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Title: Enumerations of the kolmogorov function
Authors: Beigel, R.
Buhrman, H.
Fejer, P.
Fortnow, L.
Grabowski, P.
Longpré, L.
Muchnik, A.
Stephan, F. 
Torenvliet, L.
Issue Date: Jun-2006
Source: Beigel, R., Buhrman, H., Fejer, P., Fortnow, L., Grabowski, P., Longpré, L., Muchnik, A., Stephan, F., Torenvliet, L. (2006-06). Enumerations of the kolmogorov function. Journal of Symbolic Logic 71 (2) : 501-528. ScholarBank@NUS Repository.
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 S2 P introduced by Canetti and independently Russell and Sundaram and the classes PSPACE, EXP. © 2006. Association for Symbolic Logic.
Source Title: Journal of Symbolic Logic
ISSN: 00224812
DOI: jsl/1146620156
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