Deep Inference and Probabilistic Coherence Spaces

Abstract

This paper proposes a definition of categorical model of the deep inference system BV, defined by Guglielmi. Deep inference introduces the idea of performing a deduction in the interior of a formula, at any depth. Traditional sequent calculus rules only see the roots of formulae. However in these new systems, one can rewrite at any position in the formula tree. Deep inference in particular allows the syntactic description of logics for which there is no sequent calculus. One such system is BV, which extends linear logic to include a noncommutative self-dual connective. This is the logic our paper proposes to model. Our definition is based on the notion of a linear functor, due to Cockett and Seely. A BV-category is a linearly distributive category, possibly with negation, with an additional tensor product which, when viewed as a bivariant functor, is linear with a degeneracy condition. We show that this simple definition implies all of the key isomorphisms of the theory. We consider Girard’s category of probabilistic coherence spaces and show that it contains a self-dual monoidal structure in addition to the *-autonomous structure exhibited by Girard. This structure makes the category a BV-category. We believe this structure is also of independent interest, as well-behaved noncommutative operators generally are.