An adequacy theorem between mixed powerdomains and probabilistic concurrency

2024-09-24

arXiv.org

AI Summary

This paper addresses the verification of observable properties in concurrent probabilistic programs by proposing a denotational semantics framework based on a hybrid powerdomain. Methodologically, it extends Smyth-topology-based observability theory—previously developed for deterministic and nondeterministic systems—to probabilistic concurrent systems, thereby constructing a hybrid powerdomain semantics for concurrent probabilistic Guarded Command Language (pGCL) that uniformly integrates nondeterminism and probabilistic choice; observable properties are modeled as open sets in the induced topological space. Key contributions include: (1) a proof of strict adequacy between the semantics and operational behavior; (2) the establishment of semi-decidability for observable property satisfaction, providing computational guarantees for verification; and (3) crucial semantic support for Escardó’s conjecture on the semi-decidability of probabilistic testing. The work unifies denotational modeling of probability, nondeterminism, and concurrency, advancing the theoretical foundations of probabilistic program verification.

Technology Category

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

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

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 stochasticity. The theorem itself is formulated via M. Smyth's idea of treating observable properties as open sets of a topological space. One application of our 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 intimately connected to M. Escard'o's conjecture about semi-decidability w.r.t. may and must probabilistic testing.

Problem

• Adequacy theorem for concurrent probabilistic GCL extension
• Mixed powerdomains combining non-determinism with probabilistic behavior
• Semi-decidability of concurrent programs satisfying observable properties

Innovation

• Mixed powerdomains combining non-determinism and probabilistic behavior
• Topological generalization of König's lemma for proofs
• Adequacy theorem enabling semi-decidability for observable properties