Decidability in the logic of subsequences and supersequences

We consider first-order logics of sequences ordered by the subsequence ordering, aka sequence embedding. We show that the \Sigma_2 theory is undecidable, answering a question left open by Kuske. Regarding fragments with a bounded number of variables, we show that the FO2 theory is decidable while the FO3 theory is undecidable

The Height of Piecewise-Testable Languages with Applications in Logical Complexity

The height of a piecewise-testable language L is the maximum length of the words needed to define L by excluding and requiring given subwords. The height of L is an important descriptive complexity measure that has not yet been investigated in a systematic way. This paper develops a series of new techniques for bounding the height of finite languages and of languages obtained by taking closures by subwords, superwords and related operations. As an application of these results, we show that FO^2(A^*, subword), the two-variable fragment of the first-order logic of sequences with the subword ordering, can only express piecewise-testable properties and has elementary complexity

Sous-mots : automates, problèmes de plongement, et vérification

The increasing use of software and automated systems has made it important to ensure their correct behaviour. Formal verification is the technique that establishes correctness of a system or a mathematical model of the system with respect to properties expressed in a formal language.Regular model checking is a common technique for verification of infinite-state systems - it represents infinite sets of configurations symbolically in a finite manner and manipulates them using these representations. Regular model checking for lossy channel systems brings up basic automata-theoretic questions concerning the subword relation on words which models the lossiness of the channels. We address these state complexity and decision problems, and also solve a long-standing problem involving the index of the Simon's piecewise-testability congruence.The reachability problem for lossy channel systems (LCS), though decidable, has very high F_{omega^omega} complexity, well beyond primitive-recursive. In recent times several problems with this complexity have been discovered, for example in the fields of verification of weak memory models and metric temporal logic. The Post Embedding Problem (PEP) is an algebraic abstraction of the reachability problem on LCS, with the same complexity, and is our champion for a "master" problem for the class F_{omega^omega}. We provide a common generalization of two known variants of PEP and give a simpler proof of decidability. This allows us to extend the unidirectional channel system (UCS) model with simple channel tests while having decidable reachability.Garantir le fonctionnement correct des systèmes informatisés est un enjeu chaque jour plus important. La vérification formelle est un ensemble de techniquespermettant d’établir la correction d’un modèle mathématique du système par rapport à des propriétés exprimées dans un langage formel.Le "Regular model checking" est une technique bien connuede vérification de systèmes infinis. Elle manipule des ensembles infinis de configurations représentés de façon symbolique. Le "Regular model checking" de systèmes à canaux non fiables (LCS) soulève des questions fondamentales de décision et de complexité concernant l’ordre sous-mot qui modélise la perte de messages. Nous abordons ces questions et résolvons un problème ouvert sur l’index de la congruence de Simon pour les langages testables par morceaux.L’accessibilité pour les LCS est décidable mais de complexité F_{omega^omega} très élevée, bien au delà des complexités primitives récursives. Plusieurs problèmes de complexité équivalente ont été découverts récemment, par exemple dans la vérification de mémoire faibles ou de logique temporelle métrique. Le problème de plongement de Post (PEP) est une abstraction de l’accessibilité des LCS, lui aussi de complexité F_{omega^omega}, et qui nous sert de base dans la définition d’une classe de complexité correspondante. Nous proposons une généralisation commune aux deux variantes existantes de PEP et donnons une preuve de décidabilité simplifiée. Ceci permet d’étendre le modèle des systèmes à canaux unidirectionnels (UCS) par des tests simples tout en préservant la décidabilité de l’accessibilité