Automatic Probabilistic Program Verification through Random Variable Abstraction

The weakest pre-expectation calculus has been proved to be a mature theory to analyze quantitative properties of probabilistic and nondeterministic programs. We present an automatic method for proving quantitative linear properties on any denumerable state space using iterative backwards fixed point calculation in the general framework of abstract interpretation. In order to accomplish this task we present the technique of random variable abstraction (RVA) and we also postulate a sufficient condition to achieve exact fixed point computation in the abstract domain. The feasibility of our approach is shown with two examples, one obtaining the expected running time of a probabilistic program, and the other the expected gain of a gambling strategy. Our method works on general guarded probabilistic and nondeterministic transition systems instead of plain pGCL programs, allowing us to easily model a wide range of systems including distributed ones and unstructured programs. We present the operational and weakest precondition semantics for this programs and prove its equivalence

An experimental tool for checking probabilistic program refinement

We discuss the features and current status of a software tool developed for checking refinement properties of a particular (though still quite general) class of probabilistic nondeterministic programs. The tool has been used successfully for several interesting examples, and its development is currently being expanded to encompass even more classes of probabilistic programs. The novelty of the tool lies in its expressing the refinement checking as a linear satisfiability problem, and also in producing evidence of lack of refinement by a problem of the same kind.Eje: Workshop Ingeniería de software (WIS)Red de Universidades con Carreras en Informátic

A linear process algebraic format for probabilistic systems with data

This paper presents a novel linear process algebraic format for probabilistic automata. The key ingredient is a symbolic transformation of probabilistic process algebra terms that incorporate data into this linear format while preserving strong probabilistic bisimulation. This generalises similar techniques for traditional process algebras with data, and — more importantly — treats data and data-dependent probabilistic choice in a fully symbolic manner, paving the way to the symbolic analysis of parameterised probabilistic systems

A program logic for union bounds

International audienceWe propose a probabilistic Hoare logic aHL based on the union bound, a tool from basic probability theory. While the union bound is simple, it is an extremely common tool for analyzing randomized algorithms. In formal verification terms, the union bound allows flexible and compos-itional reasoning over possible ways an algorithm may go wrong. It also enables a clean separation between reasoning about probabilities and reasoning about events, which are expressed as standard first-order formulas in our logic. Notably, assertions in our logic are non-probabilistic, even though we can conclude probabilistic facts from the judgments. Our logic can also prove accuracy properties for interactive programs, where the program must produce intermediate outputs as soon as pieces of the input arrive, rather than accessing the entire input at once. This setting also enables adaptivity, where later inputs may depend on earlier intermediate outputs. We show how to prove accuracy for several examples from the differential privacy literature, both interactive and non-interactive. 1998 ACM Subject Classification D.2.4 Software/Program Verification 1 Introduction Probabilistic computations arise naturally in many areas of computer science. For instance, they are widely used in cryptography, privacy, and security for achieving goals that lie beyond the reach of deterministic programs. However, the correctness of probabilistic programs can be quite subtle, often relying on complex reasoning about probabilistic events. Accordingly, probabilistic computations present an attractive target for formal verification. A long line of research, spanning more than four decades, has focused on expressive formalisms for reasoning about general probabilistic properties both for purely probabilistic programs and for programs that combine probabilistic and non-deterministic choice (see, e.g., [29, 34, 35]). More recent research investigates specialized formalisms that work with more restricted assertions and proof techniques, aiming to simplify formal verification. As perhaps the purest examples of this approach, some program logics prove probabilistic properties by working purely with non-probabilistic assertions; we call such systems lightweight logics. Examples include probabilistic relational Hoare logic [3] for proving the reductionist security of cryptographic constructions, and the related approximate probabilistic relational Hoare logic [4] for reasoning about differential privacy. These logics rely on the powerful abstraction of probabilistic couplings to derive probabilistic facts from non-probabilistic assertions [7]

Verification of Probabilistic Properties in HOL using the Cumulative Distribution Function

Abstract. Traditionally, computer simulation techniques are used to perform probabilistic analysis. However, they provide inaccurate results and cannot handle large-scale problems due to their enormous CPU time requirements. To overcome these limitations, we propose to complement simulation based tools with higher-order-logic theorem proving so that an integrated approach can provide exact results for the critical sections of the analysis in the most efficient manner. In this paper, we illustrate the practical effectiveness of our idea by verifying numerous probabilistic properties associated with random variables in the HOL theorem prover. Our verification approach revolves around the fact that any probabilistic property associated with a random variable can be verified using the classical Cumulative Distribution Function (CDF) properties, if the CDF relation of that random variable is known. For illustration purposes, we also present the verification of a couple of probabilistic properties, which cannot be evaluated precisely by the existing simulation techniques, associated with the Continuous Uniform random variable in HOL

A linear process-algebraic format for probabilistic systems with data (extended version)

This paper presents a novel linear process-algebraic format for probabilistic automata. The key ingredient is a symbolic transformation of probabilistic process algebra terms that incorporate data into this linear format while preserving strong probabilistic bisimulation. This generalises similar techniques for traditional process algebras with data, and - more importantly - treats data and data-dependent probabilistic choice in a fully symbolic manner, paving the way to the symbolic analysis of parameterised probabilistic systems

RDA: A Coq Library to Reason about Randomised Distributed Algorithms in the Message Passing Model

Distributed algorithms have received considerable attention and were studied intensively in the past few decades. Under some hypotheses on the distributed system, there is no deterministic solution to certain classical problems. Randomised solutions are then needed to solve those problems. Probabilistic algorithms are generally simple to formulate. However, their analysis can become very complex, especially in the field of distributed computing. In this paper, we formally model in Coq a class of randomised distributed algorithms. We develop some tools to help proving impossibility results about classical problems and analysing this class of algorithms. As case studies, we examine the handshake and maximal matching problems. We show how to use our tools to formally prove properties about algorithms solving those problems

Formal verification of higher-order probabilistic programs

Probabilistic programming provides a convenient lingua franca for writing succinct and rigorous descriptions of probabilistic models and inference tasks. Several probabilistic programming languages, including Anglican, Church or Hakaru, derive their expressiveness from a powerful combination of continuous distributions, conditioning, and higher-order functions. Although very important for practical applications, these combined features raise fundamental challenges for program semantics and verification. Several recent works offer promising answers to these challenges, but their primary focus is on semantical issues. In this paper, we take a step further and we develop a set of program logics, named PPV, for proving properties of programs written in an expressive probabilistic higher-order language with continuous distributions and operators for conditioning distributions by real-valued functions. Pleasingly, our program logics retain the comfortable reasoning style of informal proofs thanks to carefully selected axiomatizations of key results from probability theory. The versatility of our logics is illustrated through the formal verification of several intricate examples from statistics, probabilistic inference, and machine learning. We further show the expressiveness of our logics by giving sound embeddings of existing logics. In particular, we do this in a parametric way by showing how the semantics idea of (unary and relational) TT-lifting can be internalized in our logics. The soundness of PPV follows by interpreting programs and assertions in quasi-Borel spaces (QBS), a recently proposed variant of Borel spaces with a good structure for interpreting higher order probabilistic programs