Limits on the computational power of random strings

Eric Allender, Luke Friedman, William Gasarch

Research output: Chapter in Book/Report/Conference proceedingConference contribution

3 Scopus citations

Abstract

How powerful is the set of random strings? What can one say about a set A that is efficiently reducible to R, the set of Kolmogorov-random strings? We present the first upper bound on the class of computable sets in and . The two most widely-studied notions of Kolmogorov complexity are the "plain" complexity C(x) and "prefix" complexity K(x); this gives two ways to define the set "R": R C and R K . (Of course, each different choice of universal Turing machine U in the definition of C and K yields another variant or .) Previous work on the power of "R" (for any of these variants [1,2,9]) has shown . . . Since these inclusions hold irrespective of low-level details of how "R" is defined, we have e.g.: . ( is the class of computable sets.) Our main contribution is to present the first upper bounds on the complexity of sets that are efficiently reducible to . We show: . . Hence, in particular, is sandwiched between the class of sets Turing- and truth-table-reducible to R. As a side-product, we obtain new insight into the limits of techniques for derandomization from uniform hardness assumptions.

Original languageEnglish (US)
Title of host publicationAutomata, Languages and Programming - 38th International Colloquium, ICALP 2011, Proceedings
Pages293-304
Number of pages12
EditionPART 1
DOIs
StatePublished - 2011
Event38th International Colloquium on Automata, Languages and Programming, ICALP 2011 - Zurich, Switzerland
Duration: Jul 4 2011Jul 8 2011

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
NumberPART 1
Volume6755 LNCS

Other

Other38th International Colloquium on Automata, Languages and Programming, ICALP 2011
Country/TerritorySwitzerland
CityZurich
Period7/4/117/8/11

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)

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