10.4230/LIPICS.STACS.2009.1849
Hummel, Szczepan
Szczepan
Hummel
Michalewski, Henryk
Henryk
Michalewski
Niwinski, Damian
Damian
Niwinski
On the Borel Inseparability of Game Tree Languages
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
2009
Tree automata
Separation property
Borel sets
Parity games
Albers, Susanne
Susanne
Albers
Marion, Jean-Yves
Jean-Yves
Marion
2009
2009-02-19
2009-02-19
2009-02-19
en
urn:nbn:de:0030-drops-18493
10.4230/LIPIcs.STACS.2009
978-3-939897-09-5
1868-8969
10.4230/LIPIcs.STACS.2009
LIPIcs, Volume 3, STACS 2009
26th International Symposium on Theoretical Aspects of Computer Science
2013
3
46
565
576
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Albers, Susanne
Susanne
Albers
Marion, Jean-Yves
Jean-Yves
Marion
1868-8969
Leibniz International Proceedings in Informatics (LIPIcs)
2009
3
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
12 pages
250338 bytes
application/pdf
Creative Commons Attribution-NoDerivs 3.0 Unported license
info:eu-repo/semantics/openAccess
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 $n=2$.
LIPIcs, Vol. 3, 26th International Symposium on Theoretical Aspects of Computer Science, pages 565-576