Interprocedural Reachability for Flat Integer Programs

We study programs with integer data, procedure calls and arbitrary call graphs. We show that, whenever the guards and updates are given by octagonal relations, the reachability problem along control flow paths within some language w1* ... wd* over program statements is decidable in Nexptime. To achieve this upper bound, we combine a program transformation into the same class of programs but without procedures, with an Np-completeness result for the reachability problem of procedure-less programs. Besides the program, the expression w1* ... wd* is also mapped onto an expression of a similar form but this time over the transformed program statements. Several arguments involving context-free grammars and their generative process enable us to give tight bounds on the size of the resulting expression. The currently existing gap between Np-hard and Nexptime can be closed to Np-complete when a certain parameter of the analysis is assumed to be constant.Comment: 38 pages, 1 figur

A Perfect Model for Bounded Verification

A class of languages C is perfect if it is closed under Boolean operations and the emptiness problem is decidable. Perfect language classes are the basis for the automata-theoretic approach to model checking: a system is correct if the language generated by the system is disjoint from the language of bad traces. Regular languages are perfect, but because the disjointness problem for CFLs is undecidable, no class containing the CFLs can be perfect. In practice, verification problems for language classes that are not perfect are often under-approximated by checking if the property holds for all behaviors of the system belonging to a fixed subset. A general way to specify a subset of behaviors is by using bounded languages (languages of the form w1* ... wk* for fixed words w1,...,wk). A class of languages C is perfect modulo bounded languages if it is closed under Boolean operations relative to every bounded language, and if the emptiness problem is decidable relative to every bounded language. We consider finding perfect classes of languages modulo bounded languages. We show that the class of languages accepted by multi-head pushdown automata are perfect modulo bounded languages, and characterize the complexities of decision problems. We also show that bounded languages form a maximal class for which perfection is obtained. We show that computations of several known models of systems, such as recursive multi-threaded programs, recursive counter machines, and communicating finite-state machines can be encoded as multi-head pushdown automata, giving uniform and optimal underapproximation algorithms modulo bounded languages.Comment: 14 pages, 6 figure

Reasoning about reversal-bounded counter machines

International audienceIn this paper, we present a short survey on reversal-bounded counter machines. It focuses on the main techniques for model-checking such counter machines with specifications expressed with formulae from some linear-time temporal logic. All the decision procedures are designed by translation into Presburger arithmetic. We provide a proof that is alternative to Ibarra's original one for showing that reachability sets are effectively definable in Presburger arithmetic. Extensions to repeated control state reachability and to additional temporal properties are discussed in the paper. The article is written to the honor of Professor Ewa Orłowska and focuses on several topics that are developped in her works

Complexity of Pattern-based Verification for Multithreaded Programs

Pattern-based verification checks the correctness of the program executions that follow a given pattern, a regular expression over the alphabet of program transitions of the form w ∗ 1... w ∗ n. For multithreaded programs, the alphabet of the pattern is given by the synchronization operations between threads. We study the complexity of pattern-based verification for abstracted multithreaded programs in which, as usual in program analysis, conditions have been replaced by nondeterminism (the technique works also for boolean programs). While unrestricted verification is undecidable for abstracted multithreaded programs with recursive procedures and PSPACE-complete for abstracted multithreaded while-programs, we show that pattern-based verification is NP-complete for both classes. We then conduct a multiparameter analysis in which we study the complexity in the number of threads, the number of procedures per thread, the size of the procedures, and the size of the pattern. We first show that no algorithm for pattern-based verification can be polynomial in the number of threads, procedures per thread, or the size of the pattern (unless P=NP). Then, using recent results about Parikh images of regular languages and semilinear sets, we present an algorithm exponential in the number of threads, procedures per thread, and size of the pattern, but polynomial in the size of the procedures