J. Robin B. Cockett and Robert A.G. Seely, Proof theory for full intuitionistic linear logic, bilinear logic and MIX categories, Theory and Applications of Categories 3 (1997), 85--131.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....[8] in order to formulate the rules for linear implication, which makes it difficult to understand the meanings of the connectives in isolation. On the other hand, multiplicative disjunction in FILL seems closer to its classical counterpart; furthermore, FILL has a clear categorical semantics [10] that we have not yet explored for JILL. Related structural proofs of cut elimination have appeared for intuitionistic and classical logic [22] classical linear logic in an unpublished note [21] and ordered logic [25] but these did not incorporate possibility and related connectives ( #) ....

J. Robin B. Cockett and R.A.G. Seely. Proof theory for full intuitionistic linear logic, bilinear logic and MIX categories. Theory and Applications of Categories, 3:85--131, 1997.

....If normal natural deductions do not provide a good proof theoretic semantics for ILL, then what does The first thought is that a version of proof nets might provide the solution, but we know of no satisfactory treatment of proof nets for ILL. Treatments include [Lam94] BCST96] BCS96] [CS97]) We do think that a study of the syntactic system ILLF, along with the categorical semantics for ILL might suggest some suitable system, but this is pure speculation. Finally we should ask whether the problem is that ILL is not a sensible logic Perhaps we should consider a larger fragment, or ....

J. R. B. Cockett and R. A. G. Seely. Proof Theory for Full Intuitionistic Linear Logic, Bilinear Logic, and MIX Categories. Theory and Applications of Categories, 3(5):85--131, 1997.

....X, A # B, Y [#] # X, A # B, Y # X, A, B [ # X, A B # t [t] # X [f] # X, f # X, # [#] # X, A [ # X, A # X, A [ # X, A # X [K ] # X, A # X, A, A [WI ] # X, A 2.9.2 Proof Nets [ This section must be added. Relevant citations will be from among [26, 43, 57, 66, 107, 117, 119]] 2.10 Curry Howard Some logicians have found that it is possible to analyse proofs more closely by giving them names. After all, if proofs are first class entities, we will be better off if we can distinguish different proofs. I can illustrate this by looking at an example from intuitionistic ....

J. R. B. COCKETT AND R. A. G. SEELY. "Proof Theory for full intuitionistic linear logic, bilinear logic, and MIX categories". Theory and Applications of categories, 3(5):85--131, 1997. Available from ftp://triples.math.mcgill.ca/pub/rags/nets/fill.ps.gz.

....by exploiting a notion of path in a net. His paper shows that sequent proofs can be represented as two sided proofnets. He also proves a sequentialisation theorem saying that each such proofnet corresponds to a set of sequent proofs. No proof of cut elimination is given. Also Cockett and Seely [CS96] consider the proof theory of FILL (again omitting exponentials) Their goal was to obtain a categorical coherence result stating that a category generated by formulae and (equivalence classes of) proofnets is a free category in an appropriate sense. The FILL condition on their notion of proofnets ....

J. R. B. Cockett and R. A. G. Seely. Proof theory for full intuitionistic linear logic, bilinear logic, and MIX categories. Manuscript, 1996.

....the categorical aspects of this work, some discussion is necessary so as to make the philosophical basis for the logics intelligible. The setting we propose to use corresponds to the tensor par fragment of linear logic, whose categorical semantics lies in linearly distributive categories [7, 10, 13, 14, 15]. Linearly distributive categories were originally introduced by the latter two authors [13] with the idea that the tensor par fragment of linear logic [17] was crucial categorically, and that the remaining structure associated with linear logic could then be added to this basic fragment in a ....

J.R.B. Cockett and R.A.G. Seely (1997) "Proof theory for full intuitionistic linear logic, bilinear logic, and MIX categories", Theory and Applications of Categories 3 85--131.

....triples) are monoidal functors from a trivial bicategory. Let 1 be the suspension of the one object linearly distributive category (so there is one 0 cell , one 1 cell I , one 2 cell, with trivial tensor and par structure) A linear functor F : 1 B , which we shall call a linear point as in [Cockett Seely 1999] is given by a 0 cell X of B (viz. F ( a pair of 1 cells T ; S (viz. F (I) F (I) respectively) and various 2 cells corresponding to the m s needed to make F monoidal, the n s needed to make F comonoidal, and the s needed to make F linear (De nition 2.4) A moment s glance will show that ....

J.R.B. Cockett and R.A.G. Seely (1997a) Proof theory for full intuitionistic linear logic, bilinear logic, and MIX categories. Theory and Applications of Categories 3 85-131.

....obvious distributivity laws with respect to tensor and par. A word about terminology and notation. The reader will have already noticed that we have adopted the term linearly distributive category for what previously we have called weakly distributive category , continuing the practice begun in [CS97]. This we view as a minor matter. More controversial perhaps is our insistence upon the use of for par and for the coproduct sum . As category theorists we are unrepentant upon this point, and there the matter must rest. 1 Linear functors For the full de nition of a linearly distributive ....

.... autonomous categories We may summarize the preceding discussion as follows. Let sAUT be the 2 category of (commutative) autonomous categories, monoidal functors, and monoidal transformations, and let AUT be (the noncommutative analogue) the category of bilinear categories (as de ned in [CS97] for instance these are just noncommutative autonomous categories) monoidal functors F such that (F (A ) F ( A) and monoidal transformations. Proposition 7 There is an inclusion of 2 categories U : sAUT SLDC, and there is an inclusion of 2 categories U : AUT ....

J.R.B. Cockett and R.A.G. Seely \Proof theory for full intuitionistic linear logic, bilinear logic, and MIX categories", Theory and Applications of Categories 3 (1997) 85-131.

....logic, it is quite productive to ignore the closed structure entirely and instead focus on the interaction between the tensor product and its dual cotensor, par. This was one of the motivations of the latter two authors in introducing linearly distributive categories. In a sequence of papers [CS92,BCST96,BCS96,CS97a,CS97b], it has been amply demonstrated that once one understands the linearly distributive structure, the extension of crucial structural results to autonomy is straightforward. These results are achieved by introducing a graph theoretic language for specifying morphisms which is inspired by proof ....

....we repeat a frequent warning about terminology and notation from previous papers in this series. The reader will have already noticed that we have adopted the term linearly distributive category for what previously we have called weakly distributive category , continuing the practice begun in [CS97a]. More controversial perhaps is our insistence upon the use of for par , preferring for the coproduct sum . We wish to thank the anonymous referee for several helpful remarks and suggestions. 1 The core of a MIX category 1.1 Preliminaries 1.1.1 Linearly distributive categories For the full ....

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J.R.B. Cockett and R.A.G. Seely \Proof theory for full intuitionistic linear logic, bilinear logic, and MIX categories", Theory and Applications of Categories 3 (1997) 85-131.

....as above, however, on the case X = Y = Z to obtain a bicontextual weakly distributive category. It should be pointed out that one can add in a modular fashion the further operators of linear logic to the basic structure of a weakly distributive category. This is described in our previous papers [CS91, BCST, BCS92, CS96]. Thus we may also build, in a modular fashion, on bicontextual weakly distributive categories to interpret other structure associated with linear logic settings. For example, one may add negation to obtain a bicontextual autonomous category by following the recipe in [CS91] Thus, the structure ....

Cockett, J.R.B. and R.A.G. Seely "Proof theory for full intuitionistic linear logic, bilinear logic, and mix categories", preprint, McGill University, 1996.

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J. Robin B. Cockett and Robert A.G. Seely, Proof theory for full intuitionistic linear logic, bilinear logic and MIX categories, Theory and Applications of Categories 3 (1997), 85--131.

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J. Robin B. Cockett and Robert A.G. Seely, Proof theory for full intuitionistic linear logic, bilinear logic and MIX categories, Theory and Applications of Categories 3 (1997), 85--131.

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J. Robin B. Cockett and Robert A.G. Seely, Proof theory for full intuitionistic linear logic, bilinear logic and MIX categories, Theory and Applications of Categories, vol. 3 (1997), pp. 85--131.

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J.R.B. Cockett and R.A.G. Seely. Proof theory for full intuitionistic linear logic, bilinear logic, and mix categories. Theory and Applications of Categories, 3(5):85--131, 1997.

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J.R.B. Cockett and R.A.G. Seely. Proof Theory for Full Intuitionistic Linear Logic, and MIX Categories. Theory and Applications of Categories, 3(5):85-- 131, 1997.

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J.R.B. Cockett and R.A.G. Seely. Proof theory for full intuitionistic linear logic, bilinear logic, and mix categories. Theory and Applications of Categories, 3(5):85--131, 1997.

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J. Robin B. Cockett and Robert A.G. Seely, Proof theory for full intuitionistic linear logic, bilinear logic and MIX categories, Theory and Applications of Categories 3 (1997), 85--131.

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J. Robin B. Cockett and Robert A.G. Seely, Proof theory for full intuitionistic linear logic, bilinear logic and MIX categories, Theory and Applications of Categories 3 (1997), 85--131.

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J. Robin B. Cockett and Robert A.G. Seely, Proof theory for full intuitionistic linear logic, bilinear logic and MIX categories, Theory and Applications of Categories, vol. 3 (1997), pp. 85--131.

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