Abstract

We consider the hardness of enumerating k possible values for the Kolmogorov complexity function C(x) so that one of them is correct. We show several results including Any computable enumerator for C(x) must enumerate n) possibilities. If a k-enumerator (xed k) for C is reducible to an r.e. set A then A is Turing-equivalent to the halting problem. Every nonrecursive set is not weak-truth-table reducible to some 2-enumerator for C(x) or any other recursively-bounded function. The time-bounded enumeration question gives a new characterization of the class symP dened by Russell and Sundaram. Enumerating O(log n) values of the space-bounded Kolmogorov function remains hard for PSPACE. 1 Introduction The Kolmogorov complexity of a binary string x, C(x), is the size of the smallest program that outputs x. Kolmogorov complexity has its roots in the study of randomness: it is one way to measure randomness in a string. It has had a vast area of applications, including i...

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