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**Published in:** LIPIcs, Volume 82, 26th EACSL Annual Conference on Computer Science Logic (CSL 2017)

We prove that the Büchi topology, the automatic topology, the alphabetic topology and the strong alphabetic topology are Polish, and provide consequences of this.

Olivier Carton, Olivier Finkel, and Dominique Lecomte. Polishness of Some Topologies Related to Automata. In 26th EACSL Annual Conference on Computer Science Logic (CSL 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 82, pp. 22:1-22:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{carton_et_al:LIPIcs.CSL.2017.22, author = {Carton, Olivier and Finkel, Olivier and Lecomte, Dominique}, title = {{Polishness of Some Topologies Related to Automata}}, booktitle = {26th EACSL Annual Conference on Computer Science Logic (CSL 2017)}, pages = {22:1--22:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-045-3}, ISSN = {1868-8969}, year = {2017}, volume = {82}, editor = {Goranko, Valentin and Dam, Mads}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2017.22}, URN = {urn:nbn:de:0030-drops-76728}, doi = {10.4230/LIPIcs.CSL.2017.22}, annote = {Keywords: Automata and formal languages; logic in computer science; infinite words; B\"{u}chi automaton; regular omega-language; Cantor space; finer topologies; B\"{u}c} }

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**Published in:** LIPIcs, Volume 14, 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)

We prove that the determinacy of Gale-Stewart games whose winning sets are accepted by real-time 1-counter Büchi automata is equivalent to the determinacy of (effective) analytic Gale-Stewart games which is known to be a large cardinal assumption. We show also that the determinacy of Wadge games between two players in charge of omega-languages accepted by 1-counter Büchi automata is equivalent to the (effective) analytic Wadge determinacy. Using some results of set theory we prove that one can effectively construct a 1-counter Büchi automaton A and a Büchi automaton B such that: (1) There exists a model of ZFC in which Player 2 has a winning strategy in the Wadge game W(L(A), L(B)); (2) There exists a model of ZFC in which the Wadge game W(L(A), L(B)) is not determined. Moreover these are the only two possibilities, i.e. there are no models of ZFC in which Player 1 has a winning strategy in the Wadge game W(L(A), L(B)).

Olivier Finkel. The Determinacy of Context-Free Games. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 555-566, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{finkel:LIPIcs.STACS.2012.555, author = {Finkel, Olivier}, title = {{The Determinacy of Context-Free Games}}, booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)}, pages = {555--566}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-35-4}, ISSN = {1868-8969}, year = {2012}, volume = {14}, editor = {D\"{u}rr, Christoph and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.555}, URN = {urn:nbn:de:0030-drops-33897}, doi = {10.4230/LIPIcs.STACS.2012.555}, annote = {Keywords: Automata and formal languages, logic in computer science, Gale-Stewart games, Wadge games, determinacy, context-free games} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 8271, Topological and Game-Theoretic Aspects of Infinite Computations (2008)

The operation of taking the omega-power $V^omega$ of a language $V$ is a fundamental operation over finitary languages leading to omega-languages. Since the set $X^omega$ of infinite words over a finite alphabet $X$ can be equipped with the usual Cantor topology, the question of the topological complexity of omega-powers of finitary languages naturally arises and has been posed by Damian Niwinski (1990), Pierre Simonnet (1992), and Ludwig Staiger (1997). We investigate the topological complexity of omega-powers. We prove the following very surprising results which show that omega-powers exhibit a great opological complexity: for each non-null countable ordinal $xi$, there exist some $Sigma^0_xi$-complete omega-powers, and some $Pi^0_xi$-complete omega-powers. On the other hand, the Wadge hierarchy is a great refinement of the Borel hierarchy, determined by Bill Wadge. We show that, for each ordinal $xi$ greater than or equal to 3, there are uncountably many Wadge degrees of omega-powers of Borel rank $xi +1$. Using tools of effective descriptive set theory, we prove some effective versions of the above results.

Olivier Finkel and Dominique Lecomte. Topological Complexity of omega-Powers: Extended Abstract. In Topological and Game-Theoretic Aspects of Infinite Computations. Dagstuhl Seminar Proceedings, Volume 8271, pp. 1-9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{finkel_et_al:DagSemProc.08271.7, author = {Finkel, Olivier and Lecomte, Dominique}, title = {{Topological Complexity of omega-Powers: Extended Abstract}}, booktitle = {Topological and Game-Theoretic Aspects of Infinite Computations}, pages = {1--9}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2008}, volume = {8271}, editor = {Peter Hertling and Victor Selivanov and Wolfgang Thomas and William W. Wadge and Klaus Wagner}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.08271.7}, URN = {urn:nbn:de:0030-drops-16505}, doi = {10.4230/DagSemProc.08271.7}, annote = {Keywords: Infinite words, omega-languages, omega-powers, Cantor topology, topological complexity, Borel sets, Borel ranks, complete sets, Wadge hierarchy, Wadge} }

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