On the Borel Inseparability of Game Tree Languages

Szczepan Hummel; Henryk Michalewski; Damian Niwinski

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arXiv · 0902.1732

On the Borel Inseparability of Game Tree Languages

Szczepan Hummel·Henryk Michalewski·Damian Niwinski

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Abstract

The game tree languages can be viewed as an automata-theoretic counterpart of parity games on graphs. They witness the strictness of the index hierarchy of alternating tree automata, as well as the fixed-point hierarchy over binary trees. We consider a game tree language of the first non-trivial level, where Eve can force that 0 repeats from some moment on, and its dual, where Adam can force that 1 repeats from some moment on. Both these sets (which amount to one up to an obvious renaming) are complete in the class of co-analytic sets. We show that they cannot be separated by any Borel set, hence {\em a fortiori} by any weakly definable set of trees. This settles a case left open by L.Santocanale and A.Arnold, who have thoroughly investigated the separation property within the $μ$-calculus and the automata index hierarchies. They showed that separability fails in general for non-deterministic automata of type $Σ^μ_{n} $, starting from level $n=3$, while our result settles the missing case $n=2$.

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Szczepan Hummel, Henryk Michalewski, Damian Niwinski. 2009-02-10. On the Borel Inseparability of Game Tree Languages. https://arxiv.org/abs/0902.1732

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