Barmpalias, George, LewisPye, Andrew and Teutsch, Jason (2016) Lower bounds on the redundancy in computations from random oracles via betting strategies with restricted wagers. Information and Computation, 251. pp. 287300. ISSN 08905401

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Abstract
The Kučera–Gács theorem is a landmark result in algorithmic randomness asserting that every real is computable from a MartinLöf random real. If the computation of the first n bits of a sequence requires n+h(n) bits of the random oracle, then h is the redundancy of the computation. Kučera implicitly achieved redundancy nlogn while Gács used a more elaborate coding procedure which achieves redundancy View the MathML source. A similar bound is implicit in the later proof by Merkle and Mihailović. In this paper we obtain optimal strict lower bounds on the redundancy in computations from MartinLöf random oracles. We show that any nondecreasing computable function g such that ∑n2−g(n)=∞ is not a general upper bound on the redundancy in computations from MartinLöf random oracles. In fact, there exists a real X such that the redundancy g of any computation of X from a MartinLöf random oracle satisfies ∑n2−g(n)<∞. Moreover, the class of such reals is comeager and includes a View the MathML source real as well as all weakly 2generic reals. On the other hand, it has been recently shown that any real is computable from a MartinLöf random oracle with redundancy g, provided that g is a computable nondecreasing function such that ∑n2−g(n)<∞. Hence our lower bound is optimal, and excludes many slow growing functions such as logn from bounding the redundancy in computations from random oracles for a large class of reals. Our results are obtained as an application of a theory of effective betting strategies with restricted wagers which we develop.
Item Type:  Article 

Official URL:  http://www.journals.elsevier.com/informationandc... 
Additional Information:  © 2016 Elsevier Inc. 
Divisions:  Mathematics 
Subjects:  Q Science > QA Mathematics Q Science > QA Mathematics > QA75 Electronic computers. Computer science 
Sets:  Departments > Mathematics 
Date Deposited:  26 Sep 2016 16:15 
Last Modified:  20 Jan 2020 06:09 
Projects:  D1101130 
Funders:  Chinese Government, Chinese Academy of Sciences 
URI:  http://eprints.lse.ac.uk/id/eprint/67867 
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