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A Survey on Universal Computably Enumerable Equivalence Relations

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Book cover Computability and Complexity

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10010))

Abstract

We review the literature on universal computably enumerable equivalence relations, i.e. the computably enumerable equivalence relations (ceers) which are \(\Sigma ^0_1\)-complete with respect to computable reducibility on equivalence relations. Special attention will be given to the so-called uniformly effectively inseparable (u.e.i.) ceers, i.e. the nontrivial ceers yielding partitions of the natural numbers in which each pair of distinct equivalence classes is effectively inseparable (uniformly in their representatives). The u.e.i. ceers comprise infinitely many isomorphism types. The relation of provable equivalence in Peano Arithmetic plays an important role in the study and classification of the u.e.i. ceers.

Andrews was partially supported by NSF grant DMS-1201338.

Badaev was partially supported by Grant 3952/GF4 of the Science Committee of the Republic of Kazakhstan.

Sorbi is a member of INDAM-GNSAGA; he was partially supported by Grant 3952/GF4 of the Science Committee of the Republic of Kazakhstan, and by PRIN 2012 “Logica Modelli e Insiemi”.

The authors thank Keng Meng Ng for his helpful comments and suggestions. The wish also to thank an anonymous referee for having pointed out several inaccuracies in a previous version of the paper.

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Authors and Affiliations

  1. Department of Mathematics, University of Wisconsin, Madison, Wisconsin, 53706-1388, USA

    Uri Andrews

  2. Department of Fundamental Mathematics, Al-Farabi Kazakh National University, Almaty, 050040, Kazakhstan

    Serikzhan Badaev

  3. Dipartimento di Ingegneria Informatica e Scienze Matematiche, Università Degli Studi di Siena, 53100, Siena, Italy

    Andrea Sorbi

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  1. Uri Andrews
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  2. Serikzhan Badaev
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  3. Andrea Sorbi
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Correspondence to Andrea Sorbi .

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Editors and Affiliations

  1. Victoria University of Wellington, Wellington, New Zealand

    Adam Day

  2. University of Bergen, Bergen, Norway

    Michael Fellows

  3. Victoria University of Wellington, Wellington, New Zealand

    Noam Greenberg

  4. University of Auckland, Auckland, New Zealand

    Bakhadyr Khoussainov

  5. Massey University, Auckland, New Zealand

    Alexander Melnikov

  6. University of Bergen, Bergen, Norway

    Frances Rosamond

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Andrews, U., Badaev, S., Sorbi, A. (2017). A Survey on Universal Computably Enumerable Equivalence Relations. In: Day, A., Fellows, M., Greenberg, N., Khoussainov, B., Melnikov, A., Rosamond, F. (eds) Computability and Complexity. Lecture Notes in Computer Science(), vol 10010. Springer, Cham. https://doi.org/10.1007/978-3-319-50062-1_25

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  • DOI: https://doi.org/10.1007/978-3-319-50062-1_25

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