On probabilistic termination of functional programs with continuous distributions

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 nontrivial 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 Π 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.

Sound and Complete Certificates for Quantitative Termination Analysis of Probabilistic Programs

“…Comparison to [5]. The work of [5] proposes a type system for functional probabilistic programs that allows incrementally searching for type derivations and accumulating a lower bound on termination probability.…”

“…Comparison to [5]. The work of [5] proposes a type system for functional probabilistic programs that allows incrementally searching for type derivations and accumulating a lower bound on termination probability. In the limit, it finds arbitrarily tight lower bounds on termination probability, however it does not provide any completeness or precision guarantees in finite time.…”

“…Previous Qualitative Works. There are many approaches to prove a.s. termination based on weakest pre-expectation calculus [27,31,37], abstract interpretation [34], type systems [5] and martingales [7,9,11,14,25,26,32,35]. This work is closest in spirit to martingale-based approaches.…”

“…Quantitative analyses of probabilistic programs are often more challenging. There are only a few works that study the quantitative termination problem: [5,14,40]. The works [14,40] propose martingale-based proof rules for computing lower-bounds on termination probability, while [5] considers functional probabilistic programs and proposes a type system that allows incrementally searching for type derivations to accumulate a lower-bound on termination probability.…”

“…Lack of Completeness. While [5,14,40] all propose sound methods to compute lower-bounds on termination probability, none of them are theoretically complete nor do their algorithms provide relative completeness guarantees. This naturally leaves open whether one can define a complete certificate for proving termination with probability at least 1 − p ∈ [0, 1], i.e.…”