Algorithmic information theory
Algorithmic information theory
Kolmogorov complexity and Hausdorff dimension
Information and Computation
A Theory of Program Size Formally Identical to Information Theory
Journal of the ACM (JACM)
Recursively enumerable reals and Chaitin &OHgr; numbers
Theoretical Computer Science
Information and Randomness: An Algorithmic Perspective
Information and Randomness: An Algorithmic Perspective
Randomness and Recursive Enumerability
SIAM Journal on Computing
Characterization of the Computable Real Numbers by Means of Primitive Recursive Functions
CCA '00 Selected Papers from the 4th International Workshop on Computability and Complexity in Analysis
Computability and Randomness
Theoretical Computer Science
On oscillation-free chaitin h-random sequences
WTCS'12 Proceedings of the 2012 international conference on Theoretical Computer Science: computation, physics and beyond
Phase transition between unidirectionality and bidirectionality
WTCS'12 Proceedings of the 2012 international conference on Theoretical Computer Science: computation, physics and beyond
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In this paper we introduce the notion of @e-universal prefix-free Turing machine (@e is a computable real in (0,1]) and study its halting probability. The main result is the extension of the representability theorem for left-computable random reals to the case of @e-random reals: a real is left-computable @e-random iff it is the halting probability of an @e-universal prefix-free Turing machine. We also show that left-computable @e-random reals are provable @e-random in the Peano Arithmetic. The theory developed here parallels to a large extent the classical theory, but not completely. For example, random reals are Borel normal (in any base), but for @e@?(0,1), some @e-random reals do not contain even arbitrarily long runs of 0s.