Optimal asymptotic bounds on the oracle use in computations from Chaitin’s Omega

Barmpalias, George, Fang, Nan and Lewis-Pye, Andy (2016) Optimal asymptotic bounds on the oracle use in computations from Chaitin’s Omega. Journal of Computer and System Sciences, 82 (8). pp. 1283-1299. ISSN 0022-0000

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Abstract

Chaitin’s number is the halting probability of a universal prefix-free machine, and although it depends on the underlying enumeration of prefix-free machines, it is always Turing-complete. It can be observed, in fact, that for every computably enumerable (c.e.) real �, there exists a Turing functional via which computes �, and such that the number of bits of that are needed for the computation of the first n bits of � (i.e. the use on argument n) is bounded above by a computable function h(n) = n + o (n). We characterise the asymptotic upper bounds on the use of Chaitin’s in oracle computations of halting probabilities (i.e. c.e. reals). We show that the following two conditions are equivalent for any computable function h such that h(n)P

Item Type:	Article
Official URL:	http://www.journals.elsevier.com/journal-of-comput...
Additional Information:	© 2016 Elsevier
Library of Congress subject classification:	Q Science > QA Mathematics Q Science > QA Mathematics > QA75 Electronic computers. Computer science
Sets:	Departments > Mathematics
Project and Funder Information:	Project ID Funder Name Funder ID UNSPECIFIED Chinese Government UNSPECIFIED 2010Y2GB03 Chinese Academy of Sciences http://dx.doi.org/10.13039/501100002367 2014CB340302 China Basic Research Program UNSPECIFIED
Date Deposited:	17 May 2016 15:07
URL:	http://eprints.lse.ac.uk/66539/

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