Emptiness Of Alternating Tree Automata Using Games With Imperfect Information

We consider the emptiness problem for alternating tree automata, with two acceptance semantics: classical (all branches are accepted) and qualitative (almost all branches are accepted). For the classical semantics, the usual technique to tackle this problem relies on a Simulation Theorem which constructs an equivalent non-deterministic automaton from the original alternating one, and then checks emptiness by a reduction to a two-player perfect information game. However, for the qualitative semantics, no simulation of alternation by means of non-determinism is known. We give an alternative technique to decide the emptiness problem of alternating tree automata, that does not rely on a Simulation Theorem. Indeed, we directly reduce the emptiness problem to solving an imperfect information two-player parity game. Our new approach can successfully be applied to both semantics, and yields decidability results with optimal complexity; for the qualitative semantics, the key ingredient in the proof is a positionality result for stochastic games played over infinite graphs

Playing with Trees and Logic

This document proposes an overview of my research sinc

Emptiness Of Alternating Parity Tree Automata Using Games With Imperfect Information

We focus on the emptiness problem for alternating parity tree automata. The usual technique to tackle this problem first removes alternation, going to non-determinism, and then checks emptiness by reduction to a two-player perfect-information parity game. In this note, we give an alternative roadmap to this problem by providing a direct reduction to the emptiness problem to solving an imperfect-information two-player parity game.Nous considérons le problème du test du vide pour les automates d'arbres alternants à parité. La méthode usuelle pour résoudre ce problème, commence par supprimer l'alternance ce qui conduit à un automate non-déterministe dont le vide est ensuite testé par réduction à un jeux de parité à information parfaite. Dans cette note, nous proposons une approche alternative pour ce problème, en proposant une réduction directe du problème du vide à un jeu de parité à information imparfaite

Qualitative Tree Languages

Abstract—We study finite automata running over infinite binary trees and we relax the notion of accepting run by allowing a negligible set (in the sense of measure theory) of non-accepting branches. In this qualitative setting, a tree is accepted by the automaton if there exists a run over this tree in which almost every branch is accepting. This leads to a new class of tree languages, called the qualitative tree languages that enjoys many properties. Then, we replace the existential quantification — a tree is accepted if there exists some accepting run over the input tree — by a probabilistic quantification — a tree is accepted if almost every run over the input tree is accepting. Together with the qualitative acceptance and the Büchi condition, we obtain a class of probabilistic tree automata with a decidable emptiness problem. To our knowledge, this is the first positive result for a class of probabilistic automaton over infinite trees. I

Randomisation in Automata on Infinite Trees

International audienceWe study finite automata running over infinite binary trees. A run of such an automaton over an input tree is a tree labeled by control states of the automaton: the labelling is built in a top-down fashion and should be consistent with the transitions of the automaton. A branch in a run is accepting if the ω-word obtained by reading the states along the branch satisfies some acceptance condition (typically an ω-regular condition such as a Büchi or a parity condition). Finally, a tree is accepted by the automaton if there exists a run over this tree in which every branch is accepting. In this paper, we consider two relaxations of this definition introducing a qualitative aspect. First, we relax the notion of accepting run by allowing a negligible set (in the sense of measure theory) of non-accepting branches. In this qualitative setting, a tree is accepted by the automaton if there exists a run over this tree in which almost every branch is accepting. This leads to a new class of tree languages, qualitative tree languages. This class enjoys many good properties: closure under union and intersection (but not under complement), emptiness is decidable in polynomial time. A dual class, positive tree languages, is defined by requiring that an accepting run contains a non-negligeable set of branches. The second relaxation, is to replace the existential quantification (a tree is accepted if there exists some accepting run over the input tree) by a probabilistic quantification (a tree is accepted if almost every run over the input tree is accepting). For the run, we may use either classical acceptance or qualitative acceptance. In particular, for the latter, we exhibit a tight connection with partial observation Markov decision processes. Moreover, if we additionally restrict to the Büchi condition, we show that it leads to a class of probabilistic automata on infinite trees enjoying a decidable emptiness problem. To our knowledge, this is the first positive result for a class of probabilistic automaton over infinite trees