Convex Biproducts, Stochastic Matrices and Tape Diagrams

2026-07-27

AI Summary

This work proposes a unified framework for modeling convex combinations and linear structures in probabilistic computation. It introduces, for the first time, the notion of a convex biproduct category, which restricts the linear combinations of traditional biproduct categories to convex combinations, thereby naturally yielding an algebraic structure centered on stochastic (or sub-stochastic) matrices. The framework establishes a strict isomorphism between convex biproduct categories and probabilistic string diagrams, enabling a complete axiomatization of probabilistic Boolean circuits. By integrating category theory, compact closed categories, convex algebraic structures, and diagrammatic reasoning, this study provides a refined and formally rigorous algebraic and compositional foundation for probabilistic computation.

Abstract

Categories with finite biproducts play a central role in category theory, providing an abstract setting in which additive and linear structures can be studied uniformly. In this paper, we introduce categories with \emph{convex} biproducts, which intuitively restrict the linear structures to convex ones. We show that, whereas categories with finite biproducts give rise to a matrix calculus based on arbitrary linear combinations, convex biproduct categories instead induce a matrix calculus based on stochastic (more generally, substochastic) matrices. This perspective yields a refined algebraic and compositional framework tailored to probabilistic settings. We exploit this connection to establish an isomorphism that underpins probabilistic tape diagrams, a graphical formalism for bimonoidal (also known as rig) categories, and we demonstrate its effectiveness by providing a complete axiomatisation of probabilistic Boolean circuits.

Problem

• convex biproducts
• stochastic matrices
• probabilistic tape diagrams
• category theory
• bimonoidal categories

Innovation

• convex biproducts
• stochastic matrices
• tape diagrams
• probabilistic Boolean circuits
• bimonoidal categories