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June 2006 Enumerations of the Kolmogorov function
Richard Beigel, Harry Buhrman, Peter Fejer, Lance Fortnow, Piotr Grabowski, Luc Longpré, Andrej Muchnik, Frank Stephan, Leen Torenvliet
J. Symbolic Logic 71(2): 501-528 (June 2006). DOI: 10.2178/jsl/1146620156

Abstract

A recursive enumerator for a function h is an algorithm f which enumerates for an input x finitely many elements including h(x). f is a k(n)-enumerator if for every input x of length n, h(x) is among the first k(n) elements enumerated by f. If there is a k(n)-enumerator for h then h is called k(n)-enumerable. We also consider enumerators which are only A-recursive for some oracle A.

We determine exactly how hard it is to enumerate the Kolmogorov function, which assigns to each string x its Kolmogorov complexity:

  • For every underlying universal machine U, there is a constant a such that C is k(n)-enumerable only if k(n) ≥ n/a for almost all n.

  • For any given constant k, the Kolmogorov function is k-enumerable relative to an oracle A if and only if A is at least as hard as the halting problem.

  • There exists an r.e., Turing-incomplete set A such for every non-decreasing and unbounded recursive function k, the Kolmogorov function is k(n)-enumerable relative to A.

The last result is obtained by using a relativizable construction for a nonrecursive set A relative to which the prefix-free Kolmogorov complexity differs only by a constant from the unrelativized prefix-free Kolmogorov complexity.

Although every 2-enumerator for C is Turing hard for K, we show that reductions must depend on the specific choice of the 2-enumerator and there is no bound on the quantity of their queries. We show our negative results even for strong 2-enumerators as an oracle where the querying machine for any x gets directly an explicit list of all hypotheses of the enumerator for this input. The limitations are very general and we show them for any recursively bounded function g:

  • For every Turing reduction M and every non-recursive set B, there is a strong 2-enumerator f for g such that M does not Turing reduce B to f.

  • For every non-recursive set B, there is a strong 2-enumerator f for g such that B is not wtt-reducible to f.

Furthermore, we deal with the resource-bounded case and give characterizations for the class S₂p introduced by Canetti and independently Russell and Sundaram and the classes PSPACE, EXP.
  • S₂p is the class of all sets A for which there is a polynomially bounded function g such that there is a polynomial time tt-reduction which reduces A to every strong 2-enumerator for g.

  • PSPACE is the class of all sets A for which there is a polynomially bounded function g such that there is a polynomial time Turing reduction which reduces A to every strong 2-enumerator for g. Interestingly, g can be taken to be the Kolmogorov function for the conditional space bounded Kolmogorov complexity.

  • EXP is the class of all sets A for which there is a polynomially bounded function g and a machine M which witnesses A ∈ PSPACEf for all strong 2-enumerators f for g.

Finally, we show that any strong O(log n)-enumerator for the conditional space bounded Kolmogorov function must be PSPACE-hard if P=NP.

Citation

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Richard Beigel. Harry Buhrman. Peter Fejer. Lance Fortnow. Piotr Grabowski. Luc Longpré. Andrej Muchnik. Frank Stephan. Leen Torenvliet. "Enumerations of the Kolmogorov function." J. Symbolic Logic 71 (2) 501 - 528, June 2006. https://doi.org/10.2178/jsl/1146620156

Information

Published: June 2006
First available in Project Euclid: 2 May 2006

zbMATH: 1165.03025
MathSciNet: MR2225891
Digital Object Identifier: 10.2178/jsl/1146620156

Rights: Copyright © 2006 Association for Symbolic Logic

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Vol.71 • No. 2 • June 2006
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