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Enumerations of the Kolmogorov function. (English) Zbl 1165.03025
Summary: 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:
{$$\bullet$$} For every underlying universal machine $$U$$, there is a constant $$a$$ such that $$C$$ is $$k(n)$$-enumerable only if $$k(n) \geq n/a$$ for almost all $$n$$.
{$$\bullet$$} 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.
{$$\bullet$$} There exists an r.e., Turing-incomplete set $$A$$ such that 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 $$g$$:
{$$\bullet$$} 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 $$f$$.
{$$\bullet$$} 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 $$\text{S}_2^{\text{p}}$$, introduced by Canetti and independently by Russell and Sundaram, and the classes PSPACE, EXP.
{$$\bullet$$} $$\text{S}_2^{\text{p}}$$ 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 $$g$$.
{$$\bullet$$} 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.
{$$\bullet$$} EXP is the class of all sets $$A$$ for which there is a polynomially bounded function $$g$$ and a machine $$M$$ which witnesses $$A \in \text{PSPACE}^f$$ 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 $$\text{P}=\text{NP}$$.

##### MSC:
 03D80 Applications of computability and recursion theory 68Q15 Complexity classes (hierarchies, relations among complexity classes, etc.) 68Q30 Algorithmic information theory (Kolmogorov complexity, etc.)
##### Keywords:
Kolmogorov function; Kolmogorov complexity; enumeration
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