Advertisement

Archive for Mathematical Logic

, Volume 54, Issue 3–4, pp 329–358 | Cite as

Unified characterizations of lowness properties via Kolmogorov complexity

  • Takayuki Kihara
  • Kenshi Miyabe
Article
  • 49 Downloads

Abstract

Consider a randomness notion \({\mathcal{C}}\). A uniform test in the sense of \({\mathcal{C}}\) is a total computable procedure that each oracle X produces a test relative to X in the sense of \({\mathcal{C}}\). We say that a binary sequence Y is \({\mathcal{C}}\)-random uniformly relative to X if Y passes all uniform \({\mathcal{C}}\) tests relative to X. Suppose now we have a pair of randomness notions \({\mathcal{C}}\) and \({\mathcal{D}}\) where \({\mathcal{C} \subseteq \mathcal{D}}\), for instance Martin-Löf randomness and Schnorr randomness. Several authors have characterized classes of the form Low(\({\mathcal{C}, \mathcal{D}}\)) which consist of the oracles X that are so feeble that \({\mathcal{C} \subseteq \mathcal{D}^X}\). Our goal is to do the same when the randomness notion \({\mathcal{D}}\) is relativized uniformly: denote by Low \({\star(\mathcal{C},\mathcal{D})}\) the class of oracles X such that every \({\mathcal{C}}\)-random is uniformly \({\mathcal{D}}\)-random relative to X. (1) We show that \({X\in Low ^\star}\)(MLR, SR) if and only if X is c.e. tt-traceable if and only if X is anticomplex if and only if X is Martin-Löf packing measure zero with respect to all computable dimension functions. (2) We also show that \({X\in Low^\star}\) (SR, WR) if and only if X is computably i.o. tt-traceable if and only if X is not totally complex if and only if X is Schnorr Hausdorff measure zero with respect to all computable dimension functions.

Keywords

Algorithmic randomness Lowness for randomness Uniform relativization Traceability Effective dimension 

Mathematics Subject Classification

03D32 68Q30 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Barmpalias G., Downey R., Ng K.M.: Jump inversions inside effectively closed sets and applications to randomness. J. Symb. Logic 76(2), 491–518 (2011)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Bienvenu L., Miller J.S.: Randomness and lowness notions via open covers. Ann. Pure Appl. Logic 163, 506–518 (2012)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Binns S.: Small \({\Pi^{0}_{1}}\) classes. Arch. Math. Logic 45(4), 393–410 (2006)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Binns S.: Hyperimmunity in \({{2}^{\mathbb{N}}}\). Notre Dame J. Form. Log. 48(2), 293–316 (2007)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Binns S.: \({\Pi^{0}_{1}}\) classes with complex elements. J. Symb. Logic 73(4), 1341–1353 (2008)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Binns S., Kjos-Hanssen B.: Finding paths through narrow and wide trees. J. Symb. Logic 74(1), 349–360 (2009)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Brattka, V.: Computability over topological structures. In: Cooper, S.B., Goncharov, S.S. (eds.) Computability and Models, pp. 93–136. Kluwer, New York (2003)Google Scholar
  8. 8.
    Brattka, V., Hertling, P., Weihrauch, K.: A tutorial on computable analysis. In: Cooper, S.B., Löwe, B., Sorbi, A. (eds.) New Computational Paradigms, pp. 425–491. Springer, Berlin (2008)Google Scholar
  9. 9.
    Calude, C.S., Coles, R.J.: Program-size complexity of initial segments and domination relation reducibility. In: Karhumäki, J., Hauer, H., Păun, G., Rozenberg, G. (eds.) Jewels and Forever, pp. 225–237. Springer, Berlin (1999)Google Scholar
  10. 10.
    Chaitin G.J.: Nonrecursive infinite strings with simple initial segments. IBM J. Res. Dev. 21, 350–359 (1977)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Diamondstone, D., Kjos-Hanssen, B.: Members of random closed sets. In: Mathematical Theory and Computational Practice, pp. 144–153. Springer, Berlin (2009)Google Scholar
  12. 12.
    Downey R., Griffiths E., LaForte G.: On Schnorr and computable randomness, martingales, and machines. MLQ Math. Log. Q. 50(6), 613–627 (2004)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Downey R., Hirschfeldt D.R.: Algorithmic Randomness and Complexity. Springer, Berlin (2010)MATHGoogle Scholar
  14. 14.
    Downey R., Nies A., Weber R., Yu L.: Lowness and \({\Pi^{0}_{2}}\) nullsets. J. Symb. Logic 71(3), 1044–1052 (2006)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Downey R.G., Griffiths E.J.: Schnorr randomness. J. Symb. Logic 69(2), 533–554 (2004)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Downey, R.G., Hirschfeldt, D.R., Nies, A., Stephan, F.: Trivial reals. In: Downey, R.G., Ding, D., Tung, S.P., Qiu, Y.H., Yasugi, M. (eds.) Proceedings of the 7th and 8th Asian Logic Conferences, pp. 103–131. Singapore University Press and World Scientific, Singapore (2003)Google Scholar
  17. 17.
    Downey, R.G., Merkle, W., Reimann, J.: Schnorr dimension. In: Barry Cooper, S. et al. (ed.) New Computational Paradigms, First Conference on Computability in Europe, CiE 2005, Lecture Notes in Comput. Sci., vol. 3526, pp. 96–105. Springer, Berlin (2005)Google Scholar
  18. 18.
    Franklin J., Greenberg N., Stephan F., Wu G.: Anti-complex sets and reducibilities with tiny use. J. Symb. Logic 78(4), 1307–1327 (2013)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Franklin J., Stephan F.: Van Lambalgen’s theorem and high degrees. Notre Dame J. Form. Log. 52(2), 173–185 (2011)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Franklin J.N.Y.: Hyperimmune-free degrees and Schnorr triviality. J. Symb. Logic 73(3), 999–1008 (2008)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Franklin J.N.Y.: Schnorr trivial reals: a construction. Arch. Math. Log. 46(7–8), 665–678 (2008)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Franklin, J.N.Y.: Lowness and highness properties for randomness notions. In: Arai, T. et al. (ed.) Proceedings of the 10th Asian Logic Conference, pp. 124–151. World Scientific, Singapore (2010)Google Scholar
  23. 23.
    Franklin J.N.Y.: Schnorr triviality and genericity. J. Symb. Logic 75(1), 191–207 (2010)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Franklin J.N.Y., Stephan F.: Schnorr trivial sets and truth-table reducibility. J. Symb. Logic 75(2), 501–521 (2010)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Greenberg N., Miller J.S.: Lowness for Kurtz randomness. J. Symb. Logic 74, 665–678 (2009)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Higuchi K., Kihara T.: On effectively closed sets of effective strong measure zero. Ann. Pure Appl. Logic 165(9), 1445–1469 (2014)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Hirschfeldt D., Nies A., Stephan F.: Using random sets as oracles. J. Lond. Math. Soc. 75, 610–622 (2007)CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Hölzl, R., Merkle, W.: Traceable sets. Theoret. Comput. Sci. 323, 301–315 (2010)Google Scholar
  29. 29.
    Kanovich M.I.: On the complexity of enumeration and decision of predicates. Sov. Math. Dokl. 11, 17–20 (1970)Google Scholar
  30. 30.
    Kihara T., Miyabe K.: Uniform Kurtz randomness. J. Log. Comput. 24(4), 863–882 (2014)CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Kjos-Hanssen B., Merkle W., Stephan F.: Kolmogorov complexity and the recursion theorem Trans. Am. Math. Soc. 363(10), 5465–5480 (2011)CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Kjos-Hanssen B., Miller J.S., Solomon D.R.: Lowness notions, measure, and domination. J. Lond. Math. Soc. 85(3), 869–888 (2012)CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Kjos-Hanssen B., Nies A.: Superhighness. Notre Dame J. Form. Log. 50, 445–452 (2009)CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Kjos-Hanssen B., Nies A., Stephan F.: Lowness for the class of Schnorr random reals. SIAM J. Comput. 35(3), 647–657 (2005)CrossRefMathSciNetGoogle Scholar
  35. 35.
    Kučera A.: On relative randomness. Ann. Pure Appl. Logic 63, 61–67 (1993)CrossRefMATHMathSciNetGoogle Scholar
  36. 36.
    Kučera A., Terwijn S.: Lowness for the class of random sets. J. Symb. Logic 64, 1396–1402 (1999)CrossRefMATHGoogle Scholar
  37. 37.
    Merkle W., Miller J., Nies A., Reimann J., Stephan F.: Kolmogorov–Loveland randomness and stochasticity. Ann. Pure Appl. Logic 138(1–3), 183–210 (2006)CrossRefMATHMathSciNetGoogle Scholar
  38. 38.
    Miyabe, K.: Schnorr triviality and its equivalent notions. Theory Comput. Syst. (to appear)Google Scholar
  39. 39.
    Miyabe K.: Truth-table Schnorr randomness and truth-table reducible randomness. MLQ Math. Log. Q. 57(3), 323–338 (2011)CrossRefMATHMathSciNetGoogle Scholar
  40. 40.
    Miyabe, K., Rute, J.: Van Lambalgen’s theorem for uniformly relative Schnorr and computable randomness. In: Proceedings of the Twelfth Asian Logic Conference, pp. 251–270 (2013)Google Scholar
  41. 41.
    Nies A.: Lowness properties and randomness. Adv. Math. 197, 274–305 (2005)CrossRefMATHMathSciNetGoogle Scholar
  42. 42.
    Nies A.: Computability and Randomness. Oxford University Press, USA (2009)CrossRefMATHGoogle Scholar
  43. 43.
    Nies A., Stephan F., Terwijn S.: Randomness, relativization and Turing degrees. J. Symb. Logic 70, 515–535 (2005)CrossRefMATHMathSciNetGoogle Scholar
  44. 44.
    Odifreddi, P.: Classical Recursion Theory, vol. 1. North-Holland, Amsterdam (1990)Google Scholar
  45. 45.
    Odifreddi, P.: Classical Recursion Theory, vol. 2. North-Holland, Amsterdam (1999)Google Scholar
  46. 46.
    Pawlikowski J.: A characterization of strong measure zero sets. Israel J. Math. 93(1), 171–183 (1996)CrossRefMATHMathSciNetGoogle Scholar
  47. 47.
    Reimann, J.: Computability and Fractal Dimension. Ph.D. Thesis, Universität Heidelberg (2004)Google Scholar
  48. 48.
    Soare R.I.: Recursively Enumerable Sets and Degrees. Perspectives in Mathematical Logic. Springer, Berlin (1987)Google Scholar
  49. 49.
    Solovay, R.: Draft of Paper (or Series of Papers) on Chaitin’s Work. Unpublished Notes (1975), p. 215Google Scholar
  50. 50.
    Stephan F., Yu L.: Lowness for weakly 1-generic and Kurtz-random. Lect. Notes Comput. Sci. 3959, 756–764 (2006)CrossRefMathSciNetGoogle Scholar
  51. 51.
    Terwijn S.A., Zambella D.: Computational randomness and lowness. J. Symb. Logic 66(3), 1199–1205 (2001)CrossRefMATHMathSciNetGoogle Scholar
  52. 52.
    van Lambalgen, M.: Random Sequences. Ph.D. Thesis, University of Amsterdam (1987)Google Scholar
  53. 53.
    Weihrauch K.: Computable Analysis: An Introduction. Springer, Berlin (2000)Google Scholar
  54. 54.
    Weihrauch K., Grubba T.: Elementary computable topology. J. UCS 15(6), 1381–1422 (2009)MATHMathSciNetGoogle Scholar
  55. 55.
    Yu L.: When van Lambalgen’s theorem fails. Proc. Am. Math. Soc. 135(3), 861–864 (2007)CrossRefMATHGoogle Scholar
  56. 56.
    Zambella, D.: On Sequences with Simple Initial Segments. Tech. rep., Univ. Amsterdam (1990). ILLC Technical Report ML 1990-05Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.School of Information ScienceJapan Advanced Institute of Science and TechnologyNomiJapan
  2. 2.School of Science and TechnologyMeiji UniversityKawasakiJapan

Personalised recommendations