The Non-Uniform Perebor Conjecture for Time-Bounded Kolmogorov Complexity Is False

The Perebor (Russian for "brute-force search") conjectures, which date back to the 1950s and 1960s are some of the oldest conjectures in complexity theory. The conjectures are a stronger form of the NP= P conjecture (which they predate) and state that for "meta-complexity" problems, such as the Time-bounded Kolmogorov complexity Problem, and the Minimum Circuit Size Problem, there are no better algorithms than brute force search. In this paper, we disprove the non-uniform version of the Perebor conjecture for the Time-Bounded Kolmogorov complexity problem. We demonstrate that for every polynomial t, there exists of a circuit of size 24n/5+o(n) that solves the t-bounded Kolmogorov complexity problem on every instance. Our algorithm is black-box in the description of the Universal Turing Machine U employed in the definition of Kolmogorov Complexity and leverages the characterization of one-way functions through the hardness of the time-bounded Kolmogorov complexity problem of Liu and Pass (FOCS'20), and the time-space trade-off for one-way functions of Fiat and Naor (STOC'91). We additionally demonstrate that no such black-box algorithm can have circuit size smaller than 2n/2-o(n). Along the way (and of independent interest), we extend the result of Fiat and Naor and demonstrate that any efficiently computable function can be inverted (with probability 1) by a circuit of size 24n/5+o(n); as far as we know, this yields the first formal proof that a non-Trivial circuit can invert any efficient function.