On Probabilistic Termination of Functional Programs with Continuous Distributions

TL;DR

This paper introduces a new semantics and proof system for analyzing the termination probability of higher-order probabilistic programs with continuous distributions, enabling more precise and broader termination analysis.

Contribution

It presents a novel trace-based semantics for probabilistic programs, proves the complexity of almost-sure termination, and develops a compositional type system and proof method for non-affine recursive programs.

Findings

Deciding almost-sure termination is $ ext{Pi}^0_2$-complete.

The semantics can compute arbitrarily tight lower bounds on termination probability.

Prototype tools for lower bounds and termination verification have been implemented.

Abstract

We study termination of higher-order probabilistic functional programs with recursion, stochastic conditioning and sampling from continuous distributions. Reasoning about the termination probability of programs with continuous distributions is hard, because the enumeration of terminating executions cannot provide any non-trivial bounds. We present a new operational semantics based on traces of intervals, which is sound and complete with respect to the standard sampling-based semantics, in which (countable) enumeration can provide arbitrarily tight lower bounds. Consequently we obtain the first proof that deciding almost-sure termination (AST) for programs with continuous distributions is $\Pi^0_2$-complete. We also provide a compositional representation of our semantics in terms of an intersection type system. In the second part, we present a method of proving AST for non-affine programs, i.e., recursive programs that can, during the evaluation of the recursive body, make multiple recursive calls (of a first-order function) from distinct call sites. Unlike in a deterministic language, the number of recursion call sites has direct consequences on the termination probability. Our framework supports a proof system that can verify AST for programs that are well beyond the scope of existing methods. We have constructed prototype implementations of our method of computing lower bounds of termination probability, and AST verification.