Constructive Design of a Hierarchy of Semantics of a Transition System by Abstract Interpretation

We construct a hierarchy of semantics by successive abstract interpretations. Starting from the maximal trace semantics of a transition system, we derive the big-step semantics, termination and nontermination semantics, Plotkin’s natural, Smyth’s demoniac and Hoare’s angelic relational semantics and equivalent nondeterministic denotational semantics (with alternative powerdomains to the Egli-Milner and Smyth constructions), D. Scott’s deterministic denotational semantics, the generalized and Dijkstra’s conservative/liberal predicate transformer semantics, the generalized/total and Hoare’s partial correctness axiomatic semantics and the corresponding proof methods. All the semantics are presented in a uniform fixpoint form and the correspondences between these semantics are established through composable Galois connections, each semantics being formally calculated by abstract interpretation of a more concrete one using Kleene and/or Tarsk

Abstract interpretation

Abstract. Abstract interpretation has been widely used for verifying properties of computer systems. Here, we present a way to extend this framework to the case of probabilistic systems. The probabilistic abstraction framework that we propose allows us to systematically lift any classical analysis or verification method to the probabilistic setting by separating in the program semantics the probabilistic behavior from the (non-)deterministic behavior. This separation provides new insights for designing novel probabilistic static analyses and verification methods. We define the concrete probabilistic semantics and propose different ways to abstract them. We provide examples illustrating the expressiveness and effectiveness of our approach.

A gentle introduction to formal verification of computer systems by abstract interpretation

International audienceWe introduce and illustrate basic notions of abstract interpretation theory and its applications by relying on the readers general scientific culture and basic knowledge of computer programming

A Static Analyzer for Large Safety-Critical Software

We show that abstract interpretation-based static program analysis can be made efficient and precise enough to formally verify a class of properties for a family of large programs with few or no false alarms. This is achieved by refinement of a general purpose static analyzer and later adaptation to particular programs of the family by the end-user through parametrization. This is applied to the proof of soundness of data manipulation operations at the machine level for periodic synchronous safety critical embedded software. The main novelties are the design principle of static analyzers by refinement and adaptation through parametrization, the symbolic manipulation of expressions to improve the precision of abstract transfer functions, the octagon, ellipsoid, and decision tree abstract domains, all with sound handling of rounding errors in floating point computations, widening strategies (with thresholds, delayed) and the automatic determination of the parameters (parametrized packing)

Precondition Inference from Intermittent Assertions and Application to Contracts on Collections

International audienceIn the context of program design by contracts, programmers often insert assertions in their code to be optionally checked at runtime, at least during the debugging phase. These assertions would better be given as a precondition of the method/procedure in which they appear. Potential errors would be discovered earlier and, more importantly, the precondition could be used in the context of separate static program analysis as part of the abstract semantics of the code. However in the case of collections (data structures such as arrays, lists, etc) checking both the precondition and the assertions at runtime appears superfluous and costly. So the precondition is often omitted since it is checked anyway at runtime by the assertions. It follows that the static analysis can be much less precise, a fact that can be difficult to understand since ''the precondition and assertions are equivalent'' (i.e. at runtime, up to the time at which warnings are produced, but not statically) e.g. for separate static analysis. We define precisely and formally the contract inference problem from intermittent assertions on scalar variables and elements of collections inserted in the code by the programmer. Our definition excludes no good run even when a non-deterministic choice (e.g. an interactive input) could lead to a bad one. We then introduce new abstract interpretation-based methods to automatically infer both the static contract precondition of a method/procedure and the code to check it at runtime on scalar and collection variables

A Framework for Combining Algebraic and Logical Abstract Interpretations

We introduce a reduced product combining algebraic and logical abstractions to design program correctness verifiers and static analyzers by abstract interpretation. The key new idea is to show that the Nelson-Oppen procedure for combining theories in SMT-solvers computes a reduced product in an observational semantics, so that algebraic and logical abstract interpretations can naturally be combined in a classical way using a reduced product on this observational semantics. The main practical benefit is that reductions can be performed within the logical abstract domains, within the algebraic abstract domains, and also between the logical and the algebraic abstract domains, including the case of abstractions evolving during the analysis