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Phase Transition between Unidirectionality and Bidirectionality

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Computation, Physics and Beyond (WTCS 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7160))

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Abstract

The notion of weak truth-table reducibility plays an important role in recursion theory. In this paper, we introduce an elaboration of this notion, where a computable bound on the use function is explicitly specified. This elaboration enables us to deal with the notion of asymptotic behavior in a manner like in computational complexity theory, while staying in computability theory. We apply the elaboration to sets which appear in the statistical mechanical interpretation of algorithmic information theory. We demonstrate the power of the elaboration by revealing a critical phenomenon, i.e., a phase transition, in the statistical mechanical interpretation, which cannot be captured by the original notion of weak truth-table reducibility.

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References

  1. Calude, C.S.: Information and Randomness, 2nd edn. Revised and Extended. Springer, Heidelberg (2002)

    Book  MATH  Google Scholar 

  2. Calude, C.S., Hertling, P.H., Khoussainov, B., Wang, Y.: Recursively enumerable reals and Chaitin Ω numbers. Theoret. Comput. Sci. 255, 125–149 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Calude, C.S., Hay, N.J., Stephan, F.C.: Representation of left-computable ε-random reals. J. Comput. Syst. Sci. 77, 812–819 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Calude, C.S., Nies, A.: Chaitin Ω numbers and strong reducibilities. Journal of Universal Computer Science 3(11), 1162–1166 (1997)

    MathSciNet  MATH  Google Scholar 

  5. Calude, C.S., Staiger, L., Terwijn, S.A.: On partial randomness. Annals of Pure and Applied Logic 138, 20–30 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Calude, C.S., Stay, M.A.: Natural halting probabilities, partial randomness, and zeta functions. Inform. and Comput. 204, 1718–1739 (2006)

    Article  MathSciNet  MATH  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. Chaitin, G.J.: Algorithmic Information Theory. Cambridge University Press, Cambridge (1987)

    Book  MATH  Google Scholar 

  9. Downey, R.G., Hirschfeldt, D.R.: Algorithmic Randomness and Complexity. Springer, New York (2010)

    Book  MATH  Google Scholar 

  10. Kučera, A., Slaman, T.A.: Randomness and recursive enumerability. SIAM J. Comput. 31(1), 199–211 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Miller, J., Yu, L.: On initial segment complexity and degrees of randomness. Trans. Amer. Math. Soc. 360, 3193–3210 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  13. Reimann, J., Stephan, F.: On hierarchies of randomness tests. In: Proceedings of the 9th Asian Logic Conference, August 16-19, World Scientific Publishing, Novosibirsk (2005)

    Google Scholar 

  14. Solovay, R.M.: Draft of a paper (or series of papers) on Chaitin’s work... done for the most part during the period of September–December (1974); unpublished manuscript. IBM Thomas J. Watson Research Center, p. 215. Yorktown Heights, New York (May 1975)

    Google Scholar 

  15. Tadaki, K.: A generalization of Chaitin’s halting probability Ω and halting self-similar sets. Hokkaido Math. J. 31, 219–253 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. Tadaki, K.: A statistical mechanical interpretation of algorithmic information theory. In: Local Proceedings of Computability in Europe 2008 (CiE 2008), June 15-20, pp. 425–434. University of Athens, Greece (2008); An Extended Version Available at arXiv:0801.4194v1

    Google Scholar 

  17. Tadaki, K.: Chaitin Ω Numbers and Halting Problems. In: Ambos-Spies, K., Löwe, B., Merkle, W. (eds.) CiE 2009. LNCS, vol. 5635, pp. 447–456. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  18. Tadaki, K.: Partial Randomness and Dimension of Recursively Enumerable Reals. In: Královič, R., Niwiński, D. (eds.) MFCS 2009. LNCS, vol. 5734, pp. 687–699. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  19. Tadaki, K.: A computational complexity-theoretic elaboration of weak truth-table reducibility. Research Report of CDMTCS 406 (July 2011)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Research and Development Initiative, Chuo University, JST CREST, 1–13–27 Kasuga, Bunkyo-ku, Tokyo, 112-8551, Japan

    Kohtaro Tadaki

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  1. Kohtaro Tadaki
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Editors and Affiliations

  1. Department of Computer Science, University of Auckland, Private Bag 92019, 1142, Auckland, New Zealand

    Michael J. Dinneen , Bakhadyr Khoussainov  & André Nies ,  & 

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Tadaki, K. (2012). Phase Transition between Unidirectionality and Bidirectionality. In: Dinneen, M.J., Khoussainov, B., Nies, A. (eds) Computation, Physics and Beyond. WTCS 2012. Lecture Notes in Computer Science, vol 7160. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27654-5_16

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  • DOI: https://doi.org/10.1007/978-3-642-27654-5_16

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-27653-8

  • Online ISBN: 978-3-642-27654-5

  • eBook Packages: Computer ScienceComputer Science (R0)

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