Convexity and Order in Probabilistic Call-by-Name FPC

TL;DR

This paper explores Kegelspitzen as a mathematical framework modeling probabilistic computation, demonstrating their role as a denotational model for a probabilistic functional programming language and discussing recursive types.

Contribution

It establishes Kegelspitzen as a categorical model for probabilistic programming and provides insights into their interpretation of recursive types.

Findings

Kegelspitzen model stochastic matrices within domain categories

They form a denotational semantics for pPCF, a probabilistic language

Discussion on interpreting recursive types in this framework

Abstract

Kegelspitzen are mathematical structures coined by Keimel and Plotkin, in order to encompass the structure of a convex set and the structure of a dcpo. In this paper, we ask ourselves what are Kegelspitzen the model of. We adopt a categorical viewpoint and show that Kegelspitzen model stochastic matrices onto a category of domains. Consequently, Kegelspitzen form a denotational model of pPCF, an abstract functional programming language for probabilistic computing. We conclude the present work with a discussion of the interpretation of (probabilistic) recursive types, which are types for entities which might contain other entities of the same type, such as lists and trees.