Termination in Convex Sets of Distributions

Abstract

Convex algebras, also called (semi)convex sets, are at the heart of modellingprobabilistic systems including probabilistic automata. Abstractly, they arethe Eilenberg-Moore algebras of the finitely supported distribution monad.Concretely, they have been studied for decades within algebra and convexgeometry. In this paper we study the problem of extending a convex algebra by a singlepoint. Such extensions enable the modelling of termination in probabilisticsystems. We provide a full description of all possible extensions for aparticular class of convex algebras: For a fixed convex subset $D$ of a vectorspace satisfying additional technical condition, we consider the algebra ofconvex subsets of $D$. This class contains the convex algebras of convexsubsets of distributions, modelling (nondeterministic) probabilistic automata.We also provide a full description of all possible extensions for the class offree convex algebras, modelling fully probabilistic systems. Finally, we showthat there is a unique functorial extension, the so-called black-holeextension