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Conference on Computability in Europe

CiE 2019: Computing with Foresight and Industry pp 277–285Cite as

The d.r.e wtt-Degrees are Dense

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11558))

Abstract

We prove in this paper that the d.r.e. wtt-degrees are dense, improving a result of Wu and Yamaleev. Our result is a direct generalization of Ladner and Sasso’s splitting theorem for r.e. wtt-degrees. One essential feature of our construction is that the Lachlan sets are used as a help to obtain more information of d.r.e. sets.

The authors appreciate the comments and suggestions from the referees. Wang and Wu are partially supported by MOE2016-T2-1-083 (M4020333), M4011274 (RG29/14) and M4011672 (RG32/16) from Ministry of Education of Singapore. Yamaleev is supported by Russian Foundation for Basic Research (project 18-31-00420), by The President Grant of Russian Federation (project SS-5383.2012.1), and by The Ministry of Education and Science of Russian Federation (projects 14.A18.21.0360, 14.A18.21.0368, 14.A18.21.1127).

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References

  1. Ambos-Spies, K.: Contiguous r.e. degrees. In: Börger, E., Oberschelp, W., Richter, M.M., Schinzel, B., Thomas, W. (eds.) Computation and Proof Theory. LNM, vol. 1104, pp. 1–37. Springer, Heidelberg (1984). https://doi.org/10.1007/BFb0099477

    Chapter  MATH  Google Scholar 

  2. Cooper, S.B.: Degrees of unsolvability. Ph.D. thesis, Leicester University (1971)

    Google Scholar 

  3. Cooper, S.B.: The density of the low \(_2\)\(n\)-r.e. degrees. Arch. Math. Logic 31, 19–24 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cooper, S.B.: A splitting theorem for the \(n\)-r.e. degrees. Proc. Am. Math. Soc. 115, 461–471 (1992)

    MathSciNet  MATH  Google Scholar 

  5. Cooper, S.B., Harrington, L., Lachlan, A.H., Lempp, S., Soare, R.I.: The d.r.e. degrees are not dense. Ann. Pure Appl. Logic 55, 125–151 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  6. Downey, R., Stob, M.: Splitting theorems in recursion theory. Ann. Pure Appl. Logic 65, 1–106 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fang, C., Wu, G., Yamaleev, M.M.: On a problem of Ishmuhkametov. Arch. Math. Logic 52, 733–741 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fang, C., Liu, J., Wu, G., Yamaleev, M.M.: Nonexistence of minimal pairs in \(L[{\mathbf{d}}]\). In: Beckmann, A., Mitrana, V., Soskova, M. (eds.) CiE 2015. LNCS, vol. 9136, pp. 177–185. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-20028-6_18

    Chapter  Google Scholar 

  9. Friedberg, R.M., Rogers Jr., H.: Reducibility and completeness for sets of integers. Z. Math. Logik Grundlag. Math. 5, 117–125 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ishmukhametov, Sh.: On the predececcors of d.r.e. degrees. Arch. Math. Logic 5, 373–386 (1999)

    Google Scholar 

  11. Ishmukhametov, S.: On relative enumerability of turing degrees. Arch. Math. Logic 39, 145–154 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lachlan, A.H.: A recursively enumerable degree which will not split over all lesser ones. Ann. Math. Logic 9, 307–365 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ladner, R.E., Sasso, L.P.: The weak-truth-table degrees of recursively enumerable sets. Ann. Math. Logic 8, 429–448 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  14. Odifreddi, P.: Classical Recursion Theory, Volume II, vol. 143. Studies in Logic and the Foundations of Mathematics, Elsevier (1999)

    Google Scholar 

  15. Sacks, G.E.: On the degrees less than \({ 0}^{\prime }\). Ann. Math. 77, 211–231 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  16. Sacks, G.E.: The recursively enumerable degrees are dense. Ann. Math. 80, 300–312 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  17. Soare, R.I.: Recursively Enumerable Sets and Degrees. Springer, Heidelberg (1987)

    Book  MATH  Google Scholar 

  18. Stob, M.: \(wtt\)-degrees and \(T\)-degrees of recursively enumerable sets. J. Symb. Log. 48, 921–930 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  19. Wu, G., Yameleev, M.M.: There are no maximal d.c.e. wtt-degrees. In: Day, A., Fellows, M., Greenberg, N., Khoussainov, B., Melnikov, A., Rosamond, F. (eds.) Computability and Complexity. LNCS, vol. 10010, pp. 479–486. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-50062-1_28

    Chapter  Google Scholar 

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Author information

Authors and Affiliations

  1. Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, 637371, Singapore

    Shaoyi Wang & Guohua Wu

  2. Institute of Mathematics and Mechanics, Kazan Federal University, 18 Kremlyovskaya Street, 420008, Kazan, Russia

    Mars M. Yamaleev

Authors
  1. Shaoyi Wang
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  2. Guohua Wu
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  3. Mars M. Yamaleev
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Corresponding author

Correspondence to Guohua Wu .

Editor information

Editors and Affiliations

  1. Christian Albrechts University of Kiel, Kiel, Germany

    Florin Manea

  2. Durham University, Durham, UK

    Barnaby Martin

  3. Durham University, Durham, UK

    Daniël Paulusma

  4. Università degli Studi di Milano, Milan, Italy

    Giuseppe Primiero

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Wang, S., Wu, G., Yamaleev, M.M. (2019). The d.r.e wtt-Degrees are Dense. In: Manea, F., Martin, B., Paulusma, D., Primiero, G. (eds) Computing with Foresight and Industry. CiE 2019. Lecture Notes in Computer Science(), vol 11558. Springer, Cham. https://doi.org/10.1007/978-3-030-22996-2_24

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  • DOI: https://doi.org/10.1007/978-3-030-22996-2_24

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-22995-5

  • Online ISBN: 978-3-030-22996-2

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