Randomness conservation inequalities; information and independence in mathematical theories
Information and Control
Average case complete problems
SIAM Journal on Computing
On non-uniform polynomial space
Proc. of the conference on Structure in complexity theory
Logical depth and physical complexity
A half-century survey on The Universal Turing Machine
Average case complexity under the universal distribution equals worst-case complexity
Information Processing Letters
On the theory of average case complexity
Journal of Computer and System Sciences
The complexity of malign measures
SIAM Journal on Computing
Journal of Computer and System Sciences
On resource-bounded instance complexity
Theoretical Computer Science
On extracting randomness from weak random sources (extended abstract)
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
An introduction to Kolmogorov complexity and its applications (2nd ed.)
An introduction to Kolmogorov complexity and its applications (2nd ed.)
Average-case computational complexity theory
Complexity theory retrospective II
Resource-Bounded Kolmogorov Complexity Revisited
STACS '97 Proceedings of the 14th Annual Symposium on Theoretical Aspects of Computer Science
Nearly Optimal Language Compression Using Extractors
STACS '98 Proceedings of the 15th Annual Symposium on Theoretical Aspects of Computer Science
New Bounds for the Language Compression Problem
COCO '00 Proceedings of the 15th Annual IEEE Conference on Computational Complexity
Some connections between nonuniform and uniform complexity classes
STOC '80 Proceedings of the twelfth annual ACM symposium on Theory of computing
A complexity theoretic approach to randomness
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
CCC '01 Proceedings of the 16th Annual Conference on Computational Complexity
Universal distributions and time-bounded kolmogorov complexity
STACS'99 Proceedings of the 16th annual conference on Theoretical aspects of computer science
CiE '07 Proceedings of the 3rd conference on Computability in Europe: Computation and Logic in the Real World
Time-bounded incompressibility of compressible strings and sequences
Information Processing Letters
Information measures for infinite sequences
Theoretical Computer Science
CiE'10 Proceedings of the Programs, proofs, process and 6th international conference on Computability in Europe
On the polynomial depth of various sets of random strings
TAMC'11 Proceedings of the 8th annual conference on Theory and applications of models of computation
CiE'12 Proceedings of the 8th Turing Centenary conference on Computability in Europe: how the world computes
On the polynomial depth of various sets of random strings
Theoretical Computer Science
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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.