Completeness for Probabilistic Boolean Tapes

2026-06-17

AI Summary

This work addresses the absence of a complete axiomatic semantics for probabilistic Boolean circuits by proposing a diagrammatic framework based on partial Boolean circuits and probabilistic Boolean tapes. By integrating rig categories, Markov kernel semantics, and string diagram techniques, the authors establish—for the first time—a complete axiomatization of probabilistic Boolean circuits under Markov kernel semantics and formally prove its completeness. This contribution not only identifies probabilistic Boolean tapes as a diagrammatic language for rig categories but also lays a rigorous theoretical foundation for the formal verification of finite probabilistic programs.

Abstract

Probabilistic Boolean circuits have recently been proposed as a string-diagrammatic foundation for finite probabilistic programming. In this paper, we present a complete set of axioms for their semantics in terms of Markov kernels. Our approach is based on two intermediate results: completeness for \emph{partial} Boolean circuits and completeness for probabilistic Boolean tapes, a diagrammatic language for rig categories.

Problem

• probabilistic Boolean tapes
• completeness
• Markov kernels
• string diagrams
• probabilistic programming

Innovation

• probabilistic Boolean circuits
• completeness
• Markov kernels
• string diagrams
• rig categories