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Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ITP.2022.12
URN: urn:nbn:de:0030-drops-167219
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/16721/
Forster, Yannick ;
Kunze, Fabian ;
Lauermann, Nils
Synthetic Kolmogorov Complexity in Coq
Abstract
We present a generalised, constructive, and machine-checked approach to Kolmogorov complexity in the constructive type theory underlying the Coq proof assistant. By proving that nonrandom numbers form a simple predicate, we obtain elegant proofs of undecidability for random and nonrandom numbers and a proof of uncomputability of Kolmogorov complexity.
We use a general and abstract definition of Kolmogorov complexity and subsequently instantiate it to several definitions frequently found in the literature.
Whereas textbook treatments of Kolmogorov complexity usually rely heavily on classical logic and the axiom of choice, we put emphasis on the constructiveness of all our arguments, however without blurring their essence. We first give a high-level proof idea using classical logic, which can be formalised with Markov’s principle via folklore techniques we subsequently explain. Lastly, we show a strategy how to eliminate Markov’s principle from a certain class of computability proofs, rendering all our results fully constructive.
All our results are machine-checked by the Coq proof assistant, which is enabled by using a synthetic approach to computability: rather than formalising a model of computation, which is well known to introduce a considerable overhead, we abstractly assume a universal function, allowing the proofs to focus on the mathematical essence.
BibTeX - Entry
@InProceedings{forster_et_al:LIPIcs.ITP.2022.12,
author = {Forster, Yannick and Kunze, Fabian and Lauermann, Nils},
title = {{Synthetic Kolmogorov Complexity in Coq}},
booktitle = {13th International Conference on Interactive Theorem Proving (ITP 2022)},
pages = {12:1--12:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-252-5},
ISSN = {1868-8969},
year = {2022},
volume = {237},
editor = {Andronick, June and de Moura, Leonardo},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/16721},
URN = {urn:nbn:de:0030-drops-167219},
doi = {10.4230/LIPIcs.ITP.2022.12},
annote = {Keywords: Kolmogorov complexity, computability theory, random numbers, constructive matemathics, synthetic computability theory, constructive type theory, Coq}
}
