Modular verification for almost-sure termination of probabilistic programs

Huang, Mingzhang; Fu, Hongfei; Chatterjee, Krishnendu; Goharshady, Amir Kafshdar

doi:10.1145/3360555

Cited by 31 publications

(20 citation statements)

References 43 publications

(49 reference statements)

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“…When considering imperative programs, the reachability problem, and in particular the special case of termination analysis, has been widely studied over the past decades. Previous works include symbolic execution [25,26,60], termination analysis [48,59], abstract interpretation [38] and recent results on incorrectness logic [43,69].…”

Section: Introductionmentioning

confidence: 99%

Polynomial reachability witnesses via Stellensätze

Asadi

Chatterjee

et al. 2021

Proceedings of the 42nd ACM SIGPLAN International Conference on Programming Language Design and Implementation

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We consider the fundamental problem of reachability analysis over imperative programs with real variables. Previous works that tackle reachability are either unable to handle programs consisting of general loops (e.g. symbolic execution), or lack completeness guarantees (e.g. abstract interpretation), or are not automated (e.g. incorrectness logic). In contrast, we propose a novel approach for reachability analysis that can handle general and complex loops, is complete, and can be entirely automated for a wide family of programs. Through the notion of Inductive Reachability Witnesses (IRWs), our approach extends ideas from both invariant generation and termination to reachability analysis.We first show that our IRW-based approach is sound and complete for reachability analysis of imperative programs. Then, we focus on linear and polynomial programs and develop automated methods for synthesizing linear and polynomial IRWs. In the linear case, we follow the well-known approaches using Farkas' Lemma. Our main contribution is in the polynomial case, where we present a push-button semicomplete algorithm. We achieve this using a novel combination of classical theorems in real algebraic geometry, such as Putinar's Positivstellensatz and Hilbert's Strong Nullstellensatz. Finally, our experimental results show we can prove

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Section: Introductionmentioning

confidence: 99%

Polynomial reachability witnesses via Stellensätze

Asadi

Chatterjee

et al. 2021

Proceedings of the 42nd ACM SIGPLAN International Conference on Programming Language Design and Implementation

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“…Early techniques for discovering RSMs reduced the synthesis problem from the source code of the program into constraint solving [10]. These methods have lent themselves to various generalisations, including polynomial programs, programs with non-determinism, lexicographic and modular termination arguments, and persistence properties [2,[14][15][16]20,25]. Recently, for special classes of probabilistic programs or term rewriting systems, novel automated proof techniques that leverage computer algebra systems and satisfiability modulo theories (SMT) have been introduced [5,6,38,39,41].…”

Section: Introductionmentioning

confidence: 99%

Learning Probabilistic Termination Proofs

Abate

Giacobbe

Roy

2021

Computer Aided Verification

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We present the first machine learning approach to the termination analysis of probabilistic programs. Ranking supermartingales (RSMs) prove that probabilistic programs halt, in expectation, within a finite number of steps. While previously RSMs were directly synthesised from source code, our method learns them from sampled execution traces. We introduce the neural ranking supermartingale: we let a neural network fit an RSM over execution traces and then we verify it over the source code using satisfiability modulo theories (SMT); if the latter step produces a counterexample, we generate from it new sample traces and repeat learning in a counterexample-guided inductive synthesis loop, until the SMT solver confirms the validity of the RSM. The result is thus a sound witness of probabilistic termination. Our learning strategy is agnostic to the source code and its verification counterpart supports the widest range of probabilistic single-loop programs that any existing tool can handle to date. We demonstrate the efficacy of our method over a range of benchmarks that include linear and polynomial programs with discrete, continuous, state-dependent, multi-variate, hierarchical distributions, and distributions with undefined moments.

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“…The approach of [19] extended this method to include (demonic) non-determinism and continuous probability distributions, showing the completeness of the RSM-Rule for this program class. The compositional approach proposed in [19] was further strengthened in [29] to a sound approach using the notion of descent supermartingale map. In [1], the authors introduced lexicographic RSMs.…”

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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Modular verification for almost-sure termination of probabilistic programs

Cited by 31 publications

References 43 publications

Polynomial reachability witnesses via Stellensätze

Polynomial reachability witnesses via Stellensätze

Learning Probabilistic Termination Proofs

Automated Termination Analysis of Polynomial Probabilistic Programs

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