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

Symbolic methods and automata

peer reviewe

Subcubic certificates for CFL reachability

Many problems in interprocedural program analysis can be modeled as the context-free language (CFL) reachability problem on graphs and can be solved in cubic time. Despite years of efforts, there are no known truly sub-cubic algorithms for this problem. We study the related certification task: given an instance of CFL reachability, are there small and efficiently checkable certificates for the existence and for the non-existence of a path? We show that, in both scenarios, there exist succinct certificates (O(n^2) in the size of the problem) and these certificates can be checked in subcubic (matrix multiplication) time. The certificates are based on grammar-based compression of paths (for reachability) and on invariants represented as matrix inequalities (for non-reachability). Thus, CFL reachability lies in nondeterministic and co-nondeterministic subcubic time. A natural question is whether faster algorithms for CFL reachability will lead to faster algorithms for combinatorial problems such as Boolean satisfiability (SAT). As a consequence of our certification results, we show that there cannot be a fine-grained reduction from SAT to CFL reachability for a conditional lower bound stronger than n^ω, unless the nondeterministic strong exponential time hypothesis (NSETH) fails. In a nutshell, reductions from SAT are unlikely to explain the cubic bottleneck for CFL reachability. Our results extend to related subcubic equivalent problems: pushdown reachability and 2NPDA recognition; as well as to all-pairs CFL reachability. For example, we describe succinct certificates for pushdown non-reachability (inductive invariants) and observe that they can be checked in matrix multiplication time. We also extract a new hardest 2NPDA language, capturing the “hard core” of all these problems

An efficient automata approach to some problems on context-free grammars

Book and Otto (1993) solve a number of word problems for monadic string-rewriting systems using an elegant automata-based technique. In this note we observe that the technique is also very interesting from a pedagogical point of view, since it provides a uniform solution to several elementary problems on context-free languages

Actes de l'Ecole d'Eté Temps Réel 2005 - ETR'2005

Pdf des actes disponible à l'URL http://etr05.loria.fr/Le programme de l'Ecole d'été Temps Réel 2005 est construit autour d'exposés de synthèse donnés par des spécialistes du monde industriel et universitaire qui permettront aux participants de l'ETR, et notamment aux doctorants, de se forger une culture scientifique dans le domaine. Cette quatrième édition est centrée autour des grands thèmes d'importance dans la conception des systèmes temps réel : Langages et techniques de description d'architectures, Validation, test et preuve par des approches déterministes et stochastiques, Ordonnancement et systèmes d'exploitation temps réel, Répartition, réseaux temps réel et qualité de service