On the Termination Problem for Probabilistic Higher-Order Recursive Programs

Abstract

In the last two decades, there has been much progress on model checking of both probabilistic systems and higher-order programs. In spite of the emergence of higher-order probabilistic programming languages, not much has been done to combine those two approaches. In this paper, we initiate a study on the probabilistic higher-order model checking problem, by giving some first theoretical and experimental results. As a first step towards our goal, we introduce PHORS, a probabilistic extension of higher-order recursion schemes (HORS), as a model of probabilistic higher-order programs. The model of PHORS may alternatively be viewed as a higher-order extension of recursive Markov chains. We then investigate the probabilistic termination problem -or, equivalently, the probabilistic reachability problem. We prove that almost sure termination of order-2 PHORS is undecidable. We also provide a fixpoint characterization of the termination probability of PHORS, and develop a sound (but possibly incomplete) procedure for approximately computing the termination probability. We have implemented the procedure for order-2 PHORSs, and confirmed that the procedure works well through preliminary experiments that are reported at the end of the article.

On probabilistic termination of functional programs with continuous distributions

“…We use the following example programs. Wherever possible we try to use examples used in the implementation of [33]. Our results are, however, only partially comparable to [33].…”

“…For discrete distributions, this problem is r.e. (in Σ 0 1 ) as we can enumerate terminating paths until the sum of the weight of those paths exceeds [29,33]. In the presence of continuous distributions, this is no longer possible.…”

“…Even in its current, unoptimized, form our tool is able to compute meaningful lower bounds in a reasonable time. We evaluate our tool on various examples taken from [33,44,51] and [41] (possibly modified to match the CbN evaluation). The computed lower bounds can be found in table 1.…”

Intersection types and (positive) almost-sure termination

“…Formal verification techniques for probabilistic termination and complexity analysis are plentiful, and ranges from model checking [Etessami and Yannakakis 2009;Kobayashi et al 2019] to abstract interpretation [Monniaux 2001], to the ranking supermartingales [Chakarov and Sankaranarayanan 2013], to amortised analysis [Ngo et al 2018] to the interpretation method from term rewriting [Avanzini et al 2020]. The only methodology among these that, at least so far, has been employed for the analysis of higher-order probabilistic programs is the one by Kobayashi et al [Kobayashi et al 2019], which deals with probabilistic variations on higher-order recursion schemes. Some of the ideas which we introduced in the paper are indeed variations of similar ones from the imperative setting (e.g.…”

Aiming low is harder: induction for lower bounds in probabilistic program verification

“…give lower bounds on expected runtimes. [Kobayashi et al 2018] provide a semi-decision procedure for lower bounding termination probabilities of probabilistic higher-order recursive programs. [Ngo et al 2018] perform automated template-driven resource analysis, but infer upper bounds only.…”