Optimal Coding for Randomized Kolmogorov Complexity and Its Applications

Abstract: The coding theorem for Kolmogorov complexity states that any string sampled from a computable distribution has a description length close to its information content. A coding theorem for resource-bounded Kolmogorov complexity is the key to obtaining fundamental results in average-case complexity, yet whether any samplable distribution admits a coding theorem for randomized time-bounded Kolmogorov complexity ($\mathsf{rK}<sup>\mathsf{poly}$) is open and a common bottleneck in the recent literature of meta-complexity. Previous works bypassed this issue by considering probabilistic Kolmogorov complexity ($\mathsf{pK}<sup>\mathsf{poly}$), in which public random bits are assumed to be available. In this paper, we present an efficient coding theorem for randomized Kolmogorov complexity under the non-existence of one-way functions, thereby removing the common bottleneck. This enables us to prove $\mathsf{rK}<sup>\mathsf{poly}$ counterparts of virtually all the average-case results that were proved only for $\mathsf{pK}<sup>\mathsf{poly}$, and enables the resolution of the open problems of Hirahara, Ilango, Lu, Nanashima, and Oliveira (STOC'23) and Hirahara, Kabanets, Lu, and Oliveira (CCC'24). The key technical lemma is that any distribution whose next bits are efficiently predictable admits an efficient encoding and decoding scheme, which could be of independent interest to data compression.