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On diagonal functions for equivalence relations

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Abstract

We work with weakly precomplete equivalence relations introduced by Badaev. The weak precompleteness is a natural notion inspired by various fixed point theorems in computability theory. Let E be an equivalence relation on the set of natural numbers \(\omega \), having at least two classes. A total function f is a diagonal function for E if for every x, the numbers x and f(x) are not E-equivalent. It is known that in the case of c.e. relations E, the weak precompleteness of E is equivalent to the lack of computable diagonal functions for E. Here we prove that this result fails already for \(\Delta ^0_2\) equivalence relations, starting with the \(\Pi ^{-1}_2\) level. We focus on the Turing degrees of possible diagonal functions. We prove that for any noncomputable c.e. degree \({\textbf{d}}\), there exists a weakly precomplete c.e. equivalence E admitting a \({\textbf{d}}\)-computable diagonal function. We observe that a Turing degree \({\textbf{d}}\) can compute a diagonal function for every \(\Delta ^0_2\) equivalence relation E if and only if \({\textbf{d}}\) computes \({\textbf{0}}'\). On the other hand, every PA degree can compute a diagonal function for an arbitrary c.e. equivalence E. In addition, if \({\textbf{d}}\) computes diagonal functions for all c.e. E, then \({\textbf{d}}\) must be a DNC degree.

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References

  1. Andrews, U., Badaev, S.A.: On isomorphism classes of computably enumerable equivalence relations. J. Symb. Log. 85(1), 61–86 (2020)

    Article  MathSciNet  Google Scholar 

  2. Andrews, U., Badaev, S.A., Sorbi, A.: A survey on universal computably enumerable equivalence relations. In: Day, A.R., Fellows, M.R., Greenberg, N., Khoussainov, B., Melnikov, A.G., Rosamond, F.A. (eds.) Computability and Complexity—Essays Dedicated to Rodney G. Downey on the Occasion of His 60th Birthday, Volume 10010 of Lecture Notes Computer Science, pp. 418–451. Springer, Cham (2017)

    Google Scholar 

  3. Arslanov, M.M.: On some generalizations of a fixed point theorem. Sov. Math. 25(5), 1–10 (1981)

    MathSciNet  Google Scholar 

  4. Arslanov, M.M.: Completeness in the arithmetical hierarchy and fixed points. Algebra Logic 28(1), 1–9 (1989)

    Article  MathSciNet  Google Scholar 

  5. Arslanov, M.M.: Fixed-point selection functions. Lobachevskii J. Math. 42(4), 685–692 (2021)

    Article  MathSciNet  Google Scholar 

  6. Andrews, U., Sorbi, A.: Joins and meets in the structure of Ceers. Computability 8(3–4), 193–241 (2019)

    Article  MathSciNet  Google Scholar 

  7. Badaev, S.A.: On weakly precomplete positive equivalences. Sib. Math. J. 32(2), 321–323 (1991)

    Article  MathSciNet  Google Scholar 

  8. Bazhenov, N.A., Kalmurzaev, B.S.: On dark computably enumerable equivalence relations. Sib. Math. J. 59(1), 22–30 (2018)

    Article  MathSciNet  Google Scholar 

  9. Bazhenov, N.A., Kalmurzaev, B.S.: Weakly precomplete equivalence relations in the Ershov hierarchy. Algebra Logic 58(3), 199–213 (2019)

    Article  MathSciNet  Google Scholar 

  10. Bernardi, C., Sorbi, A.: Classifying positive equivalence relations. J. Symb. Log. 48(3), 529–538 (1983)

    Article  MathSciNet  Google Scholar 

  11. Badaev, S., Sorbi, A.: Weakly precomplete computably enumerable equivalence relations. Math. Log. Q. 62(1–2), 111–127 (2016)

    Article  MathSciNet  Google Scholar 

  12. Barendregt, H., Terwijn, S.A.: Fixed point theorems for precomplete numberings. Ann. Pure Appl. Logic 170(10), 1151–1161 (2019)

    Article  MathSciNet  Google Scholar 

  13. Downey, R.G., Hirschfeldt, D.R.: Algorithmic Randomness and Complexity. Springer, New York (2010)

    Book  Google Scholar 

  14. Dekker, J.C.E., Myhill, J.: Retraceable sets. Can. J. Math. 10, 357–373 (1958)

    Article  MathSciNet  Google Scholar 

  15. Ershov, Yu.L.: A hierarchy of sets. I. Algebra Logic 7(1), 25–43 (1968)

    Article  MathSciNet  Google Scholar 

  16. Ershov, Yu.L.: On a hierarchy of sets. II. Algebra Logic 7(4), 212–232 (1968)

    Article  MathSciNet  Google Scholar 

  17. Ershov, Yu.L.: On a hierarchy of sets. III. Algebra Logic 9(1), 20–31 (1970)

    Article  MathSciNet  Google Scholar 

  18. Ershov, Yu.L.: Theory of Numberings. Nauka, Moscow (1977). (in Russian)

    Google Scholar 

  19. Golov, A., Terwijn, S.A.: Fixpoints and relative precompleteness. Computability 11(2), 135–146 (2022)

    Article  MathSciNet  Google Scholar 

  20. Jockusch, C.G., Jr., Lerman, M., Soare, R.I., Solovay, R.: Recursively enumerable sets modulo iterated jumps and extensions of Arslanov’s completeness criterion. J. Symb. Log. 54(4), 1288–1323 (1989)

    Article  MathSciNet  Google Scholar 

  21. Jockusch, C.G., Jr.: Degrees of functions with no fixed points. In: Fenstad, J.E., Frolov, I.T., Hilpinen, R. (eds.) Logic, Methodology and Philosophy of Science VIII, Volume 126 of Studies in Logic and the Foundations of Mathematics, pp. 191–201. Elsevier, Amsterdam (1989)

    Google Scholar 

  22. Jockusch, C.G., Jr., Soare, R.I.: \(\Pi ^0_1\) classes and degrees of theories. Trans. Am. Math. Soc. 173, 33–56 (1972)

    Google Scholar 

  23. Kleene, S.C.: On notation for ordinal numbers. J. Symb. Log. 3(4), 150–155 (1938)

    Article  MathSciNet  Google Scholar 

  24. Kučera, A.: Measure, \(\Pi ^0_1\)-classes and complete extensions of PA. In: Ebbinghaus, H.-D., Müller, G.H., Sacks, G.E. (eds.) Recursion Theory Week. Lecture Notes in Mathematics, vol. 1141, pp. 245–259. Springer, Berlin (1985)

    Chapter  Google Scholar 

  25. Lempp, S.: Priority arguments in computability theory, model theory, and complexity theory. Lectures Notes (2012)

  26. Mal’tsev, A.I.: Sets with complete numberings. Algebra Logic 2(2), 4–29 (1963). (in Russian)

    Google Scholar 

  27. Muchnik, A.A.: Isomorphism of systems of recursively enumerable sets with effective properties. Tr. Mosk. Mat. Obs. 7, 407–412 (1958). (in Russian)

    MathSciNet  Google Scholar 

  28. Selivanov, V.L.: Precomplete numberings and functions without fixed points. Math. Notes 51(1), 95–99 (1992)

    Article  MathSciNet  Google Scholar 

  29. Selivanov, V.L.: Precomplete numberings. J. Math. Sci. 256(1), 96–124 (2021)

    Article  MathSciNet  Google Scholar 

  30. Soare, R.I.: Recursively Enumerable Sets and Degrees. Springer, Berlin (1987)

    Book  Google Scholar 

  31. Stephan, F., Yang, Y., Yu, L.: Turing degrees and the Ershov hierarchy. In: Arai, T., Brendle, J., Kikyo, H., Chong, C.T., Downey, R., Feng, Q., Ono, H. (eds.) Proceedings of the 10th Asian Logic Conference, pp. 300–321. World Scientific, Singapore (2009)

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Acknowledgements

Part of the research contained in this paper was carried out while N. Bazhenov was visiting Department of Mathematics of Nazarbayev University, Astana, and Kazakh-British Technical University, Almaty. The authors wish to thank Nazarbayev University and Kazakh-British Technical University for their hospitality. The authors are grateful to the anonymous reviewers for their helpful comments.

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Authors and Affiliations

  1. Kazakh-British Technical University, 59 Tole Bi Street, 050000, Almaty, Kazakhstan

    Serikzhan A. Badaev & Birzhan S. Kalmurzayev

  2. Sobolev Institute of Mathematics, 4 Acad. Koptyug Avenue, Novosibirsk, Russia, 630090

    Nikolay A. Bazhenov

  3. Al-Farabi Kazakh National University, 71 al Farabi Avenue, 050040, Almaty, Kazakhstan

    Birzhan S. Kalmurzayev

  4. Department of Mathematics, SSH, Nazarbayev University, 53 Qabanbay Batyr Avenue, 010000, Astana, Kazakhstan

    Manat Mustafa

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  1. Serikzhan A. Badaev
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  2. Nikolay A. Bazhenov
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  3. Birzhan S. Kalmurzayev
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  4. Manat Mustafa
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Correspondence to Nikolay A. Bazhenov.

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The work was supported by the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (Grant No.AP08856834). N. Bazhenov is supported by the Mathematical Center in Akademgorodok under the agreement No. 075-15-2022-281 with the Ministry of Science and Higher Education of the Russian Federation. M. Mustafa was supported by Nazarbayev University FDCRG (Grant No. 021220FD3851).

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Badaev, S.A., Bazhenov, N.A., Kalmurzayev, B.S. et al. On diagonal functions for equivalence relations. Arch. Math. Logic 63, 259–278 (2024). https://doi.org/10.1007/s00153-023-00896-0

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  • DOI: https://doi.org/10.1007/s00153-023-00896-0

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