Skip to main content
SpringerLink
Log in

There Are No Maximal d.c.e. wtt-degrees

  • Chapter
  • First Online:
Computability and Complexity

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

Abstract

In this article, we will study the weak-truth-table (wtt, for short) degrees of d.c.e. sets and show that there is no maximal d.c.e. wtt-degree.

Wu is partially supported by AcRF Tier 2 grants MOE2011-T2-1-071 (ARC 17/11, M45110030) and MOE2016-T2-1-083 from Ministry of Education of Singapore, and by AcRF Tier 1 grants, RG29/14, M4011274 and RG32/16, M4011672 from Ministry of Education of Singapore.

Yamaleev is supported by Russian Foundation for Basic Research (projects 15-41-02507, 15-01-08252), by research grant of Kazan Federal University, and by the subsidy allocated to Kazan Federal University for the project part of the state assignment in the sphere of scientific activities (project 1.2045.2014).

The original version of this chapter was revised. The spelling of the second author’s name was corrected. The erratum to this chapter is available at DOI: 10.1007/978-3-319-50062-1_43

An erratum to this chapter can be found at http://dx.doi.org/10.1007/978-3-319-50062-1_43

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

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). doi:10.1007/BFb0099477

    Chapter  Google Scholar 

  2. Arslanov, M.: Structural properties of the degrees below \(0^{\prime }\). Dokl. Nauk. SSSR 283, 270–273 (1985)

    MathSciNet  Google Scholar 

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

    Google Scholar 

  4. 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 

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

    MathSciNet  MATH  Google Scholar 

  6. Cooper, S.B.: Local degree theory. In: Griffor, E.R. (ed.) Handbook of Computability Theory, pp. 121–153. North-Holland, Amsterdam (1999)

    Chapter  Google Scholar 

  7. 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 

  8. Cooper, S.B., Li, A.: Non-uniformity and generalised Sacks splitting. Acta Math. Sin. 18, 327–334 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cooper, S.B., Yi, X.: Isolated d.r.e. degrees. University of Leeds, Department of Pure Mathematics, Preprint series, vol. 17 (1995)

    Google Scholar 

  10. Downey, R.: \(\Delta _2^0\) degrees and transfer theorems. Ill. J. Math. 31, 419–427 (1987)

    MATH  Google Scholar 

  11. Downey, R.: D.r.e. degrees and the nondiamond theorem. Bull. London Math. Soc. 21, 43–50 (1989)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Ershov, Y.L.: A hierarchy of sets, Part I (Russian). Algebra i Logika 7, 47–73 (1968). Algebra and Logic (English translation), 7, 24–43 (1968)

    MathSciNet  MATH  Google Scholar 

  14. Ershov, Y.L.: A hierarchy of sets, Part II (Russian). Algebra i Logika 7, 15–47 (1968). Algebra and Logic (English Translation), 7, 212–232 (1968)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Ishmukhametov, S.: D.r.e. sets, their degrees and index sets. Ph.D. thesis, Novosibirsk, Russia (1986)

    Google Scholar 

  17. Kaddah, D.: Infima in the d.r.e. degrees. Ann. Pure Appl. Logic 62, 207–263 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lachlan, A.H.: Lower bounds for pairs of recursively enumerable degrees. Proc. London Math. Soc. 16, 537–569 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  19. 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 

  20. Odifreddi, P.: Classical Recursion Theory. Studies in Logic and the Foundations of Mathematics, vol. 143. Elsevier, New York (1999)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  23. Stob, M.: \(Wtt\)-degrees and \(T\)-degrees of r.e. sets. J. Symbolic Logic 48, 921–930 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wu, G.: Isolation and lattice embeddings. J. Symbolic Logic 67, 1055–1064 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  25. Wu, G., Yamaleev, M.M.: Isolation: motivations and applications. Proc. Kazan Univ. (Phys. Math. Ser.) 52, 204–217 (2012)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Guohua Wu

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

    Mars M. Yamaleev

Authors
  1. Guohua Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mars M. Yamaleev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guohua Wu .

Editor information

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

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Wu, G., Yamaleev, M.M. (2017). 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. Lecture Notes in Computer Science(), vol 10010. Springer, Cham. https://doi.org/10.1007/978-3-319-50062-1_28

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-50062-1_28

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-50061-4

  • Online ISBN: 978-3-319-50062-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation