Relational Reasoning via Probabilistic Coupling

Barthe, Gilles; Espitau, Thomas; Grégoire, Benjamin; Hsu, Justin; Stefanesco, Léo; Strub, Pierre-Yves

doi:10.1007/978-3-662-48899-7_27

Cited by 23 publications

(42 citation statements)

References 13 publications

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“…While this might look like resorting to a general-purpose "hammer", Theorem 3 is necessary for this method to work: it is not, in fact, possible to solve general polynomial constraint systems relying just on SAT. 4 We expect, however, this direct approach to be inferior to the following observation. Theorem 1 reveals that the variables in our constraint system need not assume irrational values or have large bit representations.…”

Section: Fig 5 Np Formula For Bd-thresholdmentioning

confidence: 80%

“…An alternative line of research by Barthe et al [5] concerns formal mechanised proofs of differential privacy. Recently, that direction has been related to coupling proofs [4] -this still requires substantial effort to choose the coupling, although recent techniques have improved this [1]. We complement this line of research by taking an algorithmic verification-centred approach.…”

Section: Introductionmentioning

confidence: 99%

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Bisimilarity Distances for Approximate Differential Privacy

Chistikov

Murawski

Purser

2018

Automated Technology for Verification and Analysis

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Differential privacy is a widely studied notion of privacy for various models of computation. Technically, it is based on measuring differences between probability distributions. We study , δ-differential privacy in the setting of labelled Markov chains. While the exact differences relevant to , δ-differential privacy are not computable in this framework, we propose a computable bisimilarity distance that yields a sound technique for measuring δ, the parameter that quantifies deviation from pure differential privacy. We show this bisimilarity distance is always rational, the associated threshold problem is in NP, and the distance can be computed exactly with polynomially many calls to an NP oracle.

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Section: Fig 5 Np Formula For Bd-thresholdmentioning

confidence: 80%

Section: Introductionmentioning

confidence: 99%

Bisimilarity Distances for Approximate Differential Privacy

Chistikov

Murawski

Purser

2018

Automated Technology for Verification and Analysis

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“…Recent work by Barthe et al [4,5,9] has shown that the notion of a probabilistic coupling [43] is fundamental for relational reasoning in probabilistic program logics. Given two distributions…”

Section: Nondeterministic Couplingsmentioning

confidence: 99%

A separation logic for concurrent randomized programs

Tassarotti

Harper

2019

Proc. ACM Program. Lang.

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We present Polaris, a concurrent separation logic with support for probabilistic reasoning. As part of our logic, we extend the idea of coupling, which underlies recent work on probabilistic relational logics, to the setting of programs with both probabilistic and non-deterministic choice. To demonstrate Polaris, we verify a variant of a randomized concurrent counter algorithm and a two-level concurrent skip list. All of our results have been mechanized in Coq.1 Of course, a real scheduler is unlikely to behave in such an adversarial way. However, if the timing of operations can depend on random values, effects like this can arise even with non-adversarial schedulers, as discussed in §5. 2 However, it requires O (log 2 (n)) bits to store the count. Nevertheless, it uses less space than other alternatives for decreasing contention (e.g., having each thread maintain its own local counter).

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“…It is instructive to compare our proof with prior formalizations, and in particular with the proof in [5]. Their proof is carried out in the pRHL logic, whose [COUPLING] rule is based on the existence of a bijection that satisfies some property, rather than on our formalization based on Strassen's Theorem.…”

Section: A System For Relational Reasoningmentioning

confidence: 99%

Relational Reasoning for Markov Chains in a Probabilistic Guarded Lambda Calculus

Aguirre

Barthe

Birkedal

et al. 2018

Lecture Notes in Computer Science

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We extend the simply-typed guarded λ-calculus with discrete probabilities and endow it with a program logic for reasoning about relational properties of guarded probabilistic computations. This provides a framework for programming and reasoning about infinite stochastic processes like Markov chains. We demonstrate the logic sound by interpreting its judgements in the topos of trees and by using probabilistic couplings for the semantics of relational assertions over distributions on discrete types. The program logic is designed to support syntax-directed proofs in the style of relational refinement types, but retains the expressiveness of higher-order logic extended with discrete distributions, and the ability to reason relationally about expressions that have different types or syntactic structure. In addition, our proof system leverages a well-known theorem from the coupling literature to justify better proof rules for relational reasoning about probabilistic expressions. We illustrate these benefits with a broad range of examples that were beyond the scope of previous systems, including shift couplings and lump couplings between random walks.The goal of this paper is to develop a programming and reasoning framework for probabilistic computations over infinite objects, such as Markov chains. Although programming and reasoning frameworks for infinite objects and probabilistic computations are well-understood in isolation, their combination is challenging. In particular, one must develop a proof system that is powerful enough for proving interesting properties of probabilistic computations over infinite objects, and practical enough to support effective verification of these properties.Modelling probabilistic infinite objects A first challenge is to model probabilistic infinite objects. We focus on the case of Markov chains, due to its importance. A (discrete-time) Markov chain is a sequence of random variables {X i } over some fixed type T satisfying some independence property. Thus, the straightforward way of modelling a Markov chain is as a stream of distributions over T . Going back to the simple example outlined above, it is natural to think about this kind of discrete-time Markov chain as characterized by the sequence of positions {p i } i∈N , which in turn can be described as an infinite set indexed by the natural numbers. This suggests that a natural way to model such a Markov chain is to use streams in which each element is produced probabilistically from the previous one. However, there are some downsides to this representation. First of all, it requires explicit reasoning about probabilistic dependency, since X i+1 depends on X i . Also, we might be interested in global properties of the executions of the Markov chain, such as "The probability of passing through the initial state infinitely many times is 1". These properties are naturally expressed as properties of the whole stream. For these reasons, we want to represent Markov chains as distributions over streams. Seemingly, one downside of this repre...

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Relational Reasoning via Probabilistic Coupling

Cited by 23 publications

References 13 publications

Bisimilarity Distances for Approximate Differential Privacy

Bisimilarity Distances for Approximate Differential Privacy

A separation logic for concurrent randomized programs

Relational Reasoning for Markov Chains in a Probabilistic Guarded Lambda Calculus

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