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    Dept. of Computer Science, Rutgers University, Piscataway, NJ

    Avoiding Simplicity is Complex

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    It is a trivial observation that every decidable set has strings of length n with Kolmogorov complexity logn+O(1) if it has any strings of length n at all. Things become much more interesting when one asks whether a similar property holds when one considers resource-bounded Kolmogorov complexity. This is the question considered here: Can a feasible set A avoid accepting strings of low resource-bounded Kolmogorov complexity, while still accepting some (or many) strings of lengthn? More specifically, this paper deals with two notions of resource-bounded Kolmogorov complexity: Kt and KNt. The measure Kt was defined by Levin more than three decades ago and has been studied extensively since then. The measure KNt is a nondeterministic analog of Kt. For all strings x, Kt(x)KNt(x); the two measures are polynomially related if and only if NEXPEXP/poly (Allender et al. in J.Comput. Syst. Sci. 77:1440, 2011). Many longstanding open questions in complexity theory boil down to the question of whether there are sets in P that avoid all strings of low Kt complexity. For example, the EXP vs ZPP question is equivalent to (one version of) the question of whether avoiding simple strings is difficult: (EXP=ZPP if and only if there exist >0 and a dense set in P having no strings x with Kt(x)x (Allender et al. in SIAM J. Comput. 35:14671493, 2006)). Surprisingly, we are able to show unconditionally that avoiding simple strings (in the sense of KNt complexity) is difficult. Every dense set in NPcoNP contains infinitely many strings x such that KNt(x)x for every >0. The proof does not relativize. As an application, we are able to show that if E=NE, then accepting paths for nondeterministic exponential time machines can be found somewhat more quickly than the brute-force upper bound, if there are many accepting paths.

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    Description

    Title : Avoiding Simplicity is Complex
    Author(s) : Eric Allender, Holger Spakowski
    Abstract : It is a trivial observation that every decidable set has strings of length n with Kolmogorov complexity logn+O(1) if it has any strings of length n at all. Things become much more interesting when one asks whether a similar property holds when one considers resource-bounded Kolmogorov complexity. This is the question considered here: Can a feasible set A avoid accepting strings of low resource-bounded Kolmogorov complexity, while still accepting some (or many) strings of lengthn? More specifically, this paper deals with two notions of resource-bounded Kolmogorov complexity: Kt and KNt. The measure Kt was defined by Levin more than three decades ago and has been studied extensively since then. The measure KNt is a nondeterministic analog of Kt. For all strings x, Kt(x)KNt(x); the two measures are polynomially related if and only if NEXPEXP/poly (Allender et al. in J.Comput. Syst. Sci. 77:1440, 2011). Many longstanding open questions in complexity theory boil down to the question of whether there are sets in P that avoid all strings of low Kt complexity. For example, the EXP vs ZPP question is equivalent to (one version of) the question of whether avoiding simple strings is difficult: (EXP=ZPP if and only if there exist >0 and a dense set in P having no strings x with Kt(x)x (Allender et al. in SIAM J. Comput. 35:14671493, 2006)). Surprisingly, we are able to show unconditionally that avoiding simple strings (in the sense of KNt complexity) is difficult. Every dense set in NPcoNP contains infinitely many strings x such that KNt(x)x for every >0. The proof does not relativize. As an application, we are able to show that if E=NE, then accepting paths for nondeterministic exponential time machines can be found somewhat more quickly than the brute-force upper bound, if there are many accepting paths.
    Subject : unspecified
    Area : Other
    Language : English
    Year : 2011

    Affiliations Dept. of Computer Science, Rutgers University, Piscataway, NJ
    Journal : Theory of Computing Systems
    Issue : x
    Publisher : Springer New York
    Pages : -
    Url : http://www.springerlink.com/index/10.1007/s00224-011-9334-7
    Doi : 10.1007/s00224-011-9334-7

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