Computability
Information-based complexity
Complexity theory of real functions
Complexity theory of real functions
Information randomness & incompleteness: papers on algorithmic information theory (2nd ed.)
Information randomness & incompleteness: papers on algorithmic information theory (2nd ed.)
On the Length of Programs for Computing Finite Binary Sequences
Journal of the ACM (JACM)
Information and Randomness: An Algorithmic Perspective
Information and Randomness: An Algorithmic Perspective
Recursively Enumerable Reals and Chaitin Omega Numbers
STACS '98 Proceedings of the 15th Annual Symposium on Theoretical Aspects of Computer Science
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How fast can one approximate a real by a computable sequence of rationals? Rather surprisingly, we show that the answer to this question depends very much on the information content in the finite prefixes of the binary expansion of the real. Computable reals, whose binary expansions have a very low information content, can be approximated (very fast) with a computable convergence rate. Random reals, whose binary expansions contain very much information in their prefixes, can be approximated only very slowly by computable sequences of rationals (this is the case, for example, for Chaitin's Ω numbers) if they can be computably approximated at all. We also show that one can computably approximate any computable real very slowly, with a convergence rate slower than any computable function. However, there is still a large gap between computable reals and random reals: any computable sequence of rationals which converges (monotonically) to a random real converges slower than any computable sequence of rationals which converges (monotonically) to a computable real.