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Depth, Highness and DNR Degrees

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Fundamentals of Computation Theory (FCT 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9210))

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Abstract

We study Bennett deep sequences in the context of recursion theory; in particular we investigate the notions of \(O(1)\text {-deep}_K\), \(O(1)\text {-deep}_C\), order\(\text {-deep}_K\) and order\(\text {-deep}_C\) sequences. Our main results are that Martin-Löf random sets are not order\(\text {-deep}_C\), that every many-one degree contains a set which is not \(O(1)\text {-deep}_C\), that \(O(1)\text {-deep}_C\) sets and order\(\text {-deep}_K\) sets have high or DNR Turing degree and that no K-trival set is \(O(1)\text {-deep}_K\).

P. Moser was on Sabbatical Leave to the National University of Singapore, supported in part by SFI Stokes Professorship and Lectureship Programme. F. Stephan was supported in part by NUS grants R146-000-181-112 and R146-000-184-112.

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References

  1. Arslanov, M.M.: Degree structures in local degree theory. Complexity Logic, Recursion Theor. 187, 49–74 (1997)

    MathSciNet  Google Scholar 

  2. Antunes, L.F.C., Fortnow, L., van Melkebeek, D., Vinodchandran, N.V.: Computational depth. Theor. Comput. Sci. 354, 391–404 (2006)

    Article  MATH  Google Scholar 

  3. Antunes, L.F.C., Matos, A., Souto, A., Paul, M.B.P.: Depth as randomness deficiency. Theor. Comput. Sys. 45(4), 724–739 (2009)

    Article  MATH  Google Scholar 

  4. Baartse, M., Barmpalias, G.: On the gap between trivial and nontrivial initial segment prefix-free complexity. Theor. Comput. Sys. 52(1), 28–47 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Barmpalias, G., Lewis, A.E.M.: The importance of \(\Pi ^0_1\)-classes in effective randomness. J. Symbolic Logic 75(1), 387–400 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bennett, C.H.: Logical Depth and Physical Complexity. The Universal Turing Machine. A Half-Century Survey. Oxford University Press, New York (1988)

    Google Scholar 

  7. Chaitin, G.J.: A theory of program size formally identical to information theory. J. Assoc. Comput. Mach. 22, 329–340 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  8. Csima, B.F., Montalbán, A.: A minimal pair of K-degrees. Proc. Am. Math. soc. 134, 1499–1502 (2006)

    Article  MATH  Google Scholar 

  9. Doty, D., Moser, P.: Feasible depth. In: Cooper, S.B., Löwe, B., Sorbi, A. (eds.) CiE 2007. LNCS, vol. 4497, pp. 228–237. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  10. Downey, R.G., Hirschfeldt, D.R., Miller, J.S., Nies, A.: Relativizing chaitin’s halting probability. J. Math. Logic 5, 167–192 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Downey, R.G., Hirschfeldt, D.R.: Algorithmic Randomness and Complexity. Springer, Heidelberg (2010)

    Book  MATH  Google Scholar 

  12. Franklin, J., Stephan, F.: Schnorr trivial sets and truth-table reducibility. J. Symbolic Logic 75, 501–521 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Herbert, I.: Weak Lowness Notions for Kolmogorov Complexity. University of California, Berkeley (2013)

    Google Scholar 

  14. Hirschfeldt, D., Weber, R.: Finite Self-Information. Computability 1(1), 85–98 (2012). Manuscript

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  16. Juedes, D.W., Lathrop, J.I., Lutz, J.H.: Computational depth and reducibility. Theoret. Comput. Sci. 132, 37–70 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lathrop, J.I., Lutz, J.H.: Recursive computational depth. Inf. Comput. 153(1), 139–172 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  18. Paul, M.L., Vityányi, M.B.: An Introduction to Kolmogorov Complexity and its Applications. Springer Verlag, New York (2008)

    MATH  Google Scholar 

  19. Miller, J.S.: Every 2-random real is Kolmogorov random. J. Symbolic Logic 69(3), 907–913 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. Moser, P.: On the polynomial depth of various sets of random strings. Theoret. Comput. Sci. 477, 96–108 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Nies, A.: Computability and Randomness. Oxford University Press, New York (2009)

    Book  MATH  Google Scholar 

  22. Nies, A., Terwijn, S.A., Stephan, F.: Randomness, relativization and Turing degree. J. Symbolic Logic 70(2), 515–535 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Odifreddi, P.: Classical Recursion Theory: The Theory of Functions and Sets of Natural Numbers, vol. 1. Elsevier, The Netherlands (1989)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science, National University of Ireland, Maynooth, Co Kildare, Ireland

    Philippe Moser

  2. Department of Mathematics, The National University of Singapore, 10 Lower Kent Ridge Drive, S17, Singapore, 119076, Republic of Singapore

    Frank Stephan

Authors
  1. Philippe Moser
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  2. Frank Stephan
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Correspondence to Philippe Moser .

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Editors and Affiliations

  1. Université Paris Diderot, Paris, France

    Adrian Kosowski

  2. Université Bordeaux 1, Talence, France

    Igor Walukiewicz

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Moser, P., Stephan, F. (2015). Depth, Highness and DNR Degrees. In: Kosowski, A., Walukiewicz, I. (eds) Fundamentals of Computation Theory. FCT 2015. Lecture Notes in Computer Science(), vol 9210. Springer, Cham. https://doi.org/10.1007/978-3-319-22177-9_7

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  • DOI: https://doi.org/10.1007/978-3-319-22177-9_7

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  • Publisher Name: Springer, Cham

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