Computational depth

Abstract

Introduces computational depth, a measure for the amount of "non-random" 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 C.H. Bennett's (1988) 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); and (3) distinguishing computational depth, measuring when strings are easier to recognize than to produce (we show that if a Boolean formula has a non-negligible fraction of its satisfying assignments with low depth, then we can find a satisfying assignment efficiently).