Space-Bounded Kolmogorov Extractors

Abstract

An extractor is a function that receives some randomness and either "improves" it or produces "new" randomness. There are statistical and algorithmical specifications of this notion. We study an algorithmical one called Kolmogorov extractors and modify it to resourcebounded version of Kolmogorov complexity. Following Zimand we prove the existence of such objects with certain parameters. The utilized technique is "naive" derandomization: we replace random constructions employed by Zimand by pseudo-random ones obtained by Nisan-Wigderson generator.⋆ Supported by ANR Sycomore, NAFIT ANR-08-EMER-008-01 and RFBR 09-01-00709-a grants.and [11]. In this paper we convert Zimand's results to the space-bounded case and hence improve the respective results of Fortnow et al. Since Zimand's construction is not efficient, this conversion cannot be done straightforwardly. The technique we employ is the "naive derandomization" method introduced in [6] and [4] and later used in [13] and [14]. Originally, Zimand have characterized Kolmogorov extractors by some combinatorial properties. The existence of an object with such properties was proven implicitly. We show that such an object may be found in the output of Nisan-Wigderson pseudo-random generator. That is, to find a required object one does not need to search through all possible objects but needs only to check all seeds of the generator. This crucially decreases the required space from exponential to polynomial. The rest of the paper is organized as follows. In Sect. 2 we give formal definitions of all involved objects and formulate necessary results. In Sect. 3 we give formal definitions for space-bounded Kolmogorov extractors, formulate our existence theorems, outline the proof idea and present detailed proofs.

Short Lists with Short Programs in Short Time

“…The lower bounds, i.e., Theorem 2.7 (hence Theorem 1.4) and Theorem 1.2 (1), are proved in Section 4. In Section 5, we observe that our technique can be used to improve Muchnik's Theorem [11] (see also the works of Musatov, Romashchenko and Shen [15,13,14]), and a result concerning distinguishing complexity of Buhrman, Fortnow, and Laplante [4].…”