Equivalence of Kolmogorov complexity under circuit and Turing descriptions (Cp = Kp)

Establish whether the class Cp of length-2^n bit strings that admit polynomial-length Boolean formula descriptions (i.e., strings obtained by evaluating a succinct Boolean formula on all 2^n inputs and concatenating the outputs) equals the class Kp of length-2^n bit strings that admit polynomial-length Turing machine programs; equivalently, prove that (i) every succinct Boolean formula producing such a string can be reduced in polynomial time to a succinct Turing program that outputs the same string, and (ii) every succinct Turing program producing such a string can be reduced in polynomial time to a succinct Boolean formula that outputs the same string.

Background

The paper introduces two classes: Cp, the set of strings of length 2^n that can be generated by polynomial-length Boolean formulas (viewed as circuits) when their outputs over all inputs are concatenated, and Kp, the set of strings of length 2^n that can be generated by polynomial-length Turing machine programs. Motivated by the invariance theorem for Kolmogorov complexity across universal Turing machines and by the widespread equivalence of computational models up to polynomial overhead, the authors pose the equivalence Cp = Kp as a conjecture.

The conjecture is framed as two polynomial-time reductions between succinct circuit/Boolean formula descriptions and succinct Turing machine programs producing the same strings. The authors note that their paper does not provide a proof and later discuss potential avenues and obstacles (e.g., adapting Cook–Levin style reductions without polynomial time bounds) while emphasizing the significance of such an equivalence for bridging circuit and program views of Kolmogorov complexity.