Theory of Computing Systems

, Volume 43, Issue 3–4, pp 425–463 | Cite as

Dimension Extractors and Optimal Decompression

Article

Abstract

A dimension extractor is an algorithm designed to increase the effective dimension—i.e., the amount of computational randomness—of an infinite binary sequence, in order to turn a “partially random” sequence into a “more random” sequence. Extractors are exhibited for various effective dimensions, including constructive, computable, space-bounded, time-bounded, and finite-state dimension. Using similar techniques, the Kučera-Gács theorem is examined from the perspective of decompression, by showing that every infinite sequence S is Turing reducible to a Martin-Löf random sequence R such that the asymptotic number of bits of R needed to compute n bits of S, divided by n, is precisely the constructive dimension of S, which is shown to be the optimal ratio of query bits to computed bits achievable with Turing reductions. The extractors and decompressors that are developed lead directly to new characterizations of some effective dimensions in terms of optimal decompression by Turing reductions.

Keywords

Dimension Kolmogorov complexity Compression Martingale Algorithmic randomness Extractor 

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© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of Computer ScienceIowa State UniversityAmesUSA

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