License: 
Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2021.94
URN: urn:nbn:de:0030-drops-141630
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/14163/
Lu, Zhenjian ;
Oliveira, Igor C.
An Efficient Coding Theorem via Probabilistic Representations and Its Applications
Abstract
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 ?_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.
BibTeX - Entry
@InProceedings{lu_et_al:LIPIcs.ICALP.2021.94,
author = {Lu, Zhenjian and Oliveira, Igor C.},
title = {{An Efficient Coding Theorem via Probabilistic Representations and Its Applications}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {94:1--94:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-195-5},
ISSN = {1868-8969},
year = {2021},
volume = {198},
editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/14163},
URN = {urn:nbn:de:0030-drops-141630},
doi = {10.4230/LIPIcs.ICALP.2021.94},
annote = {Keywords: computational complexity, randomized algorithms, Kolmogorov complexity}
}
|
Keywords: |
|
computational complexity, randomized algorithms, Kolmogorov complexity |
|
Collection: |
|
48th International Colloquium on Automata, Languages, and Programming (ICALP 2021) |
|
Issue Date: |
|
2021 |
|
Date of publication: |
|
02.07.2021 |
