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**Published in:** LIPIcs, Volume 288, 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)

The Kleene-Post theorem and Post’s theorem are two central and historically important results in the development of oracle computability theory, clarifying the structure of Turing reducibility degrees. They state, respectively, that there are incomparable Turing degrees and that the arithmetical hierarchy is connected to the relativised form of the halting problem defined via Turing jumps.
We study these two results in the calculus of inductive constructions (CIC), the constructive type theory underlying the Coq proof assistant. CIC constitutes an ideal foundation for the formalisation of computability theory for two reasons: First, like in other constructive foundations, computable functions can be treated via axioms as a purely synthetic notion rather than being defined in terms of a concrete analytic model of computation such as Turing machines. Furthermore and uniquely, CIC allows consistently assuming classical logic via the law of excluded middle or weaker variants on top of axioms for synthetic computability, enabling both fully classical developments and taking the perspective of constructive reverse mathematics on computability theory.
In the present paper, we give a fully constructive construction of two Turing-incomparable degrees à la Kleene-Post and observe that the classical content of Post’s theorem seems to be related to the arithmetical hierarchy of the law of excluded middle due to Akama et. al. Technically, we base our investigation on a previously studied notion of synthetic oracle computability and contribute the first consistency proof of a suitable enumeration axiom. All results discussed in the paper are mechanised and contributed to the Coq library of synthetic computability.

Yannick Forster, Dominik Kirst, and Niklas Mück. The Kleene-Post and Post’s Theorem in the Calculus of Inductive Constructions. In 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 288, pp. 29:1-29:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{forster_et_al:LIPIcs.CSL.2024.29, author = {Forster, Yannick and Kirst, Dominik and M\"{u}ck, Niklas}, title = {{The Kleene-Post and Post’s Theorem in the Calculus of Inductive Constructions}}, booktitle = {32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)}, pages = {29:1--29:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-310-2}, ISSN = {1868-8969}, year = {2024}, volume = {288}, editor = {Murano, Aniello and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2024.29}, URN = {urn:nbn:de:0030-drops-196728}, doi = {10.4230/LIPIcs.CSL.2024.29}, annote = {Keywords: Constructive mathematics, Computability theory, Logical foundations, Constructive type theory, Interactive theorem proving, Coq proof assistant} }

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**Published in:** LIPIcs, Volume 252, 31st EACSL Annual Conference on Computer Science Logic (CSL 2023)

We present a constructive analysis and machine-checked theory of one-one, many-one, and truth-table reductions based on synthetic computability theory in the Calculus of Inductive Constructions, the type theory underlying the proof assistant Coq. We give elegant, synthetic, and machine-checked proofs of Post’s landmark results that a simple predicate exists, is enumerable, undecidable, but many-one incomplete (Post’s problem for many-one reducibility), and a hypersimple predicate exists, is enumerable, undecidable, but truth-table incomplete (Post’s problem for truth-table reducibility).
In synthetic computability, one assumes axioms allowing to carry out computability theory with all definitions and proofs purely in terms of functions of the type theory with no mention of a model of computation. Proofs can focus on the essence of the argument, without having to sacrifice formality. Synthetic computability also clears the lense for constructivisation.
Our constructively careful definition of simple and hypersimple predicates allows us to not assume classical axioms, not even Markov’s principle, still yielding the expected strong results.

Yannick Forster and Felix Jahn. Constructive and Synthetic Reducibility Degrees: Post’s Problem for Many-One and Truth-Table Reducibility in Coq. In 31st EACSL Annual Conference on Computer Science Logic (CSL 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 252, pp. 21:1-21:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{forster_et_al:LIPIcs.CSL.2023.21, author = {Forster, Yannick and Jahn, Felix}, title = {{Constructive and Synthetic Reducibility Degrees: Post’s Problem for Many-One and Truth-Table Reducibility in Coq}}, booktitle = {31st EACSL Annual Conference on Computer Science Logic (CSL 2023)}, pages = {21:1--21:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-264-8}, ISSN = {1868-8969}, year = {2023}, volume = {252}, editor = {Klin, Bartek and Pimentel, Elaine}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2023.21}, URN = {urn:nbn:de:0030-drops-174820}, doi = {10.4230/LIPIcs.CSL.2023.21}, annote = {Keywords: type theory, computability theory, constructive mathematics, Coq} }

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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} }

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**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} }

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**Published in:** LIPIcs, Volume 183, 29th EACSL Annual Conference on Computer Science Logic (CSL 2021)

"Church’s thesis" (CT) as an axiom in constructive logic states that every total function of type ℕ → ℕ is computable, i.e. definable in a model of computation. CT is inconsistent both in classical mathematics and in Brouwer’s intuitionism since it contradicts weak Kőnig’s lemma and the fan theorem, respectively. Recently, CT was proved consistent for (univalent) constructive type theory.
Since neither weak Kőnig’s lemma nor the fan theorem is a consequence of just logical axioms or just choice-like axioms assumed in constructive logic, it seems likely that CT is inconsistent only with a combination of classical logic and choice axioms. We study consequences of CT and its relation to several classes of axioms in Coq’s type theory, a constructive type theory with a universe of propositions which proves neither classical logical axioms nor strong choice axioms.
We thereby provide a partial answer to the question as to which axioms may preserve computational intuitions inherent to type theory, and which certainly do not. The paper can also be read as a broad survey of axioms in type theory, with all results mechanised in the Coq proof assistant.

Yannick Forster. Church’s Thesis and Related Axioms in Coq’s Type Theory. In 29th EACSL Annual Conference on Computer Science Logic (CSL 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 183, pp. 21:1-21:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{forster:LIPIcs.CSL.2021.21, author = {Forster, Yannick}, title = {{Church’s Thesis and Related Axioms in Coq’s Type Theory}}, booktitle = {29th EACSL Annual Conference on Computer Science Logic (CSL 2021)}, pages = {21:1--21:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-175-7}, ISSN = {1868-8969}, year = {2021}, volume = {183}, editor = {Baier, Christel and Goubault-Larrecq, Jean}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2021.21}, URN = {urn:nbn:de:0030-drops-134552}, doi = {10.4230/LIPIcs.CSL.2021.21}, annote = {Keywords: Church’s thesis, constructive type theory, constructive reverse mathematics, synthetic computability theory, Coq} }

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**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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**Published in:** LIPIcs, Volume 131, 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)

We formalise the undecidability of solvability of Diophantine equations, i.e. polynomial equations over natural numbers, in Coq’s constructive type theory. To do so, we give the first full mechanisation of the Davis-Putnam-Robinson-Matiyasevich theorem, stating that every recursively enumerable problem - in our case by a Minsky machine - is Diophantine. We obtain an elegant and comprehensible proof by using a synthetic approach to computability and by introducing Conway’s FRACTRAN language as intermediate layer.

Dominique Larchey-Wendling and Yannick Forster. Hilbert’s Tenth Problem in Coq. In 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 131, pp. 27:1-27:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{larcheywendling_et_al:LIPIcs.FSCD.2019.27, author = {Larchey-Wendling, Dominique and Forster, Yannick}, title = {{Hilbert’s Tenth Problem in Coq}}, booktitle = {4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)}, pages = {27:1--27:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-107-8}, ISSN = {1868-8969}, year = {2019}, volume = {131}, editor = {Geuvers, Herman}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2019.27}, URN = {urn:nbn:de:0030-drops-105342}, doi = {10.4230/LIPIcs.FSCD.2019.27}, annote = {Keywords: Hilbert’s tenth problem, Diophantine equations, undecidability, computability theory, reduction, Minsky machines, Fractran, Coq, type theory} }

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