Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-27T20:31:30.606Z Has data issue: false hasContentIssue false
  • English
  • Français

CHARACTERIZING LOWNESS FOR DEMUTH RANDOMNESS

Published online by Cambridge University Press:  25 June 2014

LAURENT BIENVENU ,
ROD DOWNEY ,
NOAM GREENBERG ,
ANDRÉ NIES  and
DAN TURETSKY
LAURENT BIENVENU
Affiliation:
LIAFA, CNRS & UNIVERSITY OF PARIS 7 PARIS, FRANCEE-mail: [email protected]
ROD DOWNEY
Affiliation:
SCHOOL OF MATHEMATICS, STATISTICS AND OPERATIONS RESEARCH VICTORIA UNIVERSITY OF WELLINGTON WELLINGTON, NEW ZEALANDE-mail: [email protected]
NOAM GREENBERG
Affiliation:
SCHOOL OF MATHEMATICS, STATISTICS AND OPERATIONS RESEARCH VICTORIA UNIVERSITY OF WELLINGTON WELLINGTON, NEW ZEALANDE-mail: [email protected]
ANDRÉ NIES
Affiliation:
DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF AUCKLAND, PRIVATE BAG 92019 AUCKLAND, NEW ZEALANDE-mail: [email protected]
DAN TURETSKY
Affiliation:
KURT GÖDEL RESEARCH FOR MATHEMATICAL LOGIC UNIVERSITY OF VIENNA VIENNA, AUSTRIAE-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

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.

Keywords

Primary: 03D28 and 68Q30Secondary: 03D25Demuth randomnesspartial relativizationlowness
Type
Articles
Information
The Journal of Symbolic Logic , Volume 79 , Issue 2 , June 2014 , pp. 526 - 560
Copyright
Copyright © The Association for Symbolic Logic 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Barmpalias, G., Miller, J. S., and Nies, A., Randomness notions and partial relativization. Israel Journal of Mathematics, vol. 191 (2012), no. 2, pp. 791–816.CrossRefGoogle Scholar
Bartoszyński, T., Combinatorial aspects of measure and category. Fundamenta Mathematicae, vol. 127 (1987), no. 3, pp. 225–239.CrossRefGoogle Scholar
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. 251–270, 2013.Google Scholar
Bienvenu, L., Hoelzl, R., Miller, J., and Nies, A., The Denjoy alternative for computable functions. Symposium on theoretical aspects of computer science (STACS), pp. 543–554, 2012.Google Scholar
Bienvenu, L., Hoelzl, R., Miller, J., and Nies, A., Demuth, Denjoy, and density.Journal of Mathematical Logic, to appear.Google Scholar
Bienvenu, L. and Miller, J., Randomness and lowness notions via open covers. Annals of Pure and Applied Logic, vol. 163 (2012), no. 5, pp. 506–518.CrossRefGoogle Scholar
Cole, J. and Simpson, S., Mass problems and hyperarithmeticity. Journal of Mathematical Logic, vol. 7 (2007), no. 2, pp. 125–143.CrossRefGoogle Scholar
Demuth, O., Some classes of arithmetical real numbers. Commentationes Mathematicae Universitatis Carolinae, vol. 23 (1982), no. 3, pp. 453–465.Google Scholar
Demuth, O., Remarks on the structure of tt-degrees based on constructive measure theory. Commentationes Mathematicae Universitatis Carolinae, vol. 29 (1988), no. 2, pp. 233–247.Google Scholar
Diamondstone, D., Greenberg, N., and Turetsky, D., Inherent enumerability of strong jump-traceability. Transactions of the American Mathematical Society, to appear.Google Scholar
Downey, Rod, On ${\rm{\Pi }}_1^0$classes and their ranked points. Notre Dame Journal of Formal Logic, vol. 32 (1991), no. 4, pp. 499–512.CrossRefGoogle Scholar
Downey, R. and Greenberg, N., Pseudo-jump inversion, upper cone avoidance, and strong jump-traceability. Advances in Mathematics, vol. 237 (2013), pp. 252–285.CrossRefGoogle Scholar
Downey, R., Griffiths, E., and Laforte, G., On Schnorr and computable randomness, martingales, and machines. Mathematical Logic Quarterly, vol. 50 (2004), no. 6, pp. 613–627.Google Scholar
Downey, R. and Hirschfeldt, D., Algorithmic randomness and complexity. Theory and Applications of Computability. Springer, New York, 2010.CrossRefGoogle Scholar
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. 154–166. Springer, Berlin, Heidelberg, 2009.Google Scholar
Downey, R., Nies, A., Weber, R., and Yu, L., Lowness and nullsets ${\rm{\Pi }}_2^0 $. this Journal, 71 (2006), no. 3, pp. 1044–1052.Google Scholar
Figueira, S., Nies, A., and Stephan, F., Lowness properties and approximations of the jump. Annals of Pure and Applied Logic, vol. 152 (2008), pp. 51–66.CrossRefGoogle Scholar
Franklin, J. and Diamondstone, D., Lowness for difference tests. Notre Dame Journal of Formal Logic, vol. 55 (2014), pp. 63–73.Google Scholar
Gács, P., Every sequence is reducible to a random one. Information and Control, vol. 70 (1986), pp. 186–192.Google Scholar
Greenberg, N. and Miller, J., Lowness for Kurtz randomness. this Journal, vol. 74 (2009), no. 2, pp. 665–678.Google Scholar
Greenberg, N. and Turetsky, D., Strong jump-traceability and Demuth randomness. Proceedings of the London Mathematical Society, to appear.Google Scholar
Hölzl, R., Kräling, T., Stephan, S., and Wu, G., Initial segment complexities of randomness notions. To appear.Google Scholar
Ishmukhametov, S., Weak recursive degrees and a problem of spector. Recursion theory and complexity, (Kazan, 1997), vol. 2, de Gruyter Ser. Log. Appl., pp. 81–87. de Gruyter, Berlin, 1999.CrossRefGoogle Scholar
Kautz, S., Degrees of random sets. Ph.D. Dissertation, Cornell University, 1991.Google Scholar
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. 647–657.CrossRefGoogle Scholar
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. 493–500, Springer, Berlin, 1986.Google Scholar
Kučera, A. and Terwijn, S., Lowness for the class of random sets. this Journal, vol. 64 (1999), pp. 1396–1402.Google Scholar
Kučera, A. and Nies, A., Demuth randomness and computational complexity. Annals of Pure and Applied Logic, vol. 162 (2011), pp. 504–513.CrossRefGoogle Scholar
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.Google Scholar
Miller, W. and Martin, D. A., The degree of hyperimmune sets. Zeitschrift fur mathematische Logik und Grundlage der Mathematik, vol. 14 (1968), pp. 159–166.CrossRefGoogle Scholar
Miyabe, K., Truth-table Schnorr randomness and truth-table reducible randomness. Mathematical Logic Quarterly, vol. 57 (2011), no. 3, pp. 323–338.Google Scholar
Nies, A., Reals which compute little, Logic Colloquium ’02, Lecture Notes in Logic, pp. 260–274. Springer–Verlag, 2002.Google Scholar
Nies, A., Lowness properties and randomness. Advances in Mathematics, vol. 197 (2005), pp. 274–305.CrossRefGoogle Scholar
Nies, A., Computability and randomness, Oxford Logic Guides, vol. 51, Oxford University Press, Oxford, 2009.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Shiryayev, A. N., Probability, , Graduate Texts in Mathematics, vol. 95. Springer-Verlag, New York, 1984. Translated from the Russian by R. P. Boas.Google Scholar
Stephan, F., Marin-Löf random and PA-complete sets, Logic Colloquium ’02, Lect. Notes Log., vol. 27, pp. 342–348. Assoc. Symbol. Logic, La Jolla, CA, 2006.Google Scholar
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. 756–764. Springer, Berlin, 2006.CrossRefGoogle Scholar
Terwijn, S. and Zambella, D., Algorithmic randomness and lowness. this JOURNAL, vol. 66 (2001), pp. 1199–1205.Google Scholar