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Abstract We introduce Computational Depth, a measure for theamount of "nonrandom " or "useful " information in a string by considering the difference of various Kolmogorovcomplexity measures. We investigate three instantiations of Computational Depth: ffl Basic Computational Depth, a clean notion capturingthe spirit of Bennett's Logical Depth. ffl Time-t Computational Depth and the resulting conceptof Shallow Sets, a generalization of sparse and random sets based on low depth properties of their character-istic sequences. We show that every computable set that is reducible to a shallow set has polynomial-sizecircuits. ffl Distinguishing Computational Depth, measuring whenstrings are easier to recognize than to produce. We show that if a Boolean formula has a nonnegligiblefraction of its satisfying assignments with low depth, then we can find a satisfying assignment efficiently

Year: 2009
OAI identifier: oai:CiteSeerX.psu:10.1.1.135.6453
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