Several classes of diagonally nonrecursive (DNR) functions are characterized in terms of Kolmogorov complexity. In particular, a set of natural numbers
A
A
can wtt-compute a DNR function iff there is a nontrivial recursive lower bound on the Kolmogorov complexity of the initial segments of
A
A
. Furthermore,
A
A
can Turing compute a DNR function iff there is a nontrivial
A
A
-recursive lower bound on the Kolmogorov complexity of the initial segments of
A
A
.
A
A
is PA-complete, that is,
A
A
can compute a
{
0
,
1
}
\{0,1\}
-valued DNR function, iff
A
A
can compute a function
F
F
such that
F
(
n
)
F(n)
is a string of length
n
n
and maximal
C
C
-complexity among the strings of length
n
n
.
A
≥
T
K
A \geq _T K
iff
A
A
can compute a function
F
F
such that
F
(
n
)
F(n)
is a string of length
n
n
and maximal
H
H
-complexity among the strings of length
n
n
. Further characterizations for these classes are given. The existence of a DNR function in a Turing degree is equivalent to the failure of the Recursion Theorem for this degree; thus the provided results characterize those Turing degrees in terms of Kolmogorov complexity which no longer permit the usage of the Recursion Theorem.