Quantitative analysis of assertion violations in probabilistic programs

Wang, Jinyi; Sun, Yican; Fu, Hongfei; Chatterjee, Krishnendu; Goharshady, Amir Kafshdar

doi:10.1145/3453483.3454102

Cited by 13 publications

(11 citation statements)

References 47 publications

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“…Prior works in verification have also applied the Chernoff bound to bound sums of independent random quantities (e.g., [Chakarov and Sankaranarayanan 2013;Wang et al 2021]). While independence is easier to establish, the negative association property that we need is more subtle.…”

Section: :27mentioning

confidence: 99%

A separation logic for negative dependence

Bao

Gaboardi

Hsu

et al. 2022

Proc. ACM Program. Lang.

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Formal reasoning about hashing-based probabilistic data structures often requires reasoning about random variables where when one variable gets larger (such as the number of elements hashed into one bucket), the others tend to be smaller (like the number of elements hashed into the other buckets). This is an example of negative dependence , a generalization of probabilistic independence that has recently found interesting applications in algorithm design and machine learning. Despite the usefulness of negative dependence for the analyses of probabilistic data structures, existing verification methods cannot establish this property for randomized programs. To fill this gap, we design LINA, a probabilistic separation logic for reasoning about negative dependence. Following recent works on probabilistic separation logic using separating conjunction to reason about the probabilistic independence of random variables, we use separating conjunction to reason about negative dependence. Our assertion logic features two separating conjunctions, one for independence and one for negative dependence. We generalize the logic of bunched implications (BI) to support multiple separating conjunctions, and provide a sound and complete proof system. Notably, the semantics for separating conjunction relies on a non-deterministic , rather than partial, operation for combining resources. By drawing on closure properties for negative dependence, our program logic supports a Frame-like rule for negative dependence and monotone operations. We demonstrate how LINA can verify probabilistic properties of hash-based data structures and balls-into-bins processes.

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Section: :27mentioning

confidence: 99%

A separation logic for negative dependence

Bao

Gaboardi

Hsu

et al. 2022

Proc. ACM Program. Lang.

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“…Supermartingale-Based Approaches. In addition to qualitative and quantitative termination analyses, supermartingales were also used for the formal analysis of other properties in probabilistic programs, such as, liveness and safety properties [3,8,14,42], cost analysis of probabilistic programs [36,43]. While all these works demonstrate the effectiveness of supermartingale-based techniques, below we present a more detailed comparison with other works that consider automated computation of lower bounds on termination probability.…”

Section: Related Workmentioning

confidence: 99%

Sound and Complete Certificates for Quantitative Termination Analysis of Probabilistic Programs

Chatterjee

Goharshady

Meggendorfer

et al. 2022

Lecture Notes in Computer Science

Self Cite

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We consider the quantitative problem of obtaining lower-bounds on the probability of termination of a given non-deterministic probabilistic program. Specifically, given a non-termination threshold $$p \in [0, 1],$$ p ∈ [ 0 , 1 ] , we aim for certificates proving that the program terminates with probability at least $$1-p$$ 1 - p . The basic idea of our approach is to find a terminating stochastic invariant, i.e. a subset $$ SI $$ SI of program states such that (i) the probability of the program ever leaving $$ SI $$ SI is no more than p, and (ii) almost-surely, the program either leaves $$ SI $$ SI or terminates.While stochastic invariants are already well-known, we provide the first proof that the idea above is not only sound, but also complete for quantitative termination analysis. We then introduce a novel sound and complete characterization of stochastic invariants that enables template-based approaches for easy synthesis of quantitative termination certificates, especially in affine or polynomial forms. Finally, by combining this idea with the existing martingale-based methods that are relatively complete for qualitative termination analysis, we obtain the first automated, sound, and relatively complete algorithm for quantitative termination analysis. Notably, our completeness guarantees for quantitative termination analysis are as strong as the best-known methods for the qualitative variant.Our prototype implementation demonstrates the effectiveness of our approach on various probabilistic programs. We also demonstrate that our algorithm certifies lower bounds on termination probability for probabilistic programs that are beyond the reach of previous methods.

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“…On the other hand, Wang et al [Wang et al 2021] analysed the exponential bounds, a sub-case of probabilistic imperative program verification. They found a novel fixed point theorem to compute arguably accurate upper and lower bounds of assertion violation probability of a given probabilistic imperative program.…”

Section: Related Workmentioning

confidence: 99%

“…In this paper, we revisit the verification problem of probabilistic programs. More specifically, we focus on quantitative analysis of violation probability of probabilistic programs [Smith et al 2019;Wang et al 2021] which can be described as follows: given a probabilistic program 𝑃, a precondition 𝜑 𝑒 , a postcondition 𝜑 𝑓 and a threshold 𝛽, answer the question that whether the probability of violating the Hoare-triple {𝜑 𝑒 } 𝑃 𝜑 𝑓 is no greater than 𝛽. Following the notation of [Smith et al 2019], we denote this by: ⊢ 𝛽 {𝜑 𝑒 } 𝑃 𝜑 𝑓 One of the distinct features of our approach is the combination of probabilistic program verification with trace abstraction [Heizmann et al 2009], a technique that has been successfully employed in wide areas of non-probabilistic program verification and analysis [He and Han 2020;Heizmann [n.d.]; Heizmann et al 2014;Rothenberg et al 2018].…”

Section: Introductionmentioning

confidence: 99%

ProbTA: A sound and complete proof rule for probabilistic verification

Li¹,

Han²,

He³

2022

Preprint

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We propose a sound and complete proof rule ProbTA for quantitative analysis of violation probability of probabilistic programs. Our approach extends the technique of trace abstraction with probability in the control-flow randomness style, in contrast to previous work of combining trace abstraction and probabilisitic verification which adopts the data randomness style. In our method, a program specification is proved or disproved by decomposing the program into different modules of traces. Precise quantitative analysis is enabled by novel models proposed to bridge program verification and probability theory. Based on the proof rule, we propose a new automated algorithm via CEGAR involving multiple technical issues unprecedented in non-probabilistic trace abstraction and data randomness-based approach.

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Quantitative analysis of assertion violations in probabilistic programs

Cited by 13 publications

References 47 publications

A separation logic for negative dependence

A separation logic for negative dependence

Sound and Complete Certificates for Quantitative Termination Analysis of Probabilistic Programs

ProbTA: A sound and complete proof rule for probabilistic verification

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