Alternating Tree Automata with Qualitative Semantics

Raphaël Berthon (1) , Nathanaël Fijalkow (2) , Emmanuel Filiot (3) , Shibashis Guha (1) , Bastien Maubert (4) , Aniello Murano (4) , Laureline Pinault (5) , Sophie Pinchinat (6) , Sasha Rubin (7) , Olivier Serre (8)

1 ULB - Université libre de Bruxelles
2 LaBRI - Laboratoire Bordelais de Recherche en Informatique
3 ULB - Département d'Informatique [Bruxelles]
4 University of Naples Federico II = Università degli studi di Napoli Federico II
5 PLUME - Preuves et Langages
6 UR - Université de Rennes
7 The University of Sydney
8 IRIF (UMR_8243) - Institut de Recherche en Informatique Fondamentale

We study alternating automata with qualitative semantics over infinite binary trees: Alternation means that two opposing players construct a decoration of the input tree called a run, and the qualitative semantics says that a run of the automaton is accepting if almost all branches of the run are accepting. In this article, we prove a positive and a negative result for the emptiness problem of alternating automata with qualitative semantics. The positive result is the decidability of the emptiness problem for the case of Büchi acceptance condition. An interesting aspect of our approach is that we do not extend the classical solution for solving the emptiness problem of alternating automata, which first constructs an equivalent non-deterministic automaton. Instead, we directly construct an emptiness game making use of imperfect information. The negative result is the undecidability of the emptiness problem for the case of co-Büchi acceptance condition. This result has two direct consequences: the undecidability of monadic second-order logic extended with the qualitative path-measure quantifier and the undecidability of the emptiness problem for alternating tree automata with non-zero semantics, a recently introduced probabilistic model of alternating tree automata.

Domaines: Logique en informatique [cs.LO]

Raphaël Berthon, Nathanaël Fijalkow, Emmanuel Filiot, Shibashis Guha, Bastien Maubert, et al.. Alternating Tree Automata with Qualitative Semantics. ACM Transactions on Computational Logic, 2020, 22 (1), ⟨10.1145/3431860⟩. ⟨hal-02533936⟩