Proceedings 16th Annual IEEE Conference on Computational Complexity (2001)
June 18, 2001 to June 21, 2001
Luis Antunes , University of Porto
Lance Fortnow , NEC Research Institute
Dieter Van Melkebeek , Institute for Advanced Study
Abstract: We introduce Computational Depth, a measure for the amount of "nonrandom" or "useful" information in a string by considering the difference of various Kolmogorov complexity measures. We investigate three instantiations of Computational Depth: 1) Basic Computational Depth, a clean notion capturing the spirit of Bennett's Logical Depth. 2) Time-t Computational Depth and the resulting concept of Shallow Sets, a generalization of sparse and random sets based on low depth properties of their characteristic sequences. We show that every computable set that is reducible to a shallow set has polynomial-size circuits. 3) Distinguishing Computational Depth, measuring when strings are easier to recognize than to produce. We show that if a Boolean formula has a nonnegligible fraction of its satisfying assignments with low depth, then we can find a satisfying assignment efficiently.
D. Van Melkebeek, L. Fortnow and L. Antunes, "Computational Depth," Proceedings 16th Annual IEEE Conference on Computational Complexity(CCC), Chicago, Illinois, 2001, pp. 0266.