Lower bounds on the redundancy in computations from random oracles via betting strategies with restricted wagers

Barmpalias, George, Lewis-Pye, 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. 287-300. ISSN 0890-5401

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Abstract

The Kučera–Gács theorem is a landmark result in algorithmic randomness asserting that every real is computable from a Martin-Löf 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 nlog⁡n 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 Martin-Löf 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 Martin-Löf random oracles. In fact, there exists a real X such that the redundancy g of any computation of X from a Martin-Löf 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 2-generic reals. On the other hand, it has been recently shown that any real is computable from a Martin-Löf 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 log⁡n 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/information-and-c...
Additional Information:	© 2016 Elsevier Inc.
Divisions:	Mathematics
Subjects:	Q Science > QA Mathematics Q Science > QA Mathematics > QA75 Electronic computers. Computer science
Date Deposited:	26 Sep 2016 16:15
Last Modified:	07 Jan 2024 20:24
Projects:	D1101130
Funders:	Chinese Government, Chinese Academy of Sciences
URI:	http://eprints.lse.ac.uk/id/eprint/67867

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