When Worlds Collide: Derandomization, Lower Bounds, and Kolmogorov Complexity
FST TCS '01 Proceedings of the 21st Conference on Foundations of Software Technology and Theoretical Computer Science
Computational depth: concept and applications
Theoretical Computer Science - Foundations of computation theory (FCT 2003)
ICALP'03 Proceedings of the 30th international conference on Automata, languages and programming
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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.