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On the Borel Inseparability of Game Tree Languages

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 $\mu $-calculus and the automata index hierarchies. They showed that separability fails in general for non-deterministic automata of type $\Sigma^{\mu }_{n} $, starting from level $n=3$, while our result settles the missing case $n=2$.
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Submitted on : Tuesday, February 10, 2009 - 3:47:26 PM
Last modification on : Monday, June 8, 2020 - 9:48:02 AM
Long-term archiving on: : Tuesday, June 8, 2010 - 8:37:45 PM

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  • HAL Id : inria-00360184, version 1
  • ARXIV : 0902.1732

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Szczepan Hummel, Henryk Michalewski, Damian Niwinski. On the Borel Inseparability of Game Tree Languages. 26th International Symposium on Theoretical Aspects of Computer Science STACS 2009, Feb 2009, Freiburg, Germany. pp.565-576. ⟨inria-00360184⟩

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