Skip to main content Accessibility help
×
Home

CHARACTERIZING LOWNESS FOR DEMUTH RANDOMNESS

  • LAURENT BIENVENU (a1), ROD DOWNEY (a2), NOAM GREENBERG (a3), ANDRÉ NIES (a4) and DAN TURETSKY (a5)...

Abstract

We show the existence of noncomputable oracles which are low for Demuth randomness, answering a question in [15] (also Problem 5.5.19 in [34]). We fully characterize lowness for Demuth randomness using an appropriate notion of traceability. Central to this characterization is a partial relativization of Demuth randomness, which may be more natural than the fully relativized version. We also show that an oracle is low for weak Demuth randomness if and only if it is computable.

Copyright

References

Hide All
[1]Barmpalias, G., Miller, J. S., and Nies, A., Randomness notions and partial relativization. Israel Journal of Mathematics, vol. 191 (2012), no. 2, pp. 791816.
[2]Bartoszyński, T., Combinatorial aspects of measure and category. Fundamenta Mathematicae, vol. 127 (1987), no. 3, pp. 225239.
[3]Bienvenu, L., Diamondstone, D., Greenberg, N., and Turetsky, D., Van Lambalgen’s theorem for Demuth randomness, Proceedings of the 12th Asian Logic Colloquium (Downey, R., Brendle, J., Goldblatt, R., and Kim, B., editors), World Scientific, pp. 251270, 2013.
[4]Bienvenu, L., Hoelzl, R., Miller, J., and Nies, A., The Denjoy alternative for computable functions. Symposium on theoretical aspects of computer science (STACS), pp. 543554, 2012.
[5]Bienvenu, L., Hoelzl, R., Miller, J., and Nies, A., Demuth, Denjoy, and density.Journal of Mathematical Logic, to appear.
[6]Bienvenu, L. and Miller, J., Randomness and lowness notions via open covers. Annals of Pure and Applied Logic, vol. 163 (2012), no. 5, pp. 506518.
[7]Cole, J. and Simpson, S., Mass problems and hyperarithmeticity. Journal of Mathematical Logic, vol. 7 (2007), no. 2, pp. 125143.
[8]Demuth, O., Some classes of arithmetical real numbers. Commentationes Mathematicae Universitatis Carolinae, vol. 23 (1982), no. 3, pp. 453465.
[9]Demuth, O., Remarks on the structure of tt-degrees based on constructive measure theory. Commentationes Mathematicae Universitatis Carolinae, vol. 29 (1988), no. 2, pp. 233247.
[10]Diamondstone, D., Greenberg, N., and Turetsky, D., Inherent enumerability of strong jump-traceability. Transactions of the American Mathematical Society, to appear.
[11]Downey, Rod, On ${\rm{\Pi }}_1^0$classes and their ranked points. Notre Dame Journal of Formal Logic, vol. 32 (1991), no. 4, pp. 499512.
[12]Downey, R. and Greenberg, N., Pseudo-jump inversion, upper cone avoidance, and strong jump-traceability. Advances in Mathematics, vol. 237 (2013), pp. 252285.
[13]Downey, R., Griffiths, E., and Laforte, G., On Schnorr and computable randomness, martingales, and machines. Mathematical Logic Quarterly, vol. 50 (2004), no. 6, pp. 613627.
[14]Downey, R. and Hirschfeldt, D., Algorithmic randomness and complexity. Theory and Applications of Computability. Springer, New York, 2010.
[15]Downey, R. and Ng, K. M., Lowness for Demuth randomness, Mathematical Theory and Computational Practice, Fifth Conference on Computability in Europe, CiE 2009, Heidelberg, Germany, July 19–July 24, (Ambos-Spies, Klaus, LŁowe, Benedikt, and Merkle, Wolfgang, editors), Lecture Notes in Computer Science, vol. 5635, pp. 154166. Springer, Berlin, Heidelberg, 2009.
[16]Downey, R., Nies, A., Weber, R., and Yu, L., Lowness and nullsets ${\rm{\Pi }}_2^0 $. this Journal, 71 (2006), no. 3, pp. 10441052.
[17]Figueira, S., Nies, A., and Stephan, F., Lowness properties and approximations of the jump. Annals of Pure and Applied Logic, vol. 152 (2008), pp. 5166.
[18]Franklin, J. and Diamondstone, D., Lowness for difference tests. Notre Dame Journal of Formal Logic, vol. 55 (2014), pp. 6373.
[19]Gács, P., Every sequence is reducible to a random one. Information and Control, vol. 70 (1986), pp. 186192.
[20]Greenberg, N. and Miller, J., Lowness for Kurtz randomness. this Journal, vol. 74 (2009), no. 2, pp. 665678.
[21]Greenberg, N. and Turetsky, D., Strong jump-traceability and Demuth randomness. Proceedings of the London Mathematical Society, to appear.
[22]Hölzl, R., Kräling, T., Stephan, S., and Wu, G., Initial segment complexities of randomness notions. To appear.
[23]Ishmukhametov, S., Weak recursive degrees and a problem of spector. Recursion theory and complexity, (Kazan, 1997), vol. 2, de Gruyter Ser. Log. Appl., pp. 8187. de Gruyter, Berlin, 1999.
[24]Kautz, S., Degrees of random sets. Ph.D. Dissertation, Cornell University, 1991.
[25]Kjos-Hanssen, B., Nies, A., and Stephan, F., Lowness for the class of Schnorr random sets. SIAM Journal on Computing, vol. 35 (2005), no. 3, pp. 647657.
[26]Kučera, A., An alternative, priority-free, solution to Post’s problem, Mathematical foundations of computer science, 1986, (Bratislava, 1986), (Gruska, J., editor), vol. 233, Lecture Notes in Computer Science, pp. 493500, Springer, Berlin, 1986.
[27]Kučera, A. and Terwijn, S., Lowness for the class of random sets. this Journal, vol. 64 (1999), pp. 13961402.
[28]Kučera, A. and Nies, A., Demuth randomness and computational complexity. Annals of Pure and Applied Logic, vol. 162 (2011), pp. 504513.
[29]Kučera, A. and Nies, A., Demuth’s path to randomness (extended abstract). Computation, physics and beyond. Lecture Notes in Computer Science, vol. 7160, pp. 159–173, 2012.
[30]Miller, W. and Martin, D. A., The degree of hyperimmune sets. Zeitschrift fur mathematische Logik und Grundlage der Mathematik, vol. 14 (1968), pp. 159166.
[31]Miyabe, K., Truth-table Schnorr randomness and truth-table reducible randomness. Mathematical Logic Quarterly, vol. 57 (2011), no. 3, pp. 323338.
[32]Nies, A., Reals which compute little, Logic Colloquium ’02, Lecture Notes in Logic, pp. 260274. Springer–Verlag, 2002.
[33]Nies, A., Lowness properties and randomness. Advances in Mathematics, vol. 197 (2005), pp. 274305.
[34]Nies, A., Computability and randomness, Oxford Logic Guides, vol. 51, Oxford University Press, Oxford, 2009.
[35]Nies, A., New directions in computability and randomness.Talk at the CCR 2009 in Luminy, available athttp://www.cs.auckland.ac.nz/∼nies/talklinks/Luminy.pdf, 2009.
[36]Shiryayev, A. N., Probability, , Graduate Texts in Mathematics, vol. 95. Springer-Verlag, New York, 1984. Translated from the Russian by R. P. Boas.
[37]Stephan, F., Marin-Löf random and PA-complete sets, Logic Colloquium ’02, Lect. Notes Log., vol. 27, pp. 342348. Assoc. Symbol. Logic, La Jolla, CA, 2006.
[38]Stephan, F. and Yu, L., Lowness for weakly 1-generic and Kurtz-random, Theory and applications of models of computation, Lecture Notes in Computer Science, vol. 3959, pp. 756764. Springer, Berlin, 2006.
[39]Terwijn, S. and Zambella, D., Algorithmic randomness and lowness. this JOURNAL, vol. 66 (2001), pp. 11991205.

Keywords

CHARACTERIZING LOWNESS FOR DEMUTH RANDOMNESS

  • LAURENT BIENVENU (a1), ROD DOWNEY (a2), NOAM GREENBERG (a3), ANDRÉ NIES (a4) and DAN TURETSKY (a5)...

Metrics

Full text views

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

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.