Convex Biproducts, Stochastic Matrices and Tape Diagrams
2026-07-27 • Logic in Computer Science
Logic in Computer Science
AI summaryⓘ
The authors introduce a new type of category called 'categories with convex biproducts,' which focus on combining objects in ways that reflect probabilities rather than just general linear combinations. They show that instead of usual matrices, these categories use stochastic matrices, which are important for modeling probabilistic systems. This work helps connect abstract mathematical ideas to practical tools like probabilistic tape diagrams and leads to a clear set of rules for probabilistic Boolean circuits. Overall, the authors provide a framework better suited to reasoning about probabilistic computations.
category theoryfinite biproductsconvex biproductsmatrix calculusstochastic matricessubstochastic matricesbimonoidal categoriesrig categoriesprobabilistic Boolean circuits
Authors
Filippo Bonchi, Cipriano Junior Cioffo
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.