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Title: Representation of left-computable ε-random reals
Authors: Calude, C.S.
Hay, N.J.
Stephan, F. 
Keywords: ε-random real
ε-universal prefix-free Turing machine
Halting probability
Peano Arithmetic
Issue Date: Jul-2011
Source: Calude, C.S., Hay, N.J., Stephan, F. (2011-07). Representation of left-computable ε-random reals. Journal of Computer and System Sciences 77 (4) : 812-819. ScholarBank@NUS Repository.
Abstract: In this paper we introduce the notion of ε-universal prefix-free Turing machine (ε 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 ε-random reals: a real is left-computable ε-random iff it is the halting probability of an ε-universal prefix-free Turing machine. We also show that left-computable ε-random reals are provable ε-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 ε(0,1), some ε-random reals do not contain even arbitrarily long runs of 0s. © 2010 Elsevier Inc. All rights reserved.
Source Title: Journal of Computer and System Sciences
ISSN: 00220000
DOI: 10.1016/j.jcss.2010.08.001
Appears in Collections:Staff Publications

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