Expectation Invariants for Probabilistic Program Loops as Fixed Points

Chakarov, Aleksandar; Sankaranarayanan, Sriram

doi:10.1007/978-3-319-10936-7_6

Cited by 44 publications

(50 citation statements)

References 25 publications

(36 reference statements)

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“…We evaluated our work on 13 probabilistic programs, as follows. We used 7 programs from works [4,6,8,14,18] on invariant generation. These examples are given in lines 1-7 of Table 1; we note though that BINOMIAL("p") represents our generalisation of a binomial distribution example taken from [6,8,14] to a probabilistic program with parametrised probability p. We further crafted 6 examples of our own, illustrating the distinctive features of our work.…”

Section: Implementation and Experimentsmentioning

confidence: 99%

Automatic Generation of Moment-Based Invariants for Prob-Solvable Loops

Bartocci

Kovács

Stankovič

2019

Automated Technology for Verification and Analysis

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One of the main challenges in the analysis of probabilistic programs is to compute invariant properties that summarise loop behaviours. Automation of invariant generation is still at its infancy and most of the times targets only expected values of the program variables, which is insufficient to recover the full probabilistic program behaviour. We present a method to automatically generate moment-based invariants of a subclass of probabilistic programs, called Prob-solvable loops, with polynomial assignments over random variables and parametrised distributions. We combine methods from symbolic summation and statistics to derive invariants as valid properties over higher-order moments, such as expected values or variances, of program variables. We successfully evaluated our work on several examples where full automation for computing higher-order moments and invariants over program variables was not yet possible.

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Section: Implementation and Experimentsmentioning

confidence: 99%

Automatic Generation of Moment-Based Invariants for Prob-Solvable Loops

Bartocci

Kovács

Stankovič

2019

Automated Technology for Verification and Analysis

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“…There are also automated systems for synthesizing invariants [10,3]. [8,9] use a martingale method to compute the expected time of the coupon collector process for N = 5fixing N lets them focus on a program where the outer while loop is fully unrolled. Martingales are also used by [12] for analyzing probabilistic termination.…”

Section: Related Workmentioning

confidence: 99%

“…We show soundness and relative completeness of the core abstract logic, with mechanized proofs in the Coq proof assistant. 9 Concrete logic. While the abstract logic is conceptually clean, it is inconvenient for practical formal verification-the assertions are too general and the rules involve semantic side-conditions.…”

Section: Introductionmentioning

confidence: 99%

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An Assertion-Based Program Logic for Probabilistic Programs

Barthe

Espitau

Gaboardi

et al. 2018

Programming Languages and Systems

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Research on deductive verification of probabilistic programs has considered expectation-based logics, where pre-and post-conditions are real-valued functions on states, and assertion-based logics, where pre-and post-conditions are boolean predicates on state distributions. Both approaches have developed over nearly four decades, but they have different standings today. Expectation-based systems have managed to formalize many sophisticated case studies, while assertion-based systems today have more limited expressivity and have targeted simpler examples. We present Ellora, a sound and relatively complete assertion-based program logic, and demonstrate its expressivity by verifying several classical examples of randomized algorithms using an implementation in the EasyCrypt proof assistant. Ellora features new proof rules for loops and adversarial code, and supports richer assertions than existing program logics. We also show that Ellora allows convenient reasoning about complex probabilistic concepts by developing a new program logic for probabilistic independence and distribution law, and then smoothly embedding it into Ellora. Our work demonstrates that the assertionbased approach is not fundamentally limited and suggests that some notions are potentially easier to reason about in assertion-based systems.The most mature systems for deductive verification of randomized algorithms are expectation-based techniques; seminal examples include PPDL [25] and pGCL [31]. These approaches reason about expectations, functions E from states to real numbers, 7 propagating them backwards through a program until they are transformed into a mathematical function of the input. Expectationbased systems are both theoretically elegant [21,13,32,20] and practically useful;This is the full version of the paper. 7 Treating a program as a function from input states s to output distributions µ(s), the expected value of E on µ(s) is an expectation.Implementation and case studies. We implement Ellora on top of Easy-Crypt, a general-purpose proof assistant for reasoning about probabilistic programs, and we mechanically verify a diverse collection of examples including textbook algorithms and a randomized routing procedure. We develop an Easy-Crypt formalization of probability theory from the ground up, including tools like concentration bounds (e.g., the Chernoff bound), Markov's inequality, and theorems about probabilistic independence.Embeddings. We propose a simple program logic for proving probabilistic independence. This logic is designed to reason about independence in a lightweight way, as is common in paper proofs. We prove that the logic can be embedded into Ellora, and is therefore sound. Furthermore, we prove an embedding of the Union Bound logic [4]. 9 The formalization is available at https://github.com/strub/xhl.

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“…As in the classical setting, finding such invariants is the bottleneck of proving program correctness. For some restricted classes, such as linear loop invariants, some techniques have been established [25,21,3]. To use them to synthesize polynomial loop invariants, so-called linearization can be used [1], a technique widely applied in linear algebra.…”

Section: Introductionmentioning

confidence: 99%

Finding Polynomial Loop Invariants for Probabilistic Programs

Feng

Zhang

Jansen

et al. 2017

Automated Technology for Verification and Analysis

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Abstract. Quantitative loop invariants are an essential element in the verification of probabilistic programs. Recently, multivariate Lagrange interpolation has been applied to synthesizing polynomial invariants. In this paper, we propose an alternative approach. First, we fix a polynomial template as a candidate of a loop invariant. Using Stengle's Positivstellensatz and a transformation to a sum-of-squares problem, we find sufficient conditions on the coefficients. Then, we solve a semidefinite programming feasibility problem to synthesize the loop invariants. If the semidefinite program is unfeasible, we backtrack after increasing the degree of the template. Our approach is semi-complete in the sense that it will always lead us to a feasible solution if one exists and numerical errors are small. Experimental results show the efficiency of our approach.

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Expectation Invariants for Probabilistic Program Loops as Fixed Points

Cited by 44 publications

References 25 publications

Automatic Generation of Moment-Based Invariants for Prob-Solvable Loops

Automatic Generation of Moment-Based Invariants for Prob-Solvable Loops

An Assertion-Based Program Logic for Probabilistic Programs

Finding Polynomial Loop Invariants for Probabilistic Programs

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