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.

A system of interaction and structure V: the exponentials and splitting

“…In the case of process algebras, this is almost obvious, given that NEL is Turing-complete and that exponentials have been justified since their first introduction as ways of controlling resources (i.e., messages, processes). The logic BV has also been used to define BVcategories [BPS09] for providing an axiomatic description of probabilistic coherence spaces [Gir03].…”

“…Recently, BV has been employed to axiomatise causal quantum computation better than linear logic does [BPS08], and it gave rise to a new class of categorical models [BPS09].…”

A Logical Basis for Quantum Evolution and Entanglement

“…While Girard demonstrates the * -autonomous structure, the paper [6] shows that the category properly models the additional noncommutative tensor of BV. We note that the paper [11] also has a notion of quantum coherence space, where analogous structure can be found.…”

“…The structure of BV leads to interesting category-theoretic considerations [6]. One must find a category with the following structure:…”

A system of interaction and structure IV

“…In the case of process algebras, this is almost obvious, given that NEL is Turing-complete and that exponentials have been justified since their first introduction as ways of controlling resources (i.e., messages, processes). The logic BV has also been used to define BVcategories [Blute et al 2009] for providing an axiomatic description of probabilistic coherence spaces [Girard 2003].…”