A Theory of Program Size Formally Identical to Information Theory
Journal of the ACM (JACM)
Weakly computable real numbers
Journal of Complexity
Recursively enumerable reals and Chaitin &OHgr; numbers
Theoretical Computer Science
Randomness, Computability, and Density
SIAM Journal on Computing
Randomness and Recursive Enumerability
SIAM Journal on Computing
MFCS '01 Proceedings of the 26th International Symposium on Mathematical Foundations of Computer Science
Closure Properties of Real Number Classes under CBV Functions
Theory of Computing Systems
Visualization 2001 Conference (Acm
Visualization 2001 Conference (Acm
Randomness and universal machines
Journal of Complexity
A computability theory of real numbers
CiE'06 Proceedings of the Second conference on Computability in Europe: logical Approaches to Computational Barriers
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A c.e. real x is Solovay reducible to another c.e. real y if x can be approximated at least as efficiently as y by means of increasing computable sequences of rational numbers. The Solovay reducibility classifies elegantly the relative randomness of c.e. reals. Especially, the c.e. random reals are complete unter the Solovay reducibility for c.e. reals. In this paper we investigate an extension of the Solovay reducibility to the Δ$^{\rm 0}_{\rm 2}$-reals and show that the c.e. random reals are complete under (extended) Solovay reducibility for d-c.e. reals too. Actually we show that only the d-c.e. reals can be Solovay reducible to an c.e. random real. Furthermore, we show that this fails for the class of divergence bounded computable reals which extends the class of d-c.e. reals properly. In addition, we show also that any d-c.e. random reals are either c.e. or co-c.e.