We investigate the question of whether one can characterize complexity classes in terms of efficient reducibility to the set of Kolmogorov-random strings . This question arises because and , and no larger complexity classes are known to be reducible to in this way. We show that this question cannot be posed without explicitly dealing with issues raised by the choice of universal machine in the definition of Kolmogorov complexity. What follows is a list of some of our main results.• Although (...) Kummer showed that, for every universal machine U there is a time bound t such that the halting problem is disjunctive truth-table reducible to in time t, there is no such time bound t that suffices for every universal machine U. We also show that, for some machines U, the disjunctive reduction can be computed in as little as doubly-exponential time.• Although for every universal machine U, there are very complex sets that are -reducible to , it is nonetheless true that . • Any decidable set that is polynomial-time monotone-truth-table reducible to is in .• Any decidable set that is polynomial-time truth-table reducible to via a reduction that asks at most f queries on inputs of size n lies in. (shrink)

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 ${\rm S}_{2}^{{\rm P}}$ introduced by Canetti and independently Russell and Sundaram and the classes PSPACE, EXP. • ${\rm S}_{2}^{{\rm P}}$ 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. (shrink)