We show that the spaces of transfinite words, namely ordinalindexed words, over a Noetherian space, is also Noetherian, under a natural topology which we call the regular subword topology. We characterize its sobrification and its specialization ordering, and we give an upper bound on its dimension and on its stature.

The complexity of a well-quasi-order (wqo) can be measured through three classical ordinal invariants: the width as a measure of antichains, the height as a measure of chains, and the maximal order type as a measure of bad sequences. This article considers the "finitary powerset" construction: the collection Pf(X) of finite subsets of a wqo X ordered with the Hoare embedding relation remains a wqo. The width, height and maximal order type of Pf(X) cannot be expressed as a function of the invariants of X, and we provide tight upper and lower bounds for the three invariants. The article also identifies an algebra of well-behaved wqos, that include finitary powersets as well as other more classical constructions, and for which the ordinal invariants can be computed compositionnally. This relies on a new ordinal invariant called the approximated maximal order type.

Elegant and general algorithms for handling upwards-closed and downwards-closed subsets of WQOs can be developed using the filter-based and ideal-based representation for these sets. These algorithms can be built in a generic or parameterized way, in parallel with the way complex WQOs are obtained by combining or modifying simpler WQOs.

We show that the shuffle L (sic) F of a piecewise-testable language L and a finite language F is piecewise-testable. The proof relies on a classic but little-used automata-theoretic characterization of piecewise-testable languages. We also discuss some mild generalizations of the main result, and provide bounds on the piecewise complexity of L (sic) F. (C) 2019 Elsevier B.V. All rights reserved.

We study the computational complexity of reachability, coverability and inclusion for extensions of context-free commutative grammars with integer counters and reset operations on them. Those grammars can alternatively be viewed as an extension of communication-free Petri nets. Our main results are that reachability and coverability are inter-reducible and both NP-complete. In particular, this class of commutative grammars enjoys semi-linear reachability sets. We also show that the inclusion problem is, in general, coNEXP-complete and already $\Pi_2^\text{P}$-complete for grammars with only one non-terminal symbol. Showing the lower bound for the latter result requires us to develop a novel $\Pi_2^\text{P}$-complete variant of the classic subset sum problem.

Equivalence of deterministic pushdown automata is a famous problem in theoretical computer science whose decidability has been shown by Senizergues. Our first result shows that decidability no longer holds when moving from finite words to infinite words. This solves an open problem that has recently been raised by Loding. In fact, we show that already the equivalence problem for deterministic Buchi one-counter automata is undecidable. Hence, the decidability border is rather tight when taking into account a recent result by Loding and Repke that equivalence of deterministic weak parity pushdown automata (a subclass of deterministic Buchi pushdown automata) is decidable. Another known result on finite words is that the universality problem for vector addition systems is decidable. We show undecidability when moving to infinite words. In fact, we prove that already the universality problem for nondeterministic Buchi one-counter nets (or equivalently vector addition systems with one unbounded dimension) is undecidable.

We consider first-order logic over the subword ordering on finite words where each word is available as a constant. Our first result is that the Σ 1 theory is undecidable (already over two letters). We investigate the decidability border by considering fragments where all but a certain number of variables are alternation bounded, meaning that the variable must always be quantified over languages with a bounded number of letter alternations. We prove that when at most two variables are not alternation bounded, the Σ 1 fragment is decidable, and that it becomes undecidable when three variables are not alternation bounded. Regarding higher quantifier alternation depths, we prove that the Σ 2 fragment is undecidable already for one variable without alternation bound and that when all variables are alternation bounded, the entire first-order theory is decidable.

The introduction of Well Structured Transition Systems (WSTS) in 1987 [8], i.e. transition systems that satisfies a monotony property with respect to some well-quasi-ordering (wqo), has led to an important number of decidability results of verification problems for several natural models: Petri Nets and VASS and a large number of their extensions, lossy channel systems, string rewrite systems, process algebra, communicating automaton, and so on. Surveys of results and applications obtained with this theory can be found in [9, 3, 1, 2]. The main idea behind these decidability results is a generic algorithm that explores a tree that must be finite by the wqo property: every infinite sequence of configurations of the system (xi)i∈N has an increasing pair, that is a pair i < j such that xi ≤ xj . Moreover, wqo theory has provided upper bound to these algorithms by bounding the length of so called bad sequences, (finite) sequences that do not have an increasing pair. The bounds obtained are non-primitive-recursive, which is unusual in verification. In addition, matching lower bounds has been proved for several models [19, 6].

This paper studies reachability, coverability and inclusion problems for Integer Vector Addition Systems with States (ℤ-VASS) and extensions and restrictions thereof. A ℤ-VASS comprises a finite-state controller with a finite number of counters ranging over the integers. Although it is folklore that reachability in ℤ-VASS is NP-complete, it turns out that despite their naturalness, from a complexity point of view this class has received little attention in the literature. We fill this gap by providing an in-depth analysis of the computational complexity of the aforementioned decision problems. Most interestingly, it turns out that while the addition of reset operations to ordinary VASS leads to undecidability and Ackermann-hardness of reachability and coverability, respectively, they can be added to ℤ-VASS while retaining NP-completeness of both coverability and reachability.

Représentations Effectives des Beaux Pré-Ordres Avec des motivations venant du domaine de la Vérification, nous définissons une notion de WQO effectifs pour lesquels il est possible de représenter les ensembles clos et de calculer les principales opérations ensemblistes sur ces représentations. Dans une première partie, nous montrons que de nombreuses constructions naturelles sur les WQO préservent notre notion d'effectivité, prouvant ainsi que la plupart des WQOs utilisés en pratique sont effectifs. Cette partie est basée sur un article non publié dont Jean Goubault-Larrecq, Narayan Kumar, Prateek Karandikar et Philippe Schnoebelen sont co-auteurs.Dans une seconde partie, nous étudions les conséquences qu'a notre notion sur la logique du première ordre interprété sur un WQO. Bien que le fragment existentiel positif soit décidable pour tous les WQOs effectif, les perspectives de généralisation sont limitées par le résultat suivant: le fragment existentiel de la logique du première ordre sur les mots finis, ordonnés par plongement, est déjà indécidable. Ce résultat a été publié à LICS 2017 avec Philippe Schnoebelen et Georg Zetzsche.