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**Published in:** LIPIcs, Volume 237, 13th International Conference on Interactive Theorem Proving (ITP 2022)

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.

Yannick Forster, Fabian Kunze, and Nils Lauermann. Synthetic Kolmogorov Complexity in Coq. In 13th International Conference on Interactive Theorem Proving (ITP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 237, pp. 12:1-12:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2022.12}, 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} }

Document

**Published in:** LIPIcs, Volume 193, 12th International Conference on Interactive Theorem Proving (ITP 2021)

The weak call-by-value λ-calculus Łand Turing machines can simulate each other with a polynomial overhead in time. This time invariance thesis for L, where the number of β-reductions of a computation is taken as its time complexity, is the culmination of a 25-years line of research, combining work by Blelloch, Greiner, Dal Lago, Martini, Accattoli, Forster, Kunze, Roth, and Smolka. The present paper presents a mechanised proof of the time invariance thesis for L, constituting the first mechanised equivalence proof between two standard models of computation covering time complexity.
The mechanisation builds on an existing framework for the extraction of Coq functions to L and contributes a novel Hoare logic framework for the verification of Turing machines.
The mechanised proof of the time invariance thesis establishes Łas model for future developments of mechanised computational complexity theory regarding time. It can also be seen as a non-trivial but elementary case study of time-complexity-preserving translations between a functional language and a sequential machine model. As a by-product, we obtain a mechanised many-one equivalence proof of the halting problems for Łand Turing machines, which we contribute to the Coq Library of Undecidability Proofs.

Yannick Forster, Fabian Kunze, Gert Smolka, and Maximilian Wuttke. A Mechanised Proof of the Time Invariance Thesis for the Weak Call-By-Value λ-Calculus. In 12th International Conference on Interactive Theorem Proving (ITP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 193, pp. 19:1-19:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{forster_et_al:LIPIcs.ITP.2021.19, author = {Forster, Yannick and Kunze, Fabian and Smolka, Gert and Wuttke, Maximilian}, title = {{A Mechanised Proof of the Time Invariance Thesis for the Weak Call-By-Value \lambda-Calculus}}, booktitle = {12th International Conference on Interactive Theorem Proving (ITP 2021)}, pages = {19:1--19:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-188-7}, ISSN = {1868-8969}, year = {2021}, volume = {193}, editor = {Cohen, Liron and Kaliszyk, Cezary}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2021.19}, URN = {urn:nbn:de:0030-drops-139142}, doi = {10.4230/LIPIcs.ITP.2021.19}, annote = {Keywords: formalizations of computational models, computability theory, Coq, time complexity, Turing machines, lambda calculus, Hoare logic} }

Document

**Published in:** LIPIcs, Volume 193, 12th International Conference on Interactive Theorem Proving (ITP 2021)

We mechanise the Cook-Levin theorem, i.e. the NP-completeness of SAT, in the proof assistant Coq. We use the call-by-value λ-calculus L as the model of computation to formalise time complexity, the class NP, and polynomial-time reductions. The latter two notions agree with the usual characterisations via Turing machines (TMs), as L and TMs are polynomial-time equivalent.
The use of L as the computational model, as opposed to TMs, significantly eases program verification and the derivation of resource bounds. However, for showing the NP-hardness of SAT, computations of L need to be encoded in SAT, which is complicated by L’s more complex computational structure. Thus, the polynomial-time reduction chain to SAT employs TMs as an intermediate problem, for which we neatly factor out a known textbook reduction from TMs to SAT. Still, all reduction functions are implemented and analysed in L.
To the best of our knowledge, this is the first result in computational complexity theory that has been mechanised with respect to any concrete computational model.
We discuss what makes this area of computer science hard to mechanise and highlight the design choices which enable our mechanisations.

Lennard Gäher and Fabian Kunze. Mechanising Complexity Theory: The Cook-Levin Theorem in Coq. In 12th International Conference on Interactive Theorem Proving (ITP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 193, pp. 20:1-20:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{gaher_et_al:LIPIcs.ITP.2021.20, author = {G\"{a}her, Lennard and Kunze, Fabian}, title = {{Mechanising Complexity Theory: The Cook-Levin Theorem in Coq}}, booktitle = {12th International Conference on Interactive Theorem Proving (ITP 2021)}, pages = {20:1--20:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-188-7}, ISSN = {1868-8969}, year = {2021}, volume = {193}, editor = {Cohen, Liron and Kaliszyk, Cezary}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2021.20}, URN = {urn:nbn:de:0030-drops-139154}, doi = {10.4230/LIPIcs.ITP.2021.20}, annote = {Keywords: computational model, NP completeness, Coq, Cook, Levin} }

Document

**Published in:** LIPIcs, Volume 141, 10th International Conference on Interactive Theorem Proving (ITP 2019)

We provide a plugin extracting Coq functions of simple polymorphic types to the (untyped) call-by-value lambda calculus L. The plugin is implemented in the MetaCoq framework and entirely written in Coq. We provide Ltac tactics to automatically verify the extracted terms w.r.t a logical relation connecting Coq functions with correct extractions and time bounds, essentially performing a certifying translation and running time validation. We provide three case studies: A universal L-term obtained as extraction from the Coq definition of a step-indexed self-interpreter for L, a many-reduction from solvability of Diophantine equations to the halting problem of L, and a polynomial-time simulation of Turing machines in L.

Yannick Forster and Fabian Kunze. A Certifying Extraction with Time Bounds from Coq to Call-By-Value Lambda Calculus. In 10th International Conference on Interactive Theorem Proving (ITP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 141, pp. 17:1-17:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{forster_et_al:LIPIcs.ITP.2019.17, author = {Forster, Yannick and Kunze, Fabian}, title = {{A Certifying Extraction with Time Bounds from Coq to Call-By-Value Lambda Calculus}}, booktitle = {10th International Conference on Interactive Theorem Proving (ITP 2019)}, pages = {17:1--17:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-122-1}, ISSN = {1868-8969}, year = {2019}, volume = {141}, editor = {Harrison, John and O'Leary, John and Tolmach, Andrew}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2019.17}, URN = {urn:nbn:de:0030-drops-110724}, doi = {10.4230/LIPIcs.ITP.2019.17}, annote = {Keywords: call-by-value, lambda calculus, Coq, constructive type theory, extraction, computability} }

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