Skip to main content
SpringerLink
Log in

Totally < ω ω Computably Enumerable and m-topped Degrees

  • Conference paper
Theory and Applications of Models of Computation (TAMC 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3959))

Abstract

In this paper we will discuss recent work of the authors (Downey, Greenberg and Weber [8] and Downey and Greenberg [6, 7]) devoted to understanding some new naturally definable degree classes which capture the dynamics of various natural constructions arising from disparate areas of classical computability theory.

It is quite rare in computability theory to find a single class of degrees which capture precisely the underlying dynamics of a wide class of apparently similar constructions, demonstrating that they all give the same class of degrees. A good example of this phenomenon is work pioneered by Martin [22] who identified the high c.e. degrees as the ones arising from dense simple, maximal, hh-simple and other similar kinds of c.e. sets constructions. Another example would be the example of the promptly simple degrees by Ambos-Spies, Jockusch, Shore and Soare [2]. Another more recent example of current great interest is the class of K-trivial reals of Downey, Hirscheldt, Nies and Stephan [5], and Nies [23, 24].

Research supported by the Marsden Fund of New Zealand.

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 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ambos-Spies, K., Fejer, P.: Embeddings of N5 and the contiguous degrees. Annals of Pure and Applied Logic 112, 151–188 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  2. Ambos-Spies, K., Jockusch, C., Shore, R., Soare, R.: An algebraic decomposition of recursively enumerable degrees and the coincidence of several degree classes with the promptly simple degrees. Trans. Amer. Math. Soc. 281, 109–128 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  3. Downey, R.: Lattice nonembeddings and initial segments of the recursively enumerable degrees. Annals Pure and applied Logic 49, 97–119 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  4. Downey, R.: Some Computability-Theoretical Aspects of Reals and Randomness. In: Cholak, P. (ed.) The Notre Dame Lectures. Lecture Notes in Logic, vol. 18, pp. 97–148. Association for Symbolic Logic (2005)

    Google Scholar 

  5. Downey, R., Hirschfeldt, D., Nies, A., Stephan, F.: Trivial reals. extended abstract In: Brattka, V., Schröder, M., Weihrauch, K. (eds.) Computability and Complexity in Analysis Malaga, FernUniversität, July 2002. Electronic Notes in Theoretical Computer Science, and proceedings, 294–6/2002 pp. 37–55 (2002); Final version appears in Downey, R., Decheng, D., Ping, T.S. , Hui, Q.Y., Yasuugi, M., Wu, G. (eds.) Proceedings of the 7th and 8th Asian Logic Conferences, viii+471, pp. 103–131. World Scientific, Singapore (2003)

    Google Scholar 

  6. Downey, R., Greenberg, N.: Totally ω computably enumerable degrees II: Left c.e. reals (in preparation)

    Google Scholar 

  7. Downey, R., Greenberg, N.: Totally ωω-computably enumerable degrees I: lattice embeddings (in preparation)

    Google Scholar 

  8. Downey, R., Greenberg, N., Weber, R.: Totally ω computably enumerable degrees I: bounding critical triples (submitted)

    Google Scholar 

  9. Downey, R., Jockusch, C.: T-degrees, jump classes and strong reducibilities. Trans. Amer. Math. Soc. 301, 103–136 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  10. Downey, R., Jockusch, C., Stob, M.: Array nonrecursive sets and multiple permitting arguments. In: Ambos-Spies, K., Muller, G.H., Sacks, G.E. (eds.) Recursion Theory Week. Lecture Notes in Mathematics, vol. 1432, pp. 141–174. Springer, Heidelberg (1990)

    Chapter  Google Scholar 

  11. Downey, R., Jockusch, C., Stob, M.: Array nonrecursive degrees and genericity. In: Cooper, S.B., Slaman, T.A., Wainer, S.S. (eds.) Computability, Enumerability, Unsolvability. London Mathematical Society Lecture Notes Series, vol. 224, pp. 93–105. Cambridge University Press, Cambridge (1996)

    Chapter  Google Scholar 

  12. Downey, R., LaForte, G.: Presentations of computably enumerable reals. Theoretical Computer Science 284, 539–555 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  13. Downey, R., Lempp, S.: Contiguity and distributivity in the enumerable degrees. Journal of Symbolic Logic 62, 1215–1240 (1997); Corrigendum in ibid 67, 1579–1580 (2002)

    Google Scholar 

  14. Downey, R., Shore, R.: Degree theoretical definitions of low2 recursively enumerable sets. J. Symbolic Logic 60, 727–756 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  15. Downey, R., Shore, R.: Lattice embeddings below a non-low2 recursively enumerable degree. Israel Journal of Mathematics 94, 221–246 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  16. Ishmukhametov, S.: Weak recursive degrees and a problem of Spector. In: Arslanov, M., Lempp, S. (eds.) Recursion Theory and Complexity, de Gruyter, Berlin, pp. 81–88 (1999)

    Google Scholar 

  17. Lachlan, A.: Embedding nondistributive lattices in the recursively enumerable degrees. In: Hodges, W. (ed.) Conference in Mathematical Logic, London 1970. Lecture notes in mathematics, vol. 255, pp. 149–177. Springer, Heidelberg (1972)

    Chapter  Google Scholar 

  18. Lachlan, A., Soare, R.: Not every finite lattice is embeddable in the recursively enumerable degrees. Advances in Math. 37, 78–82 (1980)

    Article  MathSciNet  Google Scholar 

  19. Lempp, S., Lerman, M.: A finite lattice without critical triple that cannot be embedded into the computably enumerable Turing degrees. Annals of Pure and Applied Logic 87, 167–185 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  20. Lempp, S., Lerman, M., Solomon, D.: Embedding finite lattices into the computably enumerable degrees - a status survey (to appear)

    Google Scholar 

  21. Lerman, M.: The embedding problem for the recursively enumerable degrees. In: Nerode, A., Shore, R. (eds.) Recursion Theory, pp. 13–20. American Math. Soc., Providence (1995)

    Google Scholar 

  22. Martin, D.: Classes of recursively enumerable sets and degrees of unsolvability. Z. Math. Logik Grundlag. Math. 12, 295–310 (1966)

    Article  MATH  Google Scholar 

  23. Nies, A.: Reals which compute little. In: Proceedings of CL 2002 (2002) (to appear)

    Google Scholar 

  24. Nies, A.: Lowness properties and randomness. Advances in Mathematics 197(1), 274–305 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  25. Nies, A., Shore, R., Slaman, T.: Interpretability and definability in the recursively enumerable degrees. Proc. Lon. Math. Soc. 3(77), 241–291 (1998)

    Article  MathSciNet  Google Scholar 

  26. Schaeffer, B.: Dynamic notions of genericity and array noncomputability. Ann. Pure Appl. Logic. 95(1-3), 37–69 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  27. Shore, R.: Natural definability in degree structures. In: Cholak, P., Lempp, S., Lerman, M., Shore, R.A. (eds.) Computability Theory and Its Applications: Current Trends and Open Problems. Contemporary Mathematics, pp. 255–272. AMS, Providence (2000)

    Google Scholar 

  28. Soare, R.: Recursively enumerable sets and degrees. Springer, Berlin (1987)

    Google Scholar 

  29. Walk, S.: Towards a definitioon of the array computable degrees, PH. D. Thesis, University of Notre Dame (1999)

    Google Scholar 

  30. Weinstein, S.: Ph. D. Thesis, University of California, Berkeley (1988)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, Statistics, and Computer Science, Victoria University, PO Box 600, Wellington, New Zealand

    Rod Downey & Noam Greenberg

Authors
  1. Rod Downey
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Noam Greenberg
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Computer Sciences Department, University of Wisconsin, 53706, Madison, WI, USA

    Jin-Yi Cai

  2. School of Mathematics, University of Leeds, LS2 9JT, Leeds, U.K.

    S. Barry Cooper

  3. State Key Lab. of Computer Science, Institute of Software, Chinese Academy of Sciences,  

    Angsheng Li

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Downey, R., Greenberg, N. (2006). Totally < ω ω Computably Enumerable and m-topped Degrees. In: Cai, JY., Cooper, S.B., Li, A. (eds) Theory and Applications of Models of Computation. TAMC 2006. Lecture Notes in Computer Science, vol 3959. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11750321_3

Download citation

  • DOI: https://doi.org/10.1007/11750321_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-34021-8

  • Online ISBN: 978-3-540-34022-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation