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Register a new userAiming Low Is Harder -- Induction for Lower Bounds in Probabilistic Program Verification
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Added by
Marcel Hark
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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We present a new inductive rule for verifying lower bounds on expected values of random variables after execution of probabilistic loops as well as on their expected runtimes. Our rule is simple in the sense that loop body semantics need to be applied only finitely often in order to verify that the candidates are indeed lower bounds. In particular, it is not necessary to find the limit of a sequence as in many previous rules.
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This paper surveys recent work on applying analysis and transformation techniques that originate in the field of constraint logic programming (CLP) to the problem of verifying software systems. We present specialisation-based techniques for translating verification problems for different programming languages, and in general software systems, into satisfiability problems for constrained Horn clauses (CHCs), a term that has become popular in the verification field to refer to CLP programs. Then, we describe static analysis techniques for CHCs that may be used for inferring relevant program properties, such as loop invariants. We also give an overview of some transformation techniques based on specialisation and fold/unfold rules, which are useful for improving the effectiveness of CHC satisfiability tools. Finally, we discuss future developments in applying these techniques.
The proceedings consist of a keynote paper by Alberto followed by 6 invited papers written by Lorenzo Clemente (U. Warsaw), Alain Finkel (U. Paris-Saclay), John Gallagher (Roskilde U. and IMDEA Software Institute) et al., Neil Jones (U. Copenhagen) et al., Michael Leuschel (Heinrich-Heine U.) and Maurizio Proietti (IASI-CNR) et al.. These invited papers are followed by 4 regular papers accepted at VPT 2020 and the papers of HCVS 2020 which consist of three contributed papers and an invited paper on the third competition of solvers for Constrained Horn Clauses. In addition, the abstracts (in HTML format) of 3 invited talks at VPT 2020 by Andrzej Skowron (U. Warsaw), Sophie Renault (EPO) and Moa Johansson (Chalmers U.), are included.
The syntax of an imperative language does not mention explicitly the state, while its denotational semantics has to mention it. In this paper we present a framework for the verification in Coq of properties of programs manipulating the global state effect. These properties are expressed in a proof system which is close to the syntax, as in effect systems, in the sense that the state does not appear explicitly in the type of expressions which manipulate it. Rather, the state appears via decorations added to terms and to equations. In this system, proofs of programs thus present two aspects: properties can be verified {em up to effects} or the effects can be taken into account. The design of our Coq library consequently reflects these two aspects: our framework is centered around the construction of two inductive and dependent types, one for terms up to effects and one for the manipulation of decorations.
It is well-known that the verification of partial correctness properties of imperative programs can be reduced to the satisfiability problem for constrained Horn clauses (CHCs). However, state-of-the-art solvers for CHCs (CHC solvers) based on predicate abstraction are sometimes unable to verify satisfiability because they look for models that are definable in a given class A of constraints, called A-definable models. We introduce a transformation technique, called Predicate Pairing (PP), which is able, in many interesting cases, to transform a set of clauses into an equisatisfiable set whose satisfiability can be proved by finding an A-definable model, and hence can be effectively verified by CHC solvers. We prove that, under very general conditions on A, the unfold/fold transformation rules preserve the existence of an A-definable model, i.e., if the original clauses have an A-definable model, then the transformed clauses have an A-definable model. The converse does not hold in general, and we provide suitable conditions under which the transformed clauses have an A-definable model iff the original ones have an A-definable model. Then, we present the PP strategy which guides the application of the transformation rules with the objective of deriving a set of clauses whose satisfiability can be proved by looking for A-definable models. PP introduces a new predicate defined by the conjunction of two predicates together with some constraints. We show through some examples that an A-definable model may exist for the new predicate even if it does not exist for its defining atomic conjuncts. We also present some case studies showing that PP plays a crucial role in the verification of relational properties of programs (e.g., program equivalence and non-interference). Finally, we perform an experimental evaluation to assess the effectiveness of PP in increasing the power of CHC solving.
This paper investigates the usage of generating functions (GFs) encoding measures over the program variables for reasoning about discrete probabilistic programs. To that end, we define a denotational GF-transformer semantics for probabilistic while-programs, and show that it instantiates Kozens seminal distribution transformer semantics. We then study the effective usage of GFs for program analysis. We show that finitely expressible GFs enable checking super-invariants by means of computer algebra tools, and that they can be used to determine termination probabilities. The paper concludes by characterizing a class of -- possibly infinite-state -- programs whose semantics is a rational GF encoding a discrete phase-type distribution.
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