Tapes as Stochastic Matrices of String Diagrams

TL;DR

This paper demonstrates that tapes, as graphical representations in certain categories, are equivalent to stochastic matrices of subdistributions, enabling a complete axiomatization of probabilistic Boolean circuits.

Contribution

It establishes an isomorphism between tapes and stochastic matrices for the subdistribution monad, extending graphical calculus to probabilistic computation.

Findings

Tapes are isomorphic to stochastic matrices of subdistributions.

Provides a complete axiomatisation of probabilistic Boolean circuits.

Extends graphical calculus to handle probabilistic models.

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.