Relational Reasoning for Markov Chains in a Probabilistic Guarded Lambda Calculus

TL;DR

This paper introduces a new program logic for reasoning about relational properties of probabilistic computations, specifically infinite stochastic processes like Markov chains, within a guarded lambda calculus framework.

Contribution

It extends the guarded lambda calculus with probabilistic reasoning, providing a sound logic that supports relational proofs for complex stochastic processes.

Findings

Logic is sound via interpretation in the topos of trees

Supports relational reasoning with probabilistic couplings

Enables proofs of advanced stochastic process properties

Abstract

We extend the simply-typed guarded $\lambda$-calculus with discrete probabilities and endow it with a program logic for reasoning about relational properties of guarded probabilistic computations. This provides a framework for programming and reasoning about infinite stochastic processes like Markov chains. We demonstrate the logic sound by interpreting its judgements in the topos of trees and by using probabilistic couplings for the semantics of relational assertions over distributions on discrete types. The program logic is designed to support syntax-directed proofs in the style of relational refinement types, but retains the expressiveness of higher-order logic extended with discrete distributions, and the ability to reason relationally about expressions that have different types or syntactic structure. In addition, our proof system leverages a well-known theorem from the coupling literature to justify better proof rules for relational reasoning about probabilistic expressions. We illustrate these benefits with a broad range of examples that were beyond the scope of previous systems, including shift couplings and lump couplings between random walks.