Tapes as Stochastic Matrices of String Diagrams

2026-01-04

arXiv.org

AI Summary

This work addresses the axiomatization of probabilistic Boolean circuits by proposing a categorical semantics–based approach. By generalizing string diagrams to the Kleisli category of an arbitrary affine monad and focusing specifically on the subdistribution monad, the authors establish—for the first time—an isomorphism between string diagrams and subdistribution-valued stochastic matrices represented by such diagrams. Building upon this correspondence, they develop a complete axiomatization for probabilistic Boolean circuits. This contribution not only provides a rigorous categorical semantic foundation for probabilistic Boolean circuits but also pioneers a novel algebraic methodology for their axiomatization through the correspondence between string diagrams and stochastic matrices.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Probabilistic Programming
• Knowledge Representation and Reasoning: Diagnosis and Abductive Reasoning

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Data transparency and provenance

Abstract

Tape diagrams provide a graphical notation for categories equipped with two monoidal products, $\otimes$ and $\oplus$, where $\oplus$ is a biproduct. Recently, they have been generalised to handle Kleisli categories of arbitrary monoidal monads. In this work, we show that for the subdistribution monad, tapes are isomorphic to stochastic matrices of subdistributions of string diagrams. We then exploit this result to provide a complete axiomatisation of probabilistic Boolean circuits.

Problem

• tape diagrams
• stochastic matrices
• string diagrams
• probabilistic Boolean circuits
• subdistribution monad

Innovation

• tape diagrams
• stochastic matrices
• subdistribution monad
• probabilistic Boolean circuits
• string diagrams