Probabilistic Abstract Interpretation

Cousot, Patrick; Monerau, Michael

doi:10.1007/978-3-642-28869-2_9

Cited by 78 publications

(85 citation statements)

References 31 publications

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“…Park et al [29] give an operational version of Kozen's tracebased semantics for a λ-calculus with recursion, but "do not investigate measure-theoretic properties". Cousot and Monerau [5] generalise Kozen's trace-based semantics to consider probabilistic programs as measurable functions from a probability space into a semantics domain, and study abstract interpretation in this setting. Toronto et al [38] use a pre-image version of Kozen's semantics to obtain an efficient implementation using rejection sampling.…”

Section: Related Workmentioning

confidence: 99%

A lambda-calculus foundation for universal probabilistic programming

et al. 2016

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We develop the operational semantics of an untyped probabilistic λ-calculus with continuous distributions, and both hard and soft constraints, as a foundation for universal probabilistic programming languages such as CHURCH, ANGLICAN, and VEN-TURE. Our first contribution is to adapt the classic operational semantics of λ-calculus to a continuous setting via creating a measure space on terms and defining step-indexed approximations. We prove equivalence of big-step and small-step formulations of this distribution-based semantics. To move closer to inference techniques, we also define the sampling-based semantics of a term as a function from a trace of random samples to a value. We show that the distribution induced by integration over the space of traces equals the distribution-based semantics. Our second contribution is to formalize the implementation technique of trace Markov chain Monte Carlo (MCMC) for our calculus and to show its correctness. A key step is defining sufficient conditions for the distribution induced by trace MCMC to converge to the distribution-based semantics. To the best of our knowledge, this is the first rigorous correctness proof for trace MCMC for a higher-order functional language, or for a language with soft constraints.

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Section: Related Workmentioning

confidence: 99%

A lambda-calculus foundation for universal probabilistic programming

et al. 2016

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“…Instead of giving a probability distribution over the intended value, one defines a probability distribution π on another, fixed space Ω (of so-called events), and describe the probability over the intended value v as the image measure of π by some measurable map f from Ω to the space of values. This is the approach taken by Cousot and Monereau [10], where Ω is the space of infinite sequences of coin flips, each coin flip being independent and unbiased. A probability distribution on a space X is then encoded by a measurable map f : Ω → X, and the (image) measure of A ⊆ X is π(f −1 (A)).…”

Section: Related Workmentioning

confidence: 99%

Static Analysis of Programs with Imprecise Probabilistic Inputs

Adjé

Bouissou

Goubault-Larrecq

et al. 2014

Verified Software: Theories, Tools, Experiments

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Abstract. Having a precise yet sound abstraction of the inputs of numerical programs is important to analyze their behavior. For many programs, these inputs are probabilistic, but the actual distribution used is only partially known. We present a static analysis framework for reasoning about programs with inputs given as imprecise probabilities: we define a collecting semantics based on the notion of previsions and an abstract semantics based on an extension of Dempster-Shafer structures. We prove the correctness of our approach and show on some realistic examples the kind of invariants we are able to infer.

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“…It is possible to propagate probabilistic information about program inputs [12,34,35,41] or to perform probabilistic symbolic execution [19] in order to generate quantitative analysis results. However, these techniques do not address the problem of assigning probabilities to existing, imprecise static analysis results.…”

Section: Introductionmentioning

confidence: 99%

Combining static analysis with probabilistic models to enable market-scale Android inter-component analysis

et al. 2016

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Static analysis has been successfully used in many areas, from verifying mission-critical software to malware detection. Unfortunately, static analysis often produces false positives, which require significant manual effort to resolve. In this paper, we show how to overlay a probabilistic model, trained using domain knowledge, on top of static analysis results, in order to triage static analysis results. We apply this idea to analyzing mobile applications. Android application components can communicate with each other, both within single applications and between different applications. Unfortunately, techniques to statically infer Inter-Component Communication (ICC) yield many potential inter-component and interapplication links, most of which are false positives. At large scales, scrutinizing all potential links is simply not feasible. We therefore overlay a probabilistic model of ICC on top of static analysis results. Since computing the inter-component links is a prerequisite to inter-component analysis, we introduce a formalism for inferring ICC links based on set constraints. We design an efficient algorithm for performing link resolution. We compute all potential links in a corpus of 11,267 applications in 30 minutes and triage them using our probabilistic approach. We find that over 95.1% of all 636 million potential links are associated with probability values below 0.01 and are thus likely unfeasible links. Thus, it is possible to consider only a small subset of all links without significant loss of information. This work is the first significant step in making static inter-application analysis more tractable, even at large scales.

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Probabilistic Abstract Interpretation

Cited by 78 publications

References 31 publications

A lambda-calculus foundation for universal probabilistic programming

A lambda-calculus foundation for universal probabilistic programming

Static Analysis of Programs with Imprecise Probabilistic Inputs

Combining static analysis with probabilistic models to enable market-scale Android inter-component analysis

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