Categories for computation in context and unified logic

In this paper we introduce context categories to provide a framework for computations in context. The structure also provides a basis for developing the categorical proof theory of Girard's unified logic. A key feature of this logic is the separation of sequents into classical and linear zones. These zones may be modelled categorically as a context/computation separation given by a fibration. The perspective leads to an analysis of the exponential structure of linear logic using strength (or context) as the primitive notion. Context is represented by the classical zone on the left of the turnstile in unified logic. To model the classical zone to the right of the turnstile, it is necessary to introduce a notion of cocontext. This results in a fibrational fork over context and cocontext and leads to the notion of a bicontext category. When we add the structure of a weakly distributive category in a suitably fork fibred manner, we obtain a model for a core fragment of unified logic. We describe the sequent calculus for the fragment of unified logic modelled by context categories; cut elimination holds for this fragment. Categorical cut elimination also is valid, but a proof of this fact is deferred to a sequel.

[1]  Valeria C V de Paiva,et al.  Term Assignment for Intuitionistic Linear Logic , 1992 .

[2]  R. A. G. Seely,et al.  Linear Logic, -Autonomous Categories and Cofree Coalgebras , 1989 .

[3]  Edmund Robinson,et al.  Premonoidal categories and notions of computation , 1997, Mathematical Structures in Computer Science.

[4]  J. Lambek,et al.  Introduction to higher order categorical logic , 1986 .

[5]  John Cartmell,et al.  Generalised algebraic theories and contextual categories , 1986, Ann. Pure Appl. Log..

[6]  Richard Blute,et al.  Linear Logic, Coherence, and Dinaturality , 1993, Theor. Comput. Sci..

[7]  Jean-Yves Girard,et al.  On the Unity of Logic , 1993, Ann. Pure Appl. Log..

[8]  J. Robin B. Cockett,et al.  ! and ? – Storage as tensorial strength , 1996, Mathematical Structures in Computer Science.

[9]  R. A. G. Seely,et al.  Weakly distributive categories , 1997 .

[10]  Robin Milner Action Calculi, or Syntactic Action Structures , 1993, MFCS.

[11]  R. Seely,et al.  Proof theory for full intuitionistic linear logic, bilinear logic, and MIX categories. , 1997 .

[12]  A. Pitts INTRODUCTION TO HIGHER ORDER CATEGORICAL LOGIC (Cambridge Studies in Advanced Mathematics 7) , 1987 .

[13]  R. Blute,et al.  Natural deduction and coherence for weakly distributive categories , 1996 .

[14]  Sally Popkorn,et al.  A Handbook of Categorical Algebra , 2009 .

[15]  Eugenio Moggi,et al.  Computational lambda-calculus and monads , 1989, [1989] Proceedings. Fourth Annual Symposium on Logic in Computer Science.

[16]  Robert Paré,et al.  Monoidal categories with natural numbers object , 1989, Stud Logica.

[17]  Gavin M. Bierman What is a Categorical Model of Intuitionistic Linear Logic? , 1995, TLCA.

[18]  Dusko Pavlovic Categorical logic of Names and Abstraction in Action Calculi , 1997, Math. Struct. Comput. Sci..