On the Optimal Compression of Sets in PSPACE

Zimand, Marius

doi:10.1007/978-3-642-22953-4_6

Cited by 3 publications

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References 20 publications

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“…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.…”

Section: Introductionmentioning

confidence: 99%

Space-Bounded Kolmogorov Extractors

Musatov

2012

Computer Science – Theory and Applications

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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.

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Section: Introductionmentioning

confidence: 99%

Space-Bounded Kolmogorov Extractors

Musatov

2012

Computer Science – Theory and Applications

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Improving the Space-Bounded Version of Muchnik’s Conditional Complexity Theorem via “Naive” Derandomization

Musatov

2012

Theory Comput Syst

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Many theorems about Kolmogorov complexity rely on existence of combinatorial objects with specific properties. Usually the probabilistic method gives such objects with better parameters than explicit constructions do. But the probabilistic method does not give "effective" variants of such theorems, i.e. variants for resource-bounded Kolmogorov complexity. We show that a "naive derandomization" approach of replacing these objects by the output of Nisan-Wigderson pseudo-random generator may give polynomial-space variants of such theorems. Specifically, we improve the preceding polynomial-space analogue of Muchnik's conditional complexity theorem. I.e., for all a and b there exists a program p of least possible length that transforms a to b and is simple conditional on b. Here all programs work in polynomial space and all complexities are measured with logarithmic accuracy instead of polylogarithmic one in the previous work.

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On Extracting Space-bounded Kolmogorov Complexity

Musatov

2014

Theory Comput Syst

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On the Optimal Compression of Sets in PSPACE

Cited by 3 publications

References 20 publications

Space-Bounded Kolmogorov Extractors

Space-Bounded Kolmogorov Extractors

Improving the Space-Bounded Version of Muchnik’s Conditional Complexity Theorem via “Naive” Derandomization

On Extracting Space-bounded Kolmogorov Complexity

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