear logic", in D.H. Pitt et al., eds., Category Theory and Computer Science, Lecture Notes in Computer Science 389, Springer -Verlag, Berlin, Heidelberg, New York, 1989.

....Valeria.Paiva cl.cam.ac.uk PUC Rio.MCC21 95 Abstract: Reddy [13] introduced an extended intuitionistic linear calculus to model some features of state manipulation. His calculus LLMS for Linear Logic Model of State includes the connective before and an its associated modality y. De Paiva [5] presents a (collection of) dialectica categorical models for Classical Linear Logic, the categories GC. These categories contain an extra tensor product functor fi and a comonad structure corresponding to a modality related to it. It is surprising that these works arising from completely ....

....Models and Concurrency. Resumo: Reddy [13] apresenta um c alculo linear intuicionista para modelar alguns aspectos de manipulac ao de estados. Seu c alculo, chamado Linear Logic Model of State (LLMS) inclui o conectivo before e uma modalidade y, associada a este conectivo. De Paiva [5] apresenta uma colec ao de modelos categ oricos dial eticos para a L ogica Linear Cl assica em termos das categorias GC. Estas categorias incluem, al em do produto tensorial Omega e a conjunc ao padr ao , um funtor produto fi e uma estrutura de comonada para uma modalidade relativa a este ....

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V. C. V. de Paiva. A dialectica-like model of linear logic. In Category Theory in Computer Science, volume 389 of LNCS, pages 341--356, September 1989.

....model for LLMS c , a dialectica style one. Dialectica models were introduced by de Paiva [3] as a model for the intuitionistic fragment of Linear Logic (the motivation for the model comes from Godel s Dialectica Interpretation, hence the name) Later, de Paiva introduced the categories GC [4, 5], a different kind of dialectica model, which following a suggestion of Girard model the whole of Linear Logic. Actually the categories GC have some extra structure (an extra tensor product and an extra modality, associated with this tensor product) and hence can be seen as a model of LLMS c . The ....

....interpretation, because we can establish some relations like [ A] A fi B] which do not represent theorems of LLMS c . 4 A Categorical Model for LLMS c To characterize LLMS c categorically, we must consider an instantiation G of the (symmetric monoidal closed) categories GC, developed in [5]. As usual, the interpretation of the formulae is given by objects of the category G and proofs of LLMS c are interpreted by morphisms of G. In the most general case a GC like category is obtained from a symmetric monoidal closed category C with products, and satisfying some other conditions ....

V. C. V. de Paiva. A dialectica-like model of linear logic. In Category Theory in Computer Science, Volume 389 of LNCS, pages 341--356, September 1989.

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Soc. V. de Paiva (1989), A Dialectica-like Model of Linear Logic. In Proceedings Conf. on Category Theory and Computer Science, Manchester, Springer-Verlag, Berlin, Heidelberg, New York.

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