Ranking and Repulsing Supermartingales for Reachability in Probabilistic Programs

Computing reachability probabilities is a fundamental problem in the analysis of randomized programs. This article aims at a comprehensive and comparative account of various martingale-based methods for over- and under-approximating reachability probabilities. Based on the existing works that stretch across different communities (formal verification, control theory, etc.), we offer a unifying account. In particular, we emphasize the role of order-theoretic fixed points—a classic topic in computer science—in the analysis of randomized programs. This leads us to two new martingale-based techniques, too. We also make an experimental comparison using our implementation of template-based synthesis algorithms for those martingales.

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“…The present article is an extended version of Reference [39] with the following additional materials:…”

Section: Introductionmentioning

confidence: 99%

Ranking and Repulsing Supermartingales for Reachability in Randomized Programs

Takisaka

Oyabu

Urabe

et al. 2021

ACM Trans. Program. Lang. Syst.

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“…The essential difference between the two approaches is not big: an invariant notion in the latter is easily seen to be an adaptation of a suitable notion of supermartingale. The work [34] presents a comprehensive account on the ordertheoretic foundation behind these techniques. These existing works are mostly focused on the following problems: deciding almost-sure termination, computing termination probabilities, and computing expected runtime.…”

Section: Introductionmentioning

confidence: 99%

Tail Probabilities for Randomized Program Runtimes via Martingales for Higher Moments

Kura

Urabe

Hasuo

2019

Lecture Notes in Computer Science

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Programs with randomization constructs is an active research topic, especially after the recent introduction of martingale-based analysis methods for their termination and runtimes. Unlike most of the existing works that focus on proving almost-sure termination or estimating the expected runtime, in this work we study the tail probabilities of runtimes-such as "the execution takes more than 100 steps with probability at most 1%." To this goal, we devise a theory of supermartingales that overapproximate higher moments of runtime. These higher moments, combined with a suitable concentration inequality, yield useful upper bounds of tail probabilities. Moreover, our vector-valued formulation enables automated template-based synthesis of those supermartingales. Our experiments suggest the method's practical use.

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“…Both proof rules are based on supermartingales and can certify AST for programs that are not necessarily PAST. The approach of [43] examined martingale-based techniques for obtaining bounds on reachability probabilities -and thus termination probabilities-from an order-theoretic viewpoint. The notions of nonnegative repulsing supermartingales and γ-scaled submartingales, accompanied by sound and complete proof rules, have also been introduced.…”

Section: Related Workmentioning

confidence: 99%

Automated Termination Analysis of Polynomial Probabilistic Programs

Marcel

Bartocci

Katoen

et al. 2021

Programming Languages and Systems

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The termination behavior of probabilistic programs depends on the outcomes of random assignments. Almost sure termination (AST) is concerned with the question whether a program terminates with probability one on all possible inputs. Positive almost sure termination (PAST) focuses on termination in a finite expected number of steps. This paper presents a fully automated approach to the termination analysis of probabilistic while-programs whose guards and expressions are polynomial expressions. As proving (positive) AST is undecidable in general, existing proof rules typically provide sufficient conditions. These conditions mostly involve constraints on supermartingales. We consider four proof rules from the literature and extend these with generalizations of existing proof rules for (P)AST. We automate the resulting set of proof rules by effectively computing asymptotic bounds on polynomials over the program variables. These bounds are used to decide the sufficient conditions – including the constraints on supermartingales – of a proof rule. Our software tool Amber can thus check AST, PAST, as well as their negations for a large class of polynomial probabilistic programs, while carrying out the termination reasoning fully with polynomial witnesses. Experimental results show the merits of our generalized proof rules and demonstrate that Amber can handle probabilistic programs that are out of reach for other state-of-the-art tools.

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Ranking and Repulsing Supermartingales for Reachability in Probabilistic Programs

Cited by 19 publications

References 29 publications

Ranking and Repulsing Supermartingales for Reachability in Randomized Programs

Ranking and Repulsing Supermartingales for Reachability in Randomized Programs

Tail Probabilities for Randomized Program Runtimes via Martingales for Higher Moments

Automated Termination Analysis of Polynomial Probabilistic Programs

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