Abstract
In this paper, we prove that if a c.e. Turing degree d is non-low2, then there are two left-c.e. reals β 0, β 1 in d, such that, if β 0 is wtt-reducible to a left-c.e. real α, then β 1 is not computable Lipschitz (cl-) reducible to α. As a corollary, d contains a left-c.e. real which is not cl-reducible to any complex (wtt-complete) left-c.e. real.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Barmpalias, G., Lewis, A.: A left-c.e. real that cannot be SW-computed by any O number. Notre Dame J. Formal Logic, 47(2), 197–209 (2006)
Barmpalias, G., Lewis, A.: Random reals and Lipschitz continuity. Math. Structures Computer Sci., 16, 737–749 (2006)
Barmpalias, G., Lewis, A.: Randomness and the linear degrees of computability. Ann. Pure Appl. Logic, 145, 252–257 (2007)
Barmpalias, G., Downey R., Greenberg, N.: Working with strong reducibilities above totally-c.e. degrees and array computable degrees. Trans. Amer. Math. Soc., 362(2), 777–813 (2010)
Downey, R., Hirschfeldt, D., LaForte, G.: Randomness and reducibility. Mathematical Foundations of Computer Science 2001, Lecture Notes in Computer Science, 2136, 316–327 (2001); Final version in Journal of Computing and System Sciences, 68, 96–114 (2004)
Downey, R., Hirschfeldt, D.: Algorithmic Randomness and Complexity, Springer-Verlag Monographs in Computer Science, New York, 2010
Fan, Y., Yu, L.: Maximal pairs of left-c.e. reals in the computably Lipschitz degrees. Ann. Pure Appl. Logic, 162(5), 357–366 (2011)
Fan, Y.: A uniform version of non-low2-ness. Ann. Pure Appl. Logic, 168(3), 738–748 (2017)
Kjos-Hanssen, K., Merkle, W., Stephan, F.: Kolmogorov complexity and the recursion theorem. Twenty-Third Annual Symposium on Theoretical Aspects of Computer Science, Marseille, France, February 23–25, 2006. Proceedings, Springer Lecture Notes in Computer Science, 3884, 149–161 (2006)
Soare, R.: Computability theory and differential geometry. Bull. Symbolic Logic, 10(4), 457–486 (2004)
Yu, L., Ding, D.: There is no SW-complete left-c.e. real. J. Symbolic Logic, 69(4), 1163–1170 (2004)
Acknowledgements
We thank the referees for their time and comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by NSFC (Grant No. 11201065)
Rights and permissions
About this article
Cite this article
Fan, Y. Non-low2-ness and computable Lipschitz reducibility. Acta. Math. Sin.-English Ser. 33, 1184–1192 (2017). https://doi.org/10.1007/s10114-017-6585-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-017-6585-5