Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-22T05:55:48.030Z Has data issue: false hasContentIssue false
  • English
  • Français

COMPUTING K-TRIVIAL SETS BY INCOMPLETE RANDOM SETS

Published online by Cambridge University Press:  13 May 2014

LAURENT BIENVENU ,
ADAM R. DAY ,
NOAM GREENBERG ,
ANTONÍN KUČERA ,
JOSEPH S. MILLER ,
ANDRÉ NIES  and
DAN TURETSKY
LAURENT BIENVENU
Affiliation:
LIAFA, CNRS & UNIVERSITÉ PARIS 7 CASE 7014, 75205 PARIS CEDEX 13 FRANCEE-mail:[email protected]
ADAM R. DAY
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]
ANTONÍN KUČERA
Affiliation:
CHARLES UNIVERSITY IN PRAGUE FACULTY OF MATHEMATICS AND PHYSICS PRAGUE, CZECH REPUBLICE-mail:[email protected]
JOSEPH S. MILLER
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF WISCONSIN MADISON, WI 53706-1388, USAE-mail:[email protected]
ANDRÉ NIES
Affiliation:
DEPARTMENT OF COMPUTER SCIENCE PRIVATE BAG 92019 AUCKLAND, NEW ZEALANDE-mail:[email protected]
DAN TURETSKY
Affiliation:
KURT GÖDEL RESEARCH CENTER FOR MATHEMATICAL LOGIC UNIVERSITY OF VIENNA VIENNA, AUSTRIAE-mail:[email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Every K-trivial set is computable from an incomplete Martin-Löf random set, i.e., a Martin-Löf random set that does not compute the halting problem.

Type
Communication
Information
Bulletin of Symbolic Logic , Volume 20 , Issue 1 , March 2014 , pp. 80 - 90
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

Arslanov, Marat M., Some generalizations of a fixed-point theorem. Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, vol. 25 (1981), no. 5, pp. 9–16.Google Scholar
Bienvenu, Laurent and Downey, Rodney G., Kolmogorov complexity and Solovay functions. Symposium on Theoretical Aspects of Computer Science, STACS, 2009, Dagstuhl Seminar Proceedings, pp. 147–158, http://drops.dagstuhl.de/opus/volltexte/2009/1810, 2009.Google Scholar
Bienvenu, Laurent, Greenberg, Noam, Kučera, Antonín, Nies, André, and Turetsky, Daniel, K-triviality, Oberwolfach randomness, and differentiability. Mathematisches ForschungsinstitutOberwolfach, Preprint Series, 2012.Google Scholar
Bienvenu, Laurent, Hoelzl, Rupert, Miller, Joseph S., and Nies, André, The Denjoy alternative for computable functions. STACS 2012, Schloss Dagstuhl, Leibniz-Zent. Inform., Wadern, pp. 543–554.Google Scholar
Bienvenu, Laurent, Hoelzl, Rupert, Miller, Joseph S., and Nies, André, Demuth, Denjoy, and density . Journal of Mathematical Logic, to appear.Google Scholar
Chaitin, Gregory J., A theory of program size formally identical to information theory. Journal of association computing machinery, 22 (1975), pp. 329–340.Google Scholar
Chaitin, Gregory J., Information-theoretic characterizations of recursive infinite strings. Theoretical Computer Science, 2 (1976), no. 1, pp. 45–48.Google Scholar
Chaitin, Gregory J., Nonrecursive infinite strings with simple initial segments. IBM Journal of Research and Development, 21 (1977), pp. 350–359 .Google Scholar
Day, Adam R. and Miller, Joseph S., Cupping with random sets. Proceedings of the American Mathematical Society, to appear.Google Scholar
Day, Adam R. and Miller, Joseph S., Density, forcing, and the covering problem, submitted, available at http://arxiv.org/abs/1304.2789.Google Scholar
Demuth, Osvald, Some classes of arithmetical real numbers. Commentationes Mathematicae Universitatis Carolinae, 23 (1962), no. 3, pp. 453–465.Google Scholar
Downey, Rodney G. and Hirschfeldt, Denis R., Algorithmic randomness and complexity. Theory and Applications of Computability, Springer, New York, 2010.Google Scholar
Downey, Rodney G., Hirschfeldt, Denis R., Nies, André, and Stephan, Frank, Trivial reals. Proceedings of the 7th and 8th Asian Logic Conferences, Singapore Univ. Press, Singapore, 2003,pp. 103–131.Google Scholar
Figueira, Santiago, Hirschfeldt, Denis, Miller, Joseph S., Ng, Keng Meng, and Nies, André, Counting the changes of random ${\rm{\Delta }}_2^0 $ sets. CiE 2010, June 2010, p. 1, available electronically athttp://link.springer.com/chapter/10.10072F978-3-642-13962-8-18.Google Scholar
Franklin, Johanna N. Y. and Ng, Keng Meng, Difference randomness. Proceedings of the American Mathematical Society, 139 (2011), no. 1, pp. 345–360.Google Scholar
Freer, Cameron, Kjos-Hanssen, Bjørn, Nies, André, and Stephan, Frank, Effective aspects of Lipschitz functions. Computability, to appear.Google Scholar
Gács, Peter, Every sequence is reducible to a random one. Information and Control, 70 (1986), pp. 186–192.Google Scholar
Greenberg, Noam and Nies, André, Benign cost functions and lowness properties. Journal of Symbolic Logic, 76 (2011), no. 1, pp. 289–312.Google Scholar
Hirschfeldt, Denis R., Nies, André, and Stephan, Frank, Using random sets as oracles. Journal of the London Mathematical Society (2), 75 (2007), no. 3, pp. 610–622.Google Scholar
Kučera, Antonín, Measure, ${\rm{\Pi }}_1^0 $-classes and complete extensions of .PA Recursion theory week, Oberwolfach, 1984, Lecture Notes in Mathematics, vol. 1141, Springer, Berlin, 1985, pp. 245–259.Google Scholar
Kučera, Antonín, An alternative, priority-free, solution to Post’s problem. Mathematical foundations of computer science, Bratislava, 1986, Lecture Notes in Computer Science, vol. 233, Springer, Berlin, 1986, pp. 493–500CrossRefGoogle Scholar
Kučera, Antonín, On relative randomness. Annals of Pure and Applied Logic, 63 (1993), no. 1, pp. 61–67, 9th International Congress of Logic, Methodology and Philosophy of Science Uppsala, 1991.Google Scholar
Kučera, Antonín and Slaman, Theodore A., Turing incomparability in Scott sets. Proceedings of the American Mathematical Society, 135 (2007), no. 11, pp. 3723–3731.Google Scholar
Kučera, Antonín and Slaman, Theodore A., Low upper bounds of ideals. Journal of Symbolic Logic, 74 (2009), pp. 517–534.Google Scholar
Kučera, Antonín and Terwijn, Sebastiaan A., Lowness for the class of random sets. Journal of Symbolic Logic, 64 (1999), pp. 1396–1402 .Google Scholar
Kučera, Antonín and Nies, André, Demuth’s path to randomness. Proceedings of the 2012 international conference on Theoretical Computer Science: computation, physics and beyond (WTCS’12), Berlin, Heidelberg, 2012, Springer-Verlag, pp. 159–173.CrossRefGoogle Scholar
Miller, Joseph S. and Nies, André, Randomness and computability: open questions. this Journal, 12 (2006), no. 3, pp. 390–410.Google Scholar
Nies, A., Reals which compute little. Logic Colloquium ’02, Lecture Notes in Logic, vol. 27, Association of Symbolic Logic, La Jolla, CA, 2002, pp. 260–274.Google Scholar
Nies, André, Lowness properties and randomness. Advances in Mathematics, 197 (2005), no. 1, pp. 274–305.Google Scholar
Nies, André, Computability and randomness, Oxford Logic Guides, vol. 51, Oxford University Press, Oxford, 2009.Google Scholar
Nies, André, Interactions of computability and randomness. Proceedings of the International Congress of Mathematicians, World Scientific, Singapore, 2010, pp. 30–57.Google Scholar
Schnorr, Claus-Peter, Process complexity and effective random tests. Journal of Computer and System Sciences, 7 (1973), pp. 376–388, Fourth Annual ACM Symposium on the Theory of Computing Denver, Colo., 1972.Google Scholar
Solovay, Robert M., Draft of paper (or series of papers) related to Chaitin’s work. IBM Thomas J. Watson Research Center, Yorktown Heights, NewYork, 1975, p. 215 .Google Scholar
Stephan, Frank, Martin-Löf random and PA-complete sets. Logic Colloquium ’02, Lecture Notes in Logic, vol. 27, Assoc. Symbol. Logic, La Jolla, CA, 2006, pp. 342–348.Google Scholar
Zambella, Domenico, On sequences with simple initial segments. ILLC technical report ML 1990-05, Univ. Amsterdam, 1990.Google Scholar