On Resource-Bounded Instance Complexity

The instance complexity of a string x with respect to a set A and time bound t, ic t (x : A), is the length of the shortest program for A that runs in time t, decides x correctly, and makes no mistakes on other strings (where \don't know" answers are permitted). Schh oning, and Watanabe states that for every recursive set A not in P and every polynomial t there is a polynomial t 0 and a constant c such that for innnitely many x, ic t (x : A) C t 0 (x)?c, where C t 0 (x) is the t 0-time bounded Kolmogorov complexity of x. In this paper the conjecture is proved for all recursive tally sets and for all recursive sets which are NP-hard under honest reductions, in particular it holds for all natural NP-hard problems. The method of proof also yields the polynomial-space bounded and the exponential-time bounded versions of the conjecture in full generality. On the other hand, the conjecture itself turns out to be oracle dependent: In any relativized world where P = NP the conjecture holds, but there are also relativized worlds where it fails, even if C-complexity is replaced by Sipser's CD-complexity. Additionally it is proved that the instance complexity measure is noncomputable and it is investigated whether for every polynomial t there is a polynomial t 0 such that C t 0-complexity is bounded above by CD t-complexity.