Computational depth: Concept and applications

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:
• Basic Computational Depth, a clean notion capturing the spirit of Bennett's Logical Depth. We show that a Turing machine M runs in time polynomial on average over the time-bounded universal distribution if and only if for all inputs x, M uses time exponential in the basic computational depth of x.
• Sublinear-time 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.
• 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.