Coupling proofs are probabilistic product programs

TL;DR

This paper introduces xpRHL, an extension of pRHL, enabling explicit construction and quantitative analysis of probabilistic couplings through probabilistic product programs, enhancing reasoning about mixing and convergence.

Contribution

It develops xpRHL, a new logic that explicitly constructs couplings as probabilistic programs and includes a novel loop rule, capturing advanced couplings like shift couplings.

Findings

Verified rapid mixing using various coupling techniques

Proved approximate equivalence of programs with loop optimizations

Demonstrated the logic's applicability to probabilistic process analysis

Abstract

Couplings are a powerful mathematical tool for reasoning about pairs of probabilistic processes. Recent developments in formal verification identify a close connection between couplings and pRHL, a relational program logic motivated by applications to provable security, enabling formal construction of couplings from the probability theory literature. However, existing work using pRHL merely shows existence of a coupling and does not give a way to prove quantitative properties about the coupling, which are need to reason about mixing and convergence of probabilistic processes. Furthermore, pRHL is inherently incomplete, and is not able to capture some advanced forms of couplings such as shift couplings. We address both problems as follows. First, we define an extension of pRHL, called xpRHL, which explicitly constructs the coupling in a pRHL derivation in the form of a probabilistic product program that simulates two correlated runs of the original program. Existing verification tools for probabilistic programs can then be directly applied to the probabilistic product to prove quantitative properties of the coupling. Second, we equip pRHL with a new rule for while loops, where reasoning can freely mix synchronized and unsynchronized loop iterations. Our proof rule can capture examples of shift couplings, and the logic is relatively complete for deterministic programs. We show soundness of xpRHL and use it to analyze two classes of examples. First, we verify rapid mixing using different tools from coupling: standard coupling, shift coupling, and path coupling, a compositional principle for combining local couplings into a global coupling. Second, we verify (approximate) equivalence between a source and an optimized program for several instances of loop optimizations from the literature.