Abstract
In this paper, we survey a few recent applications of Kolmogorov complexity to lower bounds in several models of computation. We consider KI complexity of Boolean functions, which gives the complexity of finding a bit where inputs differ, for pairs of inputs that map to different function values. This measure and variants thereof were shown to imply lower bounds for quantum and randomized decision tree complexity (or query complexity) [LM04]. We give a similar result for deterministic decision trees as well. It was later shown in [LLS05] that KI complexity gives lower bounds for circuit depth. We review those results here, emphasizing simple proofs using Kolmogorov complexity, instead of strongest possible lower bounds.
We also present a Kolmogorov complexity alternative to Yao’s min-max principle [LL04]. As an example, this is applied to randomized one-way communication complexity.
This is a preview of subscription content, log in via an institution to check access.
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
Aaronson, S.: Lower bounds for local search by quantum arguments lower bounds for local search by quantum arguments. In: Proceedings of the thirty-sixth annual ACM symposium on Theory of computing, pp. 465–474 (2004)
Ambainis, A.: Quantum lower bounds by quantum arguments. Journal of Computer and System Sciences 64, 750–767 (2002)
Ambainis, A.: Polynomial degree vs. quantum query complexity. In: Proceedings of 44th IEEE Symposium on Foundations of Computer Science, pp. 230–239 (2003)
Barnum, H., Saks, M., Szegedy, M.: Quantum decision trees and semidefinite programming. In: Proceedings of the 18th IEEE Conference on Computational Complexity, pp. 179–193 (2003)
Bar-Yossef, Z., Jayram, T.S., Kumar, R., Sivakumar, D.: Information theory methods in communication complexity. In: Proceedings of the 17th Annual IEEE Conference on Computational Complexity, pp. 93–102 (2002)
Kremer, I., Nisan, N., Ron, D.: On randomized one-round communication complexity. Computational Complexity 8(1), 21–49 (1999)
Karchmer, M., Wigderson, A.: Monotone connectivity circuits require super-logarithmic depth. In: Proceedings of the 20th STOC, pp. 539–550 (1988)
Laplante, S., Lee, T.: A few short lower bounds in one-way communication complexity (July 2004) (unpublished manuscript )
Laplante, S., Lee, T., Szegedy, M.: The quantum adversary method and classical formula size lower bounds. In: Proceedings of the Twentieth Annual IEEE Conference on Computational Complexity, pp. 76–90 (2005)
Laplante, S., Magniez, F.: Lower bounds for randomized and quantum query complexity using kolmogorov arguments. In: Proceedings of the Nineteenth Annual IEEE Conference on Computational Complexity, pp. 294–304 (2004)
Spira, P.: On time-hardware complexity tradeoffs for Boolean functions. In: Proceedings of the 4th Hawaii Symposium on System Sciences, pp. 525–527. Western Periodicals Company, North Hollywood (1971)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Laplante, S. (2006). Lower Bounds Using Kolmogorov Complexity. In: Beckmann, A., Berger, U., Löwe, B., Tucker, J.V. (eds) Logical Approaches to Computational Barriers. CiE 2006. Lecture Notes in Computer Science, vol 3988. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11780342_32
Download citation
DOI: https://doi.org/10.1007/11780342_32
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-35466-6
Online ISBN: 978-3-540-35468-0
eBook Packages: Computer ScienceComputer Science (R0)