Synthetic Kolmogorov Complexity in Coq
In: ITP 2022 - 13th International Conference on Interactive Theorem Proving ; https://inria.hal.science/hal-03596267 ; ITP 2022 - 13th International Conference on Interactive Theorem Proving, Aug 2022, Haifa, Israel, 2022
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International audience ; 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.
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Synthetic Kolmogorov Complexity in Coq
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Autor/in / Beteiligte Person: | Forster, Yannick ; Kunze, Fabian ; Lauermann, Nils ; Saarland University Saarbrücken ; Gallinette : vers une nouvelle génération d'assistant à la preuve (GALLINETTE) ; Inria Rennes – Bretagne Atlantique ; Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire des Sciences du Numérique de Nantes (LS2N) ; Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-IMT Atlantique (IMT Atlantique) ; Institut Mines-Télécom Paris (IMT)-Institut Mines-Télécom Paris (IMT)-NANTES UNIVERSITÉ - École Centrale de Nantes (Nantes Univ - ECN) ; Nantes Université (Nantes Univ)-Nantes Université (Nantes Univ)-Nantes université - UFR des Sciences et des Techniques (Nantes univ - UFR ST) ; Nantes Université - pôle Sciences et technologie ; Nantes Université (Nantes Univ)-Nantes Université (Nantes Univ)-Nantes Université - pôle Sciences et technologie ; Nantes Université (Nantes Univ)-Centre National de la Recherche Scientifique (CNRS)-IMT Atlantique (IMT Atlantique) ; Nantes Université (Nantes Univ) ; University of Cambridge UK (CAM) ; European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 101024493. ; 13th International Conference on Interactive Theorem Proving (ITP 2022) |
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Zeitschrift: | ITP 2022 - 13th International Conference on Interactive Theorem Proving ; https://inria.hal.science/hal-03596267 ; ITP 2022 - 13th International Conference on Interactive Theorem Proving, Aug 2022, Haifa, Israel, 2022 |
Veröffentlichung: | HAL CCSD, 2022 |
Medientyp: | Konferenz |
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