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Randomness and the linear degrees of computability

Lewis-Pye, Andrew and Barmpalias, George (2007) Randomness and the linear degrees of computability. Annals of Pure and Applied Logic, 145 (3). pp. 252-257. ISSN 0168-0072

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Identification Number: 10.1016/j.apal.2006.08.001


We show that there exists a real α such that, for all reals β, if α is linear reducible to β (α≤ℓβ, previously denoted as α≤swβ) then β≤Tα. In fact, every random real satisfies this quasi-maximality property. As a corollary we may conclude that there exists no ℓ-complete Δ2 real. Upon realizing that quasi-maximality does not characterize the random reals–there exist reals which are not random but which are of quasi-maximal ℓ-degree – it is then natural to ask whether maximality could provide such a characterization. Such hopes, however, are in vain since no real is of maximal ℓ-degree.

Item Type: Article
Official URL:
Additional Information: © 2006 Elsevier B.V.
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Date Deposited: 06 Aug 2013 11:13
Last Modified: 04 Jan 2024 22:03

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