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On Work of Barmpalias and Lewis-Pye: A Derivation on the D.C.E. Reals

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Computability and Complexity

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

Abstract

Let \(\alpha \) and \(\beta \) be (Martin-Löf) random left-c.e. reals with left-c.e. approximations \(\{\alpha _s\}_{s\in \omega }\) and \(\{\beta _s\}_{s\in \omega }\).

2010 Mathematics Subject Classification. Primary 03D32; Secondary 68Q30, 13N15.

The author was partially supported by grant #358043 from the Simons Foundation.

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Notes

  1. 1.

    For reasons that will become clear, we use different notation than Barmpalias and Lewis-Pye [2]. They write \(\mathcal {D}(\alpha ,\beta )\) instead of \(\partial \alpha /\partial \beta \).

  2. 2.

    However, we will show that \(\partial \) maps outside of the d.c.e. reals, so it does not make them a differential field.

  3. 3.

    In fact, Rettinger and Zheng [12, 14] extended Solovay reducibility to the d.c.e. reals and showed that their notion retains this basic property, putting all randoms in the top degree.

  4. 4.

    There is not broad agreement in the literature on what to call left-c.e. reals. They are often called “c.e. reals”, as in Downey, Hirschfeldt, and Nies [7], or “left computable”, as in Ambos-Spies, Weihrauch, and Zheng [1]. Several other names have been used, including “lower semicomputable”. Both Downey and Hirschfeldt [6] and Nies [10] use “left-c.e.”, so perhaps a consensus is forming.

  5. 5.

    D.c.e. is short for “difference of computably enumerable”, which is admittedly an imperfect name because it is too easy to confuse d.c.e. reals with d.c.e. sets. As with “left-c.e.”, various other terms have been used in the literature. Many sources, including Ambos-Spies, Weihrauch, and Zheng [1], call them “weakly computable” real numbers, which is not particularly descriptive. On the other hand, Downey and Hirschfeldt [6] call them “left-d.c.e.”, while admitting that “d.l.c.e.” would make somewhat more sense. Indeed, Nies [10] calls them “difference left-c.e.”.

  6. 6.

    An alternate proof might appeal to those familiar with Solovay reducibility: we can show that if \(\partial \alpha \ne 0\), then we can extract good approximations of \(\Omega \) from good approximations of \(\alpha \); hence, if \(\alpha \) were not random, then we could derandomize \(\Omega \).

References

  1. Ambos-Spies, K., Weihrauch, K., Zheng, X.: Weakly computable real numbers. J. Complexity 16(4), 676–690 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Barmpalias, G., Lewis-Pye, A.: Differences of halting probabilities (submitted)

    Google Scholar 

  3. Calude, C.S., Hertling, P.H., Khoussainov, B., Wang, Y.: Recursively enumerable reals and Chaitin \(\Omega \) numbers. Theoret. Comput. Sci. 255(1–2), 125–149 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. 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 

  5. Demut, O.: Constructive pseudonumbers. Comment. Math. Univ. Carolinae 16, 315–331 (1975)

    MathSciNet  Google Scholar 

  6. Rodney, G.: Algorithmic Randomness and Complexity. Theory and Applications of Computability. Springer, New York (2010)

    MATH  Google Scholar 

  7. Downey, R.G., Hirschfeldt, D.R., Nies, A.: Randomness, computability, and density. SIAM J. Comput. 31(4), 1169–1183 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Ng, K.M.: Some properties of d.c.e. reals and their degrees. Master’s thesis, National University of Singapore (2006)

    Google Scholar 

  10. Nies, A.: Computability and Randomness. Oxford Logic Guides, vol. 51. Oxford University Press, Oxford (2009)

    Book  MATH  Google Scholar 

  11. Raichev, A.: Relative randomness and real closed fields. J. Symbolic Logic 70(1), 319–330 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Rettinger, R., Zheng, X.: Solovay reducibility on d-c.e real numbers. In: Wang, L. (ed.) COCOON 2005. LNCS, vol. 3595, pp. 359–368. Springer, Heidelberg (2005). doi:10.1007/11533719_37

    Chapter  Google Scholar 

  13. Solovay, R.M.: Draft of paper (or series of papers) related to Chaitin’s work. IBM Thomas J. Watson Research Center, Yorktown Heights, NY, 215 pages (1975)

    Google Scholar 

  14. Zheng, X., Rettinger, R.: On the extensions of Solovay-Reducibility. In: Chwa, K.-Y., Munro, J.I.J. (eds.) COCOON 2004. LNCS, vol. 3106, pp. 360–369. Springer, Heidelberg (2004). doi:10.1007/978-3-540-27798-9_39

    Chapter  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Wisconsin–Madison, 480 Lincoln Dr., Madison, Wisconsin, 53706, USA

    Joseph S. Miller

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  1. Joseph S. Miller
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Correspondence to Joseph S. Miller .

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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

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Miller, J.S. (2017). On Work of Barmpalias and Lewis-Pye: A Derivation on the D.C.E. Reals. 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_39

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