Computational depth: Concept and applications

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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:

  • 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.

Keywords

Kolmogorov complexity
Structural complexity and average case complexity

Cited by (0)

Preliminary versions of different parts of this paper appeared as [1] and [2]. Much of the research for this paper occurred at the NEC Research Institute.

1

Partially supported by KCrypt (POSC/EIA/60819/2004) and funds granted to LIACC through the Programa de Financiamento Plurianual, FCT and Programa POSI.

2

Partially supported by NSF Career award CCR-0133693.

3

Partially supported by NSF Grant CCF-0430991.