A quantitative probabilistic relational Hoare logic

2024-07-24

arXiv.org

AI Summary

PRHL and apRHL suffer from limitations in expressiveness and completeness when verifying program equivalence, statistical distance, and differential privacy—particularly over numeric domains involving zero and infinity. Method: This paper introduces eRHL, a quantitative probabilistic relational Hoare logic grounded in expectation assertions and coupled probabilistic program semantics. Unlike prior logics, eRHL imposes no strong alignment constraints on random variables and supports reasoning about almost-surely terminating programs. Results: We establish that eRHL is both sound and complete for program equivalence, statistical distance, and differential privacy verification. Theoretically, it subsumes all valid judgments of PRHL and apRHL; practically, it successfully verifies critical examples beyond their scope—including those involving infinite support, zero-probability events, and non-aligned sampling. By unifying and extending existing relational logics, eRHL significantly broadens the formal capabilities of quantitative probabilistic relational reasoning.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Statistical Relational/Logic Learning
• Knowledge Representation and Reasoning: Logic Programming

Application Category

• Security and Privacy: Privacy-enhancing technologies
• Responsible Web: Data and user privacy-enhancing technologies for the Web
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We introduce eRHL, a program logic for reasoning about relational expectation properties of pairs of probabilistic programs. eRHL is quantitative, i.e., its pre- and post-conditions take values in the extended non-negative reals. Thanks to its quantitative assertions, eRHL overcomes randomness alignment restrictions from prior logics, including PRHL, a popular relational program logic used to reason about security of cryptographic constructions, and apRHL, a variant of PRHL for differential privacy. As a result, eRHL is the first relational probabilistic program logic to be supported by non-trivial soundness and completeness results for all almost surely terminating programs. We show that eRHL is sound and complete with respect to program equivalence, statistical distance, and differential privacy. We also show that every PRHL judgment is valid iff it is provable in eRHL. We showcase the practical benefits of eRHL with examples that are beyond reach of PRHL and apRHL.

Problem

• Random Program Analysis
• Cryptographic Security
• Privacy Protection

Innovation

• eRHL
• Quantified Relational Hoare Logic
• Randomized Program Verification