When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2009.1849
URN: urn:nbn:de:0030-drops-18493
URL: https://drops.dagstuhl.de/opus/volltexte/2009/1849/
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### On the Borel Inseparability of Game Tree Languages

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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 $\mu$-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 $n=2$.

### BibTeX - Entry

@InProceedings{hummel_et_al:LIPIcs:2009:1849,
author =	{Szczepan Hummel and Henryk Michalewski and Damian Niwinski},
title =	{{On the Borel Inseparability of Game Tree Languages}},
booktitle =	{26th International Symposium on Theoretical Aspects of Computer Science},
pages =	{565--576},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-939897-09-5},
ISSN =	{1868-8969},
year =	{2009},
volume =	{3},
editor =	{Susanne Albers and Jean-Yves Marion},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},