Randomness conservation inequalities; information and independence in mathematical theories
Information and Control
Logical depth and physical complexity
A half-century survey on The Universal Turing Machine
Almost everywhere high nonuniform complexity
Journal of Computer and System Sciences
Computational depth and reducibility
Theoretical Computer Science
Information and Computation
Compressibility and Resource Bounded Measure
SIAM Journal on Computing
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
SIAM Journal on Computing
Computational depth: concept and applications
Theoretical Computer Science - Foundations of computation theory (FCT 2003)
Effective Strong Dimension in Algorithmic Information and Computational Complexity
SIAM Journal on Computing
Martingale families and dimension in P
Theoretical Computer Science
CiE '07 Proceedings of the 3rd conference on Computability in Europe: Computation and Logic in the Real World
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
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We introduce a general framework for defining the depth of an infinite binary sequence with respect to a class of observers. We show that our general framework captures all depth notions introduced in computability/complexity theory so far. We review most such notions, show how they are particular cases of our general depth framework, and review some classical results about the different depth notions. We use our framework to define new notions of polynomial depth (called monotone poly depth), based on a polynomial version of monotone Kolmogorov complexity. We show that monotone poly depth satisfies all desirable properties of depth notions. We give two natural examples of deep sets, by showing that both the set of Levin random strings and the set of Kolmogorov random strings are monotone poly deep.