Corto Mascle

Richard Mandel, Corto Mascle, Georg Zetzsche

Indexed languages are a classical notion in formal language theory, which has attracted attention in recent decades due to its role in higher-order model checking: They are precisely the languages accepted by order-2 pushdown automata. The downward closure of an indexed language -- the set of all (scattered) subwords of its members -- is well-known to be a regular over-approximation. It is known since 2015 that the downward closure of a given indexed language is effectively computable. However, the algorithm comes with no complexity bounds, and it has remained open whether a primitive-recursive construction exists. We settle this question and provide a triply (resp. quadruply) exponential construction of a non-deterministic (resp. deterministic) automaton. We also prove (asymptotically) matching lower bounds. For the upper bounds, we rely on recent advances in semigroup theory, which let us compute bounded-size summaries of words with respect to a finite semigroup. By replacing stacks with their summaries, we are able to transform an indexed grammar into a context-free one with the same downward closure, and then apply existing bounds for context-free grammars.

Antonio Casares, Olivier Idir, Denis Kuperberg et al.

We present a polynomial-time algorithm minimising the number of states of history-deterministic generalised coBüchi automata, building on the work of Abu Radi and Kupferman on coBüchi automata. On the other hand, we establish that the minimisation problem for both deterministic and history-deterministic generalised Büchi automata is NP-complete, as well as the problem of minimising at the same time the number of states and colours of history-deterministic generalised coBüchi automata.

Michael Blondin, Michaël Cadilhac, Benjamin Courchesne et al.

Population protocols are a distributed computation model in which a collection of anonymous, finite-state agents interact in randomly chosen pairs and update their states according to a fixed transition function. The computation is defined by the eventual stabilization of the population to a consensus that represents the output. In practice, it is natural to allow each agent to carry a unique identifier and compare it with that of another agent before interacting. We model this extension by having agents be totally ordered and interactions between two agents to be fireable only if their pair of identifiers falls in some condition set. For instance, $\mathsf{PP}[<]$ allows for two agents to interact only if the first one appears before the second one. We study population protocols over ordered agents $\mathsf{PP}[N]$ where $N$ is a set of predicates available to restrict transition firing. We also study $\textsf{IO-PP}[N]$, the immediate observation fragment of $\mathsf{PP}[N]$ where only one agent changes state per interaction. Our main result is that $\textsf{IO-PP}[<]$ recognizes exactly the unambiguous star-free languages, which admits many other characterizations, such as two-variable first-order logic or two-way deterministic partially-ordered automata. We also provide a logic and an automaton model that fits in $\mathsf{PP}[<]$. We further show that if the successor predicate appears in a set $N$ of $\mathsf{NSPACE}(n)$-computable predicates, then $\textsf{IO-PP}[N]=\mathsf{PP}[N]=\mathsf{NSPACE}(n)$. Finally, we investigate the problem of deciding whether a given population protocol always stabilizes to a consensus. While this problem is decidable for unordered population protocols, we show that this is undecidable already for $\mathsf{PP}[<]$ and $\textsf{IO-PP}[+1]$, but conditionally decidable for $\textsf{IO-PP}[<]$.

Corto Mascle, Nathanaël Fijalkow, Guillaume Lagarde

We study the problem of learning linear temporal logic (LTL) formulas from examples, as a first step towards expressing a property separating positive and negative instances in a way that is comprehensible for humans. In this paper we initiate the study of the computational complexity of the problem. Our main results are hardness results: we show that the LTL learning problem is NP-complete, both for the full logic and for almost all of its fragments. This motivates the search for efficient heuristics, and highlights the complexity of expressing separating properties in concise natural language.

Yoan Géran, Bastien Laboureix, Corto Mascle et al.

We introduce a new formalisation of languages, called keyboards. We consider a set of elementary operations (writing/erasing a letter, going to the right or to the left,...) and we define a keyboard as a set of finite sequences of such operations, called keys. The corresponding language is the set of words obtained by applying some sequence of those keys. Unlike classical models of computation, every key can be applied anytime. We define various classes of languages based on different sets of elementary operations, and compare their expressive powers. We also compare them to well-known classes of languages (Chomsky hierarchy). We obtain a strict hierarchy of languages, whose expressivity is orthogonal to the one of the aforementionned classical models. -- Nous introduisons une nouvelle représentation de langages, les claviers. On se munit d'un ensemble d'opérations élémentaires (ajout, effacement d'une lettre, déplacement à droite, à gauche, ...), et on définit un clavier comme un ensemble de suites finies d'opérations élémentaires, appelées touches. Son langage sera l'ensemble des mots obtenus en appliquant une suite quelconque de touches. Contrairement à des modèles de calcul classiques, toutes les touches peuvent être appliquées à tout moment. En premier lieu nous définissons différentes classes de claviers en faisant varier l'ensemble des opérations élémentaires autorisées, et nous comparons l'expressivité des classes de langages obtenues. Nous comparons également ces classes à la hiérarchie de Chomsky. Nous obtenons que toutes les classes étudiées sont différentes, et nous caractérisons les classes inclues dans les rationnels et les algébriques. L'expressivité des claviers semble orthogonale à celle des modèles évoqués précédemment.