Abstract
We show the following results for polynomial-time reducibility to R C , the set of Kolmogorov random strings.
-
1
If P ≠ NP, then SAT does not dtt-reduce to R C .
-
1
If PH does not collapse, then SAT does not n α-tt-reduce to R C for any α< 1.
-
1
If PH does not collapse, then SAT does not n α-T-reduce to R C for any \(\alpha < {{1}\over{2}}\).
-
1
There is a problem in E that does not dtt-reduce to R C .
-
1
There is a problem in E that does not n α-tt-reduce to R C , for any α< 1.
-
1
There is a problem in E that does not n α-T-reduce to R C , for any \(\alpha < {{1}\over{2}}\).
These results hold for both the plain and prefix-free variants of Kolmogorov complexity and are also independent of the choice of the universal machine.
This research was supported in part by NSF grant 0652601 and by an NWO travel grant. Part of this research was done while the author was on sabbatical at CWI.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Allender, E., Buhrman, H., Koucký, M.: What can be efficiently reduced to the Kolmogorov-random strings? Annals of Pure and Applied Logic 138, 2–19 (2006)
Allender, E., Buhrman, H., Koucký, M., van Melkebeek, D., Ronneburger, D.: Power from random strings. SIAM Journal on Computing 35, 1467–1493 (2006)
Buhrman, H., Fortnow, L., Koucký, M., Loff, B.: Derandomizing from random strings. Technical Report 0912.3162, arXiv.org e-Print archive (2009)
Buhrman, H., Hitchcock, J.M.: NP-hard sets are exponentially dense unless NP ⊆ coNP/poly. In: Proceedings of the 23rd Annual IEEE Conference on Computational Complexity, pp. 1–7. IEEE Computer Society, Los Alamitos (2008)
Fortnow, L., Santhanam, R.: Infeasibility of instance compression and succinct PCPs for NP. In: Proceedings of the 40th Annual ACM Symposium on Theory of Computing, pp. 133–142 (2008)
Hitchcock, J.M.: Online learning and resource-bounded dimension: Winnow yields new lower bounds for hard sets. SIAM Journal on Computing 36(6), 1696–1708 (2007)
Kummer, M.: On the complexity of random strings. In: Proceedings of the 13th Annual Symposium on Theoretical Aspects of Computer Science, pp. 25–36. Springer, Heidelberg (1996)
Littlestone, N.: Learning quickly when irrelevant attributes abound: A new linear-threshold algorithm. Machine Learning 2(4), 285–318 (1988)
Ladner, R.E., Lynch, N.A., Selman, A.L.: A comparison of polynomial-time reducibilities. Theoretical Computer Science 1(2), 103–123 (1975)
Lutz, J.H.: Dimension in complexity classes. SIAM Journal on Computing 32(5), 1236–1259 (2003)
Li, M., Vitányi, P.M.B.: An Introduction to Kolmogorov Complexity and its Applications, 3rd edn. Springer, Heidelberg (2008)
Ukkonen, E.: Two results on polynomial time truth-table reductions to sparse sets. SIAM Journal on Computing 12(3) (1983)
Yap, C.K.: Some consequences of non-uniform conditions on uniform classes. Theoretical Computer Science 26, 287–300 (1983)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Hitchcock, J.M. (2010). Lower Bounds for Reducibility to the Kolmogorov Random Strings. In: Ferreira, F., Löwe, B., Mayordomo, E., Mendes Gomes, L. (eds) Programs, Proofs, Processes. CiE 2010. Lecture Notes in Computer Science, vol 6158. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13962-8_22
Download citation
DOI: https://doi.org/10.1007/978-3-642-13962-8_22
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-13961-1
Online ISBN: 978-3-642-13962-8
eBook Packages: Computer ScienceComputer Science (R0)