Archive | 2021
An Efficient Coding Theorem via Probabilistic Representations and its Applications
Abstract
A probabilistic representation of a string x ∈ {0, 1} is given by the code of a randomized algorithm that outputs x with high probability (Oliveira, ICALP 2019, [30]). We employ probabilistic representations to establish the first unconditional Coding Theorem in time-bounded Kolmogorov complexity. More precisely, we show that if a distribution ensemble Dm can be uniformly sampled in time T (m) and generates a string x ∈ {0, 1}∗ with probability at least δ, then x admits a time-bounded probabilistic representation of complexity O(log(1/δ) + log(T ) + log(m)). Under mild assumptions, a representation of this form can be computed from x and the code of the sampler in time polynomial in n = |x|. We derive consequences of this result relevant to the study of data compression, pseudodeterministic algorithms, time hierarchies for sampling distributions, and complexity lower bounds. In particular, we describe an instance-based search-to-decision reduction for Levin’s Kt complexity (Levin, Information and Control 1984, [23]) and its probabilistic analogue rKt [30]. As a consequence, if a string x admits a succinct time-bounded representation, then a near-optimal representation can be generated from x with high probability in polynomial time. This partially addresses in a time-bounded setting a question from [23] on the efficiency of computing an optimal encoding of a string. 2012 ACM Subject Classification Theory of computation