A Deductive Verification Infrastructure for Probabilistic Programs

This paper presents a quantitative program verification infrastructure for discrete probabilistic programs. Our infrastructure can be viewed as the probabilistic analogue of Boogie: its central components are an intermediate verification language (IVL) together with a real-valued logic. Our IVL provides a programming-language-style for expressing verification conditions whose validity implies the correctness of a program under investigation. As our focus is on verifying quantitative properties such as bounds on expected outcomes, expected run-times, or termination probabilities, off-the-shelf IVLs based on Boolean first-order logic do not suffice. Instead, a paradigm shift from the standard Boolean to a real-valued domain is required. Our IVL features quantitative generalizations of standard verification constructs such as assume- and assert-statements. Verification conditions are generated by a weakest-precondition-style semantics, based on our real-valued logic. We show that our verification infrastructure supports natural encodings of numerous verification techniques from the literature. With our SMT-based implementation, we automatically verify a variety of benchmarks. To the best of our knowledge, this establishes the first deductive verification infrastructure for expectation-based reasoning about probabilistic programs

Continuous probability distributions in model-based specification languages

PhD ThesisModel-based speci cation languages provide a means for obtaining assurance of dependability of complex computer-based systems, but provide little support for modelling and analysing fault behaviour, which is inherently probabilistic in nature. In particular, the need for a detailed account of the role of continuous probability has been largely overlooked. This thesis addresses the role of continuous probability in model-based speci cation languages. A model-based speci cation language (sGCL) that supports continuous probability distributions is de ned. The use of sGCL and how it interacts with engineering practices is also explored. In addition, a re nement ordering for continuous probability distributions is given, and the challenge of combining non-determinism and continuous probability is discussed in depth. The thesis is presented in three parts. The rst uses two case studies to explore the use of probability in formal methods. The rst case study, on ash memory, is used to present the capabilities of probabilistic formal methods and to determine the kinds of questions that require continuous probability distributions to answer. The second, on an emergency brake system, illustrates the strengths and weaknesses of existing languages and provides a basis for exploring a prototype language that includes continuous probability. The second part of the thesis gives the formal de nition of sGCL's syntax and semantics. The semantics is made up of two parts, the proof theory (transformer semantics) and the underpinning mathematics (relational semantics). The additional language constructs and semantical features required to include non-determinism as well as continuous probability are also discussed. The most challenging aspect lies in proving the consistency of the semantics when non-determinism is also included. The third part uses a nal case study, on an aeroplane pitch monitor, to demonstrate the use of sGCL. The new analysis techniques provided by sGCL, and how they t in with engineering practices, are explored.EPSRC: The School of Computing Science, Newcastle University: DEPLOY project

Relatively Complete Verification of Probabilistic Programs: An Expressive Language for Expectation-Based Reasoning

We study a syntax for specifying quantitative “assertions” - functions mapping program states to numbers - for probabilistic program verification. We prove that our syntax is expressive in the following sense: Given any probabilistic program C, if a function f is expressible in our syntax, then the function mapping each initial state σ to the expected value of f evaluated in the final states reached after termination C on σ (also called the weakest preexpectation wp[C](f)) is also expressible in our syntax. As a consequence, we obtain a relatively complete verification system for verifying expected values and probabilities in the sense of Cook: Apart from a single reasoning step about the inequality of two functions given as syntactic expressions in our language, given f, g, and C, we can check whether g ≤ wp[C](f)

A Deductive Verification Infrastructure for Probabilistic Programs

This paper presents a quantitative program verification infrastructure for discrete probabilistic programs. Our infrastructure can be viewed as the probabilistic analogue of Boogie: its central components are an intermediate verification language (IVL) together with a real-valued logic. Our IVL provides a programming-language-style for expressing verification conditions whose validity implies the correctness of a program under investigation. As our focus is on verifying quantitative properties such as bounds on expected outcomes, expected run-times, or termination probabilities, off-the-shelf IVLs based on Boolean first-order logic do not suffice. Instead, a paradigm shift from the standard Boolean to a real-valued domain is required. Our IVL features quantitative generalizations of standard verification constructs such as assume- and assert-statements. Verification conditions are generated by a weakest-precondition-style semantics, based on our real-valued logic. We show that our verification infrastructure supports natural encodings of numerous verification techniques from the literature. With our SMT-based implementation, we automatically verify a variety of benchmarks. To the best of our knowledge, this establishes the first deductive verification infrastructure for expectation-based reasoning about probabilistic programs

Model exploration and analysis for quantitative safety refinement in probabilistic B

The role played by counterexamples in standard system analysis is well known; but less common is a notion of counterexample in probabilistic systems refinement. In this paper we extend previous work using counterexamples to inductive invariant properties of probabilistic systems, demonstrating how they can be used to extend the technique of bounded model checking-style analysis for the refinement of quantitative safety specifications in the probabilistic B language. In particular, we show how the method can be adapted to cope with refinements incorporating probabilistic loops. Finally, we demonstrate the technique on pB models summarising a one-step refinement of a randomised algorithm for finding the minimum cut of undirected graphs, and that for the dependability analysis of a controller design.Comment: In Proceedings Refine 2011, arXiv:1106.348

Quantitative Safety: Linking Proof-Based Verification with Model Checking for Probabilistic Systems

This paper presents a novel approach for augmenting proof-based verification with performance-style analysis of the kind employed in state-of-the-art model checking tools for probabilistic systems. Quantitative safety properties usually specified as probabilistic system invariants and modeled in proof-based environments are evaluated using bounded model checking techniques. Our specific contributions include the statement of a theorem that is central to model checking safety properties of proof-based systems, the establishment of a procedure; and its full implementation in a prototype system (YAGA) which readily transforms a probabilistic model specified in a proof-based environment to its equivalent verifiable PRISM model equipped with reward structures. The reward structures capture the exact interpretation of the probabilistic invariants and can reveal succinct information about the model during experimental investigations. Finally, we demonstrate the novelty of the technique on a probabilistic library case study

Proofs of randomized algorithms in Coq

International audienceRandomized algorithms are widely used for finding efficiently approximated solutions to complex problems, for instance primality testing and for obtaining good average behavior. Proving properties of such algorithms requires subtle reasoning both on algorithmic and probabilistic aspects of programs. Thus, providing tools for the mechanization of reasoning is an important issue. This paper presents a new method for proving properties of randomized algorithms in a proof assistant based on higher-order logic. It is based on the monadic interpretation of randomized programs as probabilistic distributions. It does not require the definition of an operational semantics for the language nor the development of a complex formalization of measure theory. Instead it uses functional and algebraic properties of unit interval. Using this model, we show the validity of general rules for estimating the probability for a randomized algorithm to satisfy specified properties. This approach addresses only discrete distributions and gives rules for analysing general recursive functions. We apply this theory to the formal proof of a program implementing a Bernoulli distribution from a coin flip and to the (partial) termination of several programs. All the theories and results presented in this paper have been fully formalized and proved in the Coq proof assistant

Using Theorem Proving to Verify Expectation and Variance for Discrete Random Variables

Statistical quantities, such as expectation (mean) and variance, play a vital role in the present age probabilistic analysis. In this paper, we present some formalization of expectation theory that can be used to verify the expectation and variance characteristics of discrete random variables within the HOL theorem prover. The motivation behind this is the ability to perform error free probabilistic analysis, which in turn can be very useful for the performance and reliability analysis of systems used in safety-critical domains, such as space travel, medicine and military. We first present a formal definition of expectation of a function of a discrete random variable. Building upon this definition, we formalize the mathematical concept of variance and verify some classical properties of expectation and variance in HOL. We then utilize these formal definitions to verify the expectation and variance characteristics of the Geometric random variable. In order to demonstrate the practical effectiveness of the formalization presented in this paper, we also present the probabilistic analysis of the Coupon Collector’s problem in HOL