Skip to main content
×
×
Home

USING ALMOST-EVERYWHERE THEOREMS FROM ANALYSIS TO STUDY RANDOMNESS

  • KENSHI MIYABE (a1), ANDRÉ NIES (a2) and JING ZHANG (a3)
Abstract
Abstract

We study algorithmic randomness notions via effective versions of almost-everywhere theorems from analysis and ergodic theory. The effectivization is in terms of objects described by a computably enumerable set, such as lower semicomputable functions. The corresponding randomness notions are slightly stronger than Martin–Löf (ML) randomness.

We establish several equivalences. Given a ML-random real z, the additional randomness strengths needed for the following are equivalent. (1)

all effectively closed classes containing z have density 1 at z.

(2)

all nondecreasing functions with uniformly left-c.e. increments are differentiable at z.

(3)

z is a Lebesgue point of each lower semicomputable integrable function.

We also consider convergence of left-c.e. martingales, and convergence in the sense of Birkhoff’s pointwise ergodic theorem. Lastly, we study randomness notions related to density of ${\rm{\Pi }}_n^0$ and ${\rm{\Sigma }}_1^1$ classes at a real.

Copyright
References
Hide All
[1] Bienvenu L., Day A., Greenberg N., Kučera A., Miller J., Nies A., and Turetsky D., Computing K-trivial sets by incomplete random sets, this Bulletin, vol. 20 (2014), pp. 8090.
[2] Bienvenu L., Day A. R., Hoyrup M., Mezhirov I., and Shen A., A constructive version of Birkhoff’s ergodic theorem for Martin-Löf random points . Information and Computation, vol. 210 (2012), pp. 2130.
[3] Bienvenu L., Greenberg N., Kučera A., Nies A., and Turetsky D., Coherent randomness tests and computing the K-trivial sets. Journal of the European Mathematical Society , 2015, to appear.
[4] Bienvenu L., Hölzl R., Miller J., and Nies A., Denjoy, Demuth, and Density . Journal of Mathematical Logic, 1450004, 2014, 35 p.
[5] Birkhoff G., The mean ergodic theorem . Duke Mathematical Journal, vol. 5 (1939), no. 1, pp. 1920.
[6] Bogachev V. I., Measure Theory, vol. I, II, Springer-Verlag, Berlin, 2007.
[7] Brattka V., Miller J., and Nies A., Randomness and differentiability . Transactions of the AMS, vol. 368 (2016), pp. 581605. ArXiv version at http://arxiv.org/abs/1104.4465.
[8] Carothers N. L., Real Analysis, Cambridge University Press, Cambridge, 2000.
[9] Day A. R. and Miller J. S., Cupping with random sets . Proceedings of the American Mathematical Society, vol. 142 (2014), no. 8, pp. 28712879.
[10] Day A. R. and Miller J. S., Density, forcing and the covering problem . Mathematical Research Letters, vol. 22 (2015), no. 3, pp. 719727.
[11] Demuth O., The differentiability of constructive functions of weakly bounded variation on pseudo numbers . Commentationes Mathematicae Universitatis Carolinae, vol. 16 (1975), no. 3, pp. 583599 (In Russian).
[12] Downey R. and Hirschfeldt D., Algorithmic Randomness and Complexity, Springer-Verlag, Berlin, 2010, 855 p.
[13] Downey R., Nies A., Weber R., and Yu L., Lowness and ${\rm{\Pi }}_2^0$ nullsets, this Journal, vol. 71 (2006), no. 3, pp. 10441052.
[14] Durrett R., Probability: Theory and Examples, second ed., Duxbury Press, Belmont, CA, 1996.
[15] Figueira S., Hirschfeldt D., Miller J., Ng Selwyn, and Nies A, Counting the changes of random ${\rm{\Delta }}_2^0$ sets . Journal of Logic and Computation, vol. 25 (2015), pp. 10731089. Journal version of conference paper at CiE 2010.
[16] Franklin J., Greenberg N., Miller J. S., and Ng K. M., Martin-Löf random points satisfy Birkhoff’s ergodic theorem for effectively closed sets . Proceedings of the American Mathematical Society, vol. 140 (2012), no. 10, pp. 36233628.
[17] Franklin J. and Towsner H., Randomness and non-ergodic systems . Moscow Mathematical Journal, vol. 14 (2014), pp. 711714.
[18] Freer C., Kjos-Hanssen B., Nies A., and Stephan F., Algorithmic aspects of lipschitz functions . Computability, vol. 3 (2014), no. 1, pp. 4561.
[19] Gács P., Hoyrup M., and Rojas C., Randomness on computable probability spaces - a dynamical point of view . Theory of Computing Systems, vol. 48 (2011), no. 3, 465485.
[20] Hoyrup M. and Rojas C., Computability of probability measures and Martin-Löf randomness over metric spaces . Information and Computation, vol. 207 (2009), no. 7, pp. 830847.
[21] Kautz S., Degrees of Random Sets, Ph.D. dissertation, Cornell University, Ithaca, NY, 1991.
[22] Khan M., Lebesgue density and ${\rm{\Pi }}_1^0$ -classes. Journal of Symbolic Logic , to appear.
[23] Krengel U., Ergodic Theorems, W. de Gruyter, Boston, 1985.
[24] Kurtz S., Randomness and genericity in the degrees of unsolvability, Ph.D. dissertation, University of Illinois, Urbana, 1981.
[25] Lebesgue H., Leçons sur l’Intégration et la recherche des fonctions primitives . Gauthier-Villars, Paris, 1904.
[26] Lebesgue H., Sur les intégrales singulières . Annales de la Faculte des Sciences de Toulouse sciences Mathematics Science Physics (3), vol. 1 (1909), pp. 25117.
[27] Lebesgue H., Sur l’intégration des fonctions discontinues . Annales scientifiques de l’ Ecole normale supérieure, vol. 27 (1910), pp. 361450.
[28] Li M. and Vitányi P., An Introduction to Kolmogorov Complexity and its Applications, second ed., Graduate Texts in Computer Science, Springer-Verlag, New York, 1997.
[29] Miller J. S., Lebesgue density in ${\rm{\Pi }}_1^0$ -classes. Slides, Available at http://www-2.dc.uba.ar/ccr/talks/miller0201.pdf, February 2013.
[30] Miller J. S. and Nies A., Randomness and computability: Open questions, this Bulletin, vol. 12 (2006), no. 3, pp. 390410.
[31] Miyabe K., Characterization of Kurtz randomness by a differentiation theorem . Theory of Computing Systems, vol. 52 (2013), no. 1, pp. 113132.
[32] Morayne M. and Solecki S., Martingale proof of the existence of Lebesgue points . Real Analysis Exchange, vol. 15 (1989/90), no. 1, pp. 401406.
[33] Nies A., Computability and Randomness , Oxford Logic Guides, vol. 51, Oxford University Press, Oxford, 2009, 444 p. Paperback version 2011.
[34] Nies A., Differentiability of polynomial time computable functions , 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014) (Mayr E. W. and Portier N., editors), Leibniz International Proceedings in Informatics (LIPIcs), vol. 25, Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 2014, pp. 602613.
[35] Pathak N., Rojas C., and Simpson S. G., Schnorr randomness and the Lebesgue differentiation theorem . Proceedings of the American Mathematical Society, vol. 142 (2014), no. 1, pp. 335349.
[36] Pour-El M. and Richards J., Computability in Analysis and Physics, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1989.
[37] Rudin W., Real and Complex Analysis, third ed., McGraw-Hill, New York, 1987.
[38] Schnorr C. P., Zufälligkeit und Wahrscheinlichkeit , Eine algorithmische Begründung der Wahrscheinlichkeitstheorie, vol. 218, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1971.
[39] V’yugin V., Ergodic theorems for individual random sequences . Theoretical Computer Science, vol. 207 (1998), no. 2, pp. 343361.
[40] Weihrauch K., Computable Analysis, Springer, Berlin, 2000.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Bulletin of Symbolic Logic
  • ISSN: 1079-8986
  • EISSN: 1943-5894
  • URL: /core/journals/bulletin-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 31 *
Loading metrics...

Abstract views

Total abstract views: 135 *
Loading metrics...

* Views captured on Cambridge Core between 10th October 2016 - 16th December 2017. This data will be updated every 24 hours.