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Game Characterizations and Lower Cones in the Weihrauch Degrees

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Unveiling Dynamics and Complexity (CiE 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10307))

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Abstract

We introduce generalized Wadge games and show that each lower cone in the Weihrauch degrees is characterized by such a game. These generalized Wadge games subsume (a variant of) the original Wadge game, the eraser and backtrack games as well as Semmes’s tree games. In particular, we propose that the lower cones in the Weihrauch degrees are the answer to Andretta’s question on which classes of functions admit game characterizations. We then discuss some applications of such generalized Wadge games.

This research was partially done whilst the authors were visiting fellows at the Isaac Newton Institute for Mathematical Sciences in the programme ‘Mathematical, Foundational and Computational Aspects of the Higher Infinite’. The research benefited from the Royal Society International Exchange Grant Infinite games in logic and Weihrauch degrees. The first author was partially supported by a CAPES Science Without Borders grant (9625/13-5), and the second author was partially supported by the ERC inVEST (279499) project.

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Notes

  1. 1.

    A key lemma for the proof of this theorem goes back to helpful comments by Takayuki Kihara.

  2. 2.

    While Pequignot only introduces the notion for second countable \(T_0\) spaces, the extension to all represented spaces is immediate. Note that one needs to take into account that for general represented spaces, the Borel sets can show unfamiliar properties, e.g., even singletons can fail to be Borel (cf. also [37, 38]).

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Acknowledgments

We are grateful to Benedikt Löwe, Luca Motto Ros, Takayuki Kihara and Raphaël Carroy for helpful and inspiring discussions. We would also like to thank the anonymous referees for the many corrections which have significantly improved the paper.

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Authors and Affiliations

  1. Institute for Logic, Language, and Computation, University of Amsterdam, Amsterdam, The Netherlands

    Hugo Nobrega

  2. Département D’informatique, Université Libre de Bruxelles, Brussels, Belgium

    Arno Pauly

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  1. Hugo Nobrega
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  2. Arno Pauly
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Correspondence to Hugo Nobrega .

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  1. University of Turku, Turku, Finland

    Jarkko Kari

  2. Christian-Albrechts-University of Kiel, Kiel, Germany

    Florin Manea

  3. Åbo Akademi University, Turku, Finland

    Ion Petre

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Nobrega, H., Pauly, A. (2017). Game Characterizations and Lower Cones in the Weihrauch Degrees. In: Kari, J., Manea, F., Petre, I. (eds) Unveiling Dynamics and Complexity. CiE 2017. Lecture Notes in Computer Science(), vol 10307. Springer, Cham. https://doi.org/10.1007/978-3-319-58741-7_31

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  • DOI: https://doi.org/10.1007/978-3-319-58741-7_31

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