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Documents authored by Hummel, Szczepan

Document
DOI: 10.4230/LIPIcs.CSL.2015.534
On Unambiguous Regular Tree Languages of Index (0,2)

Authors: Jacques Duparc, Kevin Fournier, and Szczepan Hummel

Published in: LIPIcs, Volume 41, 24th EACSL Annual Conference on Computer Science Logic (CSL 2015)

Abstract
Unambiguous automata are usually seen as a natural class of automata in-between deterministic and nondeterministic ones. We show that in case of infinite tree languages, the unambiguous ones are topologically far more complicated than the deterministic ones. We do so by providing operations that generate a family (A_{alpha})_{alpha < phi_2(0)} of unambiguous automata such that: 1. It respects the strict Wadge ordering: alpha < beta if and only if A_{alpha} <_W A_{beta}. This can be established without the help of any determinacy principle, simply by providing effective winning strategies in the underlying games. 2. Its length (phi_2(0)) is the first fixpoint of the ordinal function that itself enumerates all fixpoints of the ordinal exponentiation x |-> omega^x: an ordinal tremendously larger than (omega^(omega))^3 +3 which is the height of the Wadge hierarchy of deterministic tree languages as uncovered by Filip Murlak. 3. The priorities of all these parity automata only range from 0 to 2.
Cite as

Jacques Duparc, Kevin Fournier, and Szczepan Hummel. On Unambiguous Regular Tree Languages of Index (0,2). In 24th EACSL Annual Conference on Computer Science Logic (CSL 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 41, pp. 534-548, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{duparc_et_al:LIPIcs.CSL.2015.534,
  author =	{Duparc, Jacques and Fournier, Kevin and Hummel, Szczepan},
  title =	{{On Unambiguous Regular Tree Languages of Index (0,2)}},
  booktitle =	{24th EACSL Annual Conference on Computer Science Logic (CSL 2015)},
  pages =	{534--548},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-90-3},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{41},
  editor =	{Kreutzer, Stephan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2015.534},
  URN =		{urn:nbn:de:0030-drops-54369},
  doi =		{10.4230/LIPIcs.CSL.2015.534},
  annote =	{Keywords: Tree Automata, Unambiguity, Wadge Hierarchy.}
}
Document
DOI: 10.4230/LIPIcs.STACS.2009.1849
On the Borel Inseparability of Game Tree Languages

Authors: Szczepan Hummel, Henryk Michalewski, and Damian Niwinski

Published in: LIPIcs, Volume 3, 26th International Symposium on Theoretical Aspects of Computer Science (2009)

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 $n=2$.
Cite as

Szczepan Hummel, Henryk Michalewski, and Damian Niwinski. On the Borel Inseparability of Game Tree Languages. In 26th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 3, pp. 565-576, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{hummel_et_al:LIPIcs.STACS.2009.1849,
  author =	{Hummel, Szczepan and Michalewski, Henryk and Niwinski, Damian},
  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 =	{Albers, Susanne and Marion, Jean-Yves},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2009.1849},
  URN =		{urn:nbn:de:0030-drops-18493},
  doi =		{10.4230/LIPIcs.STACS.2009.1849},
  annote =	{Keywords: Tree automata, Separation property, Borel sets, Parity games}
}
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