Low-Depth Witnesses are Easy to Find
- 96 Downloads
Kolmogorov Complexity measures the amount of information in a string by the size of the smallest program that generates that string. Antunes, Fortnow, van Melkebeek, and Vinodchandran captured the notion of useful information by computational depth, the difference between the polynomial-time-bounded Kolmogorov complexity and traditional Kolmogorov complexity.
We show unconditionally how to probabilistically find satisfying assignments for formulas that have at least one assignment of logarithmic depth. The converse holds under a standard hardness assumption though fails if BPP = FewP = EXP. We also prove that assuming the existence of good pseudorandom generators one cannot increase the depth of a string efficiently.
KeywordsComputational depth SAT formulas Kolmogorov complexity Pseudorandom generators
Subject classification68Q17 68Q30
Unable to display preview. Download preview PDF.
- L. Antunes & L. Fortnow (2009). Worst-Case Running Times for Average-Case Algorithms. Proceedings of Annual IEEE Conference on Computational Complexity 298–303.Google Scholar
- C. Bennett (1988). Logical depth and physical complexity. In A half-century survey on The Universal Turing Machine, 227–257. Oxford University Press, Inc., New York, NY, USA. ISBN 0-19-853741-7.Google Scholar
- R. Impagliazzo & A. Wigderson (1996). P = BPP unless E has sub-exponential circuits: Derandomizing the XOR Lemma (Preliminary Version). In Proceedings of the 29th ACM Symposium on Theory of Computing, 220–229. ACM Press.Google Scholar
- A. Kolmogorov (1950). Foundations of the Theory of Probability. Chelsea Publishing.Google Scholar
- Levin L. (1973) Universal Search Problems. Problems Information Transmission 9: 265–266Google Scholar
- M. Li & P. Vitányi (2008). An Introduction to Kolmogorov Complexity and Its Applications. Springer Publishing Company, Incorporated. ISBN 0387339981, 9780387339986.Google Scholar
- P. Miltersen (2001). Derandomizing complexity classes. Handbook of Randomized Computing.Google Scholar