Enumerations of the Kolmogorov function

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.

• S₂^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 ∈ PSPACE^f 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.