A Decision Procedure for Probabilistic Kleene Algebra with Angelic Nondeterminism

2025-07-15

AI Summary

This paper addresses the decidability problem for the equational theory of probabilistic Kleene algebras with angelic nondeterminism (PKAₐ). Prior to this work, no effective decision procedure existed for PKAₐ’s equational logic—a significant theoretical gap. We propose the first sound, complete, and decidable framework for verifying equational identities in PKAₐ. Our method models PKAₐ terms as probabilistic transition systems equipped with powerdomain semantics, then integrates structural induction with fixed-point theory to construct a semantic equivalence-checking algorithm. We formally prove the algorithm’s soundness and completeness within a rigorous proof-theoretic setting. This yields the first full decidability result for PKAₐ’s equational theory. The framework provides a foundational tool for probabilistic program logics, verification of randomized algorithms, and modeling systems exhibiting hybrid uncertainty—combining probabilistic choice, angelic nondeterminism, and sequential composition under a unified algebraic semantics.

Technology Category

• Reasoning under Uncertainty: Probabilistic Programming
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Probabilistic Circuits and Graphical Models

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: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Algorithmic accountability and transparency on the web

Abstract

We give a decision procedure and proof of correctness for the equational theory of probabilistic Kleene algebra with angelic nondeterminism introduced in Ong, Ma, and Kozen (2025).

Problem

• Decision procedure for probabilistic Kleene algebra
• Handling angelic nondeterminism in algebra
• Proving equational theory correctness

Innovation

• Decision procedure for probabilistic Kleene algebra
• Proof of correctness for equational theory
• Handles angelic nondeterminism in algebra