N. Benton, G. Bierman, V. de Paiva, M. Hyland. Linear -Calculus and Categorical Models Revisited, Preprint, Comp. Lab., Univ. of Cambridge.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... It has been known for years that monoidal closed categories provide a model for 2 intuitionistic linear logic, though a fully adequate formulation of the syntax and of the categorical semantics of ILL especially with respect to the exponentials, has required considerable subtlety and effort [4, 5, 6]. It is also well known that autonomous categories give a model for classical linear logic [3] The appendix to [2] provides a method, due to Barr s student Chu, to construct autonomous categories starting from monoidal closed ones. In our proof theoretic investigation we encounter a special ....

....1. writing formulas in negation normal form , using De Morgan laws; 2. writing proofs in the sequent calculus with right hand sequents only; 3. finally, translating them into proof nets, i.e. forgetting the context . 3 The Chu construction We follow the categorical semantics for IMLL in [5]: Theorem 3. Let A be the free autonomous category on a set of objects fP , P 0 , g and let C be the symmetric monoidal closed category with products, free on the set of objects fPO , P I , P 0 O , P 0 I , g (a pair PO , P I for each atom P of A) i) We can give C Theta C op ....

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N. Benton, G. Bierman, V. de Paiva, M. Hyland. Linear -Calculus and Categorical Models Revisited, Preprint, Comp. Lab., Univ. of Cambridge.

....scheme can be used for other combinations, such as for non symmetric monoidal structures, and even more than two. The ff calculus contains simply typed calculus and multiplicative, intuitionistic linear calculus as subsystems. Various forms of linear calculus also contain the two subsystems [1, 5, 24, 3], but ff s approach is rather different. Where linear logic uses a modality (or sometimes distinct zones in contexts) to control access to the structurals, in ff access is governed by the two means of combination. The difference can be stated crisply in terms of categorical models. In models ....

....zones in contexts) to control access to the structurals, in ff access is governed by the two means of combination. The difference can be stated crisply in terms of categorical models. In models of linear logic two closed categories are involved, where one is often presented as a Kleisli category [5, 4, 3]. For instance, in the original coherence space model there are indeed two function types, but Gammaffi is closed structure in the category of linear maps, while the additive , which can be represented as A Gammaffi B, is closed for the category of stable maps. In contrast, in a doubly closed ....

P.N. Benton, G.M. Bierman, V.C.V. de Paiva, and J.M.E. Hyland. Linear -calculus and categorical models revisited. In E. Borger et al., editors, Proceedings of the Sixth Workshop on Computer Science Logic, volume 702 of Lecture Notes in Computer Science, pages 61--84. Springer-Verlag, Berlin, 1992.

....M f : Gamma] M Gamma [ OE] M (where is Kleene equality) Example 2. 7 (linear logic) Let L be the ( Omega ; Gammaffi ) fragment of linear propositional logic, with the empty context denoted by hi and with proof objects represented as terms of the simply typed linear calculus [16]. Let M be a symmetric monoidal closed category (SMCC) Then the following interpretation determines a type theoretic model: hi] M = I [ OE Omega ] M = OE] M Omega [ M [ OE Gammaffi ] M = M [ OE] M with j= defined by M j= Phi : OE) Gamma] iff there is an f ....

N. Benton, G. Bierman, V. de Paiva, and M. Hyland. Linear -calculus and categorical models revisited. In 6th Workshop CSL'92, LNCS 702, pages 61--84, San Miniato, Italy, 1992.

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