2026-07-25T21:35:58Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:14163 2024-03-06T10:53:38Z ddc:004 open_access

An Efficient Coding Theorem via Probabilistic Representations and Its Applications Lu, Zhenjian Oliveira, Igor C. computational complexity randomized algorithms Kolmogorov complexity A probabilistic representation of a string x ∈ {0,1}ⁿ is given by the code of a randomized algorithm that outputs x with high probability [Igor C. Oliveira, 2019]. 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 𝒟_m 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 [Leonid A. Levin, 1984] and its probabilistic analogue rKt [Igor C. Oliveira, 2019]. 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 [Leonid A. Levin, 1984] on the efficiency of computing an optimal encoding of a string. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Zhenjian Lu and Igor C. Oliveira 2021 Is Part Of LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICALP.2021.94 urn:nbn:de:0030-drops-141630 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.94 eng https://creativecommons.org/licenses/by/4.0/legalcode