Deep Inference and Probabilistic Coherence Spaces

                                                 Rick Blute

           (joint work with Prakash Panangaden and Sergey Slavnov.)

We propose a definition of categorical model of the deep inference system BV. BV was designed
as a logical framework for the consideration of Retoré's extension
of multiplicative linear logic to include a self-dual  noncommutative connective.

Our definition is based on Cockett and Seely's notion of a linear functor.
A BV-category is a *-autonomous category 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 the isomorphisms required of the theory.

We show that coherence spaces, with Retoré's noncommutative tensor, form a model.
We then consider Girard's category of probabilistic coherence spaces, and show that it
contains an additional monoidal structure making it a BV-category.

We discuss possible applications of such categories to  the discrete quantum causal dynamics
 of Blute-Ivanov-Panangaden. The latter is ongoing work with Guglielmi, Panangaden,
and  Strassburger.