Interpreting De Finetti's theorem in the Category of Integrable Cones (long version)

TL;DR

This paper connects De Finetti's theorem in category theory with the free exponential construction in Linear Logic, using probabilistic semantics to characterize total elements in probabilistic coherence spaces.

Contribution

It formalizes the relationship between stochastic kernels and integrable cones, linking probabilistic semantics with Linear Logic constructions.

Findings

Established a formal connection between two probabilistic semantics frameworks.

Characterized total elements of the probabilistic coherence space !Bool.

Linked De Finetti's theorem with the free exponential in Linear Logic.

Abstract

We establish a connection between two results in the literature on probabilistic semantics: a formulation of De Finetti's theorem in the language of category theory due to Jacobs and Staton, and the generic construction of the free exponential of Linear Logic by Melli\`es et al, that has been instantiated in the model of probabilistic coherence spaces by Crubill\'e et al. The structural proximity of these two constructions is manifest, but making this connection formal requires technical developments on the relationship between the category of stochastic kernels and the category of integrable cones, two well-known categories in probabilistic semantics. We then use this connection to give a characterization of the total elements of the probabilistic coherence space !Bool.