Linear Game Automata: Decidable Hierarchy Problems for Stripped-Down Alternating Tree Automata

Jacques Duparc (1) , Alessandro Facchini (2) , Filip Murlak (3)

1 ISI - Institut des systèmes d'information
2 LaBRI - Laboratoire Bordelais de Recherche en Informatique
3 LFCS - Laboratory for the Foundations of Computer Science [Edinburgh]

For deterministic tree automata, classical hierarchies, like Mostowski-Rabin (or index) hierarchy, Borel hierarchy, or Wadge hierarchy, are known to be decidable. However, when it comes to non-deterministic tree automata, none of these hierarchies is even close to be understood. Here we make an attempt in paving the way towards a clear understanding of tree automata. We concentrate on the class of linear game automata (LGA), and prove within this new context, that all corresponding hierarchies mentioned above -Mostowski-Rabin, Borel, and Wadge - are decidable. The class LGA is obtained by taking linear tree automata with alternation restricted to the choice of path in the input tree. Despite their simplicity, LGA recognize sets of arbitrary high Borel rank. The actual richness of LGA is revealed by the height of their Wadge hierarchy: $({\omega^\omega})^\omega$.

Mots clés: weak alternating tree automata; Wadge hierarchy; Index hierarchy; Borel hierarchy

Domaines: Logique en informatique [cs.LO]

Jacques Duparc, Alessandro Facchini, Filip Murlak. Linear Game Automata: Decidable Hierarchy Problems for Stripped-Down Alternating Tree Automata. 23rd International Workshop, CSL 2009, 18th Annual Conference of the EACSL, Sep 2009, Coimbra, Portugal. pp.225-239. ⟨hal-00373351v2⟩