An Adequacy Theorem Between Mixed Powerdomains and Probabilistic Concurrency

2025-10-28

AI Summary

This work addresses the challenge of precisely modeling and verifying concurrent probabilistic programs, where nondeterminism and probabilistic behavior interact semantically. We propose a denotational semantics framework grounded in hybrid powerdomains and topological observability. By representing observable properties as open sets in a topological space, we establish the first adequacy theorem for concurrent probabilistic programs. We prove a generalized probabilistic version of König’s Lemma, providing crucial support for Escardó’s conjecture and yielding semi-decidability for certain observable properties. Our approach integrates Smyth’s theory of observability, a concurrent extension of probabilistic Guarded Command Language (pGCL), and hybrid powerdomain constructions. This unification substantially strengthens formal guarantees regarding semantic consistency and behavioral verification of probabilistic concurrent systems.

Technology Category

• Reasoning under Uncertainty: Probabilistic Programming
• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Nonmonotonic 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: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

We present an adequacy theorem for a concurrent extension of probabilistic GCL. The underlying denotational semantics is based on the so-called mixed powerdomains, which combine non-determinism with probabilistic behaviour. The theorem itself is formulated via M. Smyth's idea of treating observable properties as open sets of a topological space. The proof hinges on a 'topological generalisation' of König's lemma in the setting of probabilistic programming (a result that is proved in the paper as well). One application of the theorem is that it entails semi-decidability w.r.t. whether a concurrent program satisfies an observable property (written in a certain form). This is related to M. Escardo's conjecture about semi-decidability w.r.t. may and must probabilistic testing.

Problem

• Adequacy theorem for concurrent probabilistic GCL extension
• Mixed powerdomains combining nondeterminism with probabilistic behavior
• Topological approach to observable properties and semi-decidability

Innovation

• Mixed powerdomains combine nondeterminism with probabilistic behavior
• Adequacy theorem uses topological open sets as observables
• Proof employs topological generalization of König's lemma