Incompleteness theorems for random reals
Advances in Applied Mathematics
On the continued fraction representation of computable real numbers
Theoretical Computer Science
Information randomness & incompleteness: papers on algorithmic information theory (2nd ed.)
Information randomness & incompleteness: papers on algorithmic information theory (2nd ed.)
An introduction to Kolmogorov complexity and its applications (2nd ed.)
An introduction to Kolmogorov complexity and its applications (2nd ed.)
Relatively recursive reals and real functions
Theoretical Computer Science - Special issue on real numbers and computers
Theoretical Computer Science - Special issue on computability and complexity in analysis
A Theory of Program Size Formally Identical to Information Theory
Journal of the ACM (JACM)
Weakly computable real numbers
Journal of Complexity
Information and Randomness: An Algorithmic Perspective
Information and Randomness: An Algorithmic Perspective
A characterization of c.e. random reals
Theoretical Computer Science
Presentations of computably enumerable reals
Theoretical Computer Science
MFCS '01 Proceedings of the 26th International Symposium on Mathematical Foundations of Computer Science
Recursively Enumerable Reals and Chaitin Omega Numbers
STACS '98 Proceedings of the 15th Annual Symposium on Theoretical Aspects of Computer Science
Algorithmic information theory
IBM Journal of Research and Development
Process complexity and effective random tests
Journal of Computer and System Sciences
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We study effectively given positive reals (more specifically, computably enumerable reals) under a measure of relative randomness introduced by Solovay [32] and studied by Calude, Hertling, Khoussainov, and Wang [7], Calude [3], Slaman [28], and Coles, Downey, and LaForte [14], among others. This measure is called domination or Solovay reducibility, and is defined by saying that α dominates β if there are a constant c and a partial computable function ϕ such that for all positive rationals q q) ↓ q) ≤ c(α - q). The intuition is that an approximating sequence for α generates one for β whose rate of convergence is not much slower than that of the original sequence. It is not hard to show that if α dominates β then the initial segment complexity of α is at least that of β. In this paper we are concerned with structural properties of the degree structure generated by Solovay reducibility. We answer a long-standing question in this area of investigation by establishing the density of the Solovay degrees. We also provide a new characterization of the random c.e. reals in terms of splittings in the Solovay degrees. Specifically, we show that the Solovay degrees of computably enumerable reals are dense, that any incomplete Solovay degree splits over any lesser degree, and that the join of any two incomplete Solovay degrees is incomplete, so that the complete Solovay degree does not split at all. The methodology is of some technical interest, since it includes a priority argument in which the injuries are themselves controlled by randomness considerations.