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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.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)

Gödel published his groundbreaking first incompleteness theorem in 1931, stating that a large class of formal logics admits independent sentences which are neither provable nor refutable. This result, in conjunction with his second incompleteness theorem, established the impossibility of concluding Hilbert’s program, which pursued a possible path towards a single formal system unifying all of mathematics. Using a technical trick to refine Gödel’s original proof, the incompleteness result was strengthened further by Rosser in 1936 regarding the conditions imposed on the formal systems.
Computability theory, which also originated in the 1930s, was quickly applied to formal logics by Turing, Kleene, and others to yield incompleteness results similar in strength to Gödel’s original theorem, but weaker than Rosser’s refinement. Only much later, Kleene found an improved but far less well-known proof based on computational notions, yielding a result as strong as Rosser’s.
In this expository paper, we work in constructive type theory to reformulate Kleene’s incompleteness results abstractly in the setting of synthetic computability theory and assuming a form of Church’s thesis, an axiom internalising the fact that all functions definable in such a setting are computable. Our novel, greatly condensed reformulation showcases the simplicity of the computational argument while staying formally entirely precise, a combination hard to achieve in typical textbook presentations. As an application, we instantiate the abstract result to first-order logic in order to derive essential incompleteness and, along the way, essential undecidability of Robinson arithmetic.
This paper is accompanied by a Coq mechanisation covering all our results and based on existing libraries of undecidability proofs and first-order logic, complementing the extensive work on mechanised incompleteness using the Gödel-Rosser approach. In contrast to the related mechanisations, our development follows Kleene’s ideas and utilises Church’s thesis for additional simplicity.

Dominik Kirst and Benjamin Peters. Gödel’s Theorem Without Tears - Essential Incompleteness in Synthetic Computability. In 31st EACSL Annual Conference on Computer Science Logic (CSL 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 252, pp. 30:1-30:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{kirst_et_al:LIPIcs.CSL.2023.30, author = {Kirst, Dominik and Peters, Benjamin}, title = {{G\"{o}del’s Theorem Without Tears - Essential Incompleteness in Synthetic Computability}}, booktitle = {31st EACSL Annual Conference on Computer Science Logic (CSL 2023)}, pages = {30:1--30:18}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2023.30}, URN = {urn:nbn:de:0030-drops-174911}, doi = {10.4230/LIPIcs.CSL.2023.30}, annote = {Keywords: incompleteness, undecidability, synthetic computability theory} }

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

We develop and mechanize compact proofs of the undecidability of various problems for dyadic first-order logic over a small logical fragment. In this fragment, formulas are restricted to only a single binary relation, and a minimal set of logical connectives. We show that validity, satisfiability, and provability, along with finite satisfiability and finite validity are undecidable, by directly reducing from a suitable binary variant of Diophantine constraints satisfiability. Our results improve upon existing work in two ways: First, the reductions are direct and significantly more compact than existing ones. Secondly, the undecidability of the small logic fragment of dyadic first-order logic was not mechanized before. We contribute our mechanization to the Coq Library of Undecidability Proofs, utilizing its synthetic approach to computability theory.

Johannes Hostert, Andrej Dudenhefner, and Dominik Kirst. Undecidability of Dyadic First-Order Logic in Coq. In 13th International Conference on Interactive Theorem Proving (ITP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 237, pp. 19:1-19:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{hostert_et_al:LIPIcs.ITP.2022.19, author = {Hostert, Johannes and Dudenhefner, Andrej and Kirst, Dominik}, title = {{Undecidability of Dyadic First-Order Logic in Coq}}, booktitle = {13th International Conference on Interactive Theorem Proving (ITP 2022)}, pages = {19:1--19:19}, 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/entities/document/10.4230/LIPIcs.ITP.2022.19}, URN = {urn:nbn:de:0030-drops-167280}, doi = {10.4230/LIPIcs.ITP.2022.19}, annote = {Keywords: undecidability, synthetic computability, first-order logic, Coq} }

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

The back-and-forth method is a well-known technique to establish isomorphisms of countable structures. In this proof pearl, we formalise this method abstractly in the framework of constructive type theory, emphasising the computational interpretation of the constructed isomorphisms. As prominent instances, we then deduce Cantor’s and Myhill’s isomorphism theorems on dense linear orders and one-one interreducible sets, respectively. By exploiting the symmetry of the abstract argument, our approach yields a particularly compact mechanisation of the method itself as well as its two instantiations, all implemented using the Coq proof assistant. As adequate for a proof pearl, we attempt to make the text and mechanisation accessible for a general mathematical audience.

Dominik Kirst. Computational Back-And-Forth Arguments in Constructive Type Theory. In 13th International Conference on Interactive Theorem Proving (ITP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 237, pp. 22:1-22:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{kirst:LIPIcs.ITP.2022.22, author = {Kirst, Dominik}, title = {{Computational Back-And-Forth Arguments in Constructive Type Theory}}, booktitle = {13th International Conference on Interactive Theorem Proving (ITP 2022)}, pages = {22:1--22:12}, 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/entities/document/10.4230/LIPIcs.ITP.2022.22}, URN = {urn:nbn:de:0030-drops-167311}, doi = {10.4230/LIPIcs.ITP.2022.22}, annote = {Keywords: back-and-forth method, computable isomorphisms, Coq} }

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**Published in:** LIPIcs, Volume 228, 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022)

Tennenbaum’s theorem states that the only countable model of Peano arithmetic (PA) with computable arithmetical operations is the standard model of natural numbers. In this paper, we use constructive type theory as a framework to revisit and generalize this result.
The chosen framework allows for a synthetic approach to computability theory, by exploiting the fact that, externally, all functions definable in constructive type theory can be shown computable. We internalize this fact by assuming a version of Church’s thesis expressing that any function on natural numbers is representable by a formula in PA. This assumption allows for a conveniently abstract setup to carry out rigorous computability arguments and feasible mechanization.
Concretely, we constructivize several classical proofs and present one inherently constructive rendering of Tennenbaum’s theorem, all following arguments from the literature. Concerning the classical proofs in particular, the constructive setting allows us to highlight differences in their assumptions and conclusions which are not visible classically. All versions are accompanied by a unified mechanization in the Coq proof assistant.

Marc Hermes and Dominik Kirst. An Analysis of Tennenbaum’s Theorem in Constructive Type Theory. In 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 228, pp. 9:1-9:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{hermes_et_al:LIPIcs.FSCD.2022.9, author = {Hermes, Marc and Kirst, Dominik}, title = {{An Analysis of Tennenbaum’s Theorem in Constructive Type Theory}}, booktitle = {7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022)}, pages = {9:1--9:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-233-4}, ISSN = {1868-8969}, year = {2022}, volume = {228}, editor = {Felty, Amy P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2022.9}, URN = {urn:nbn:de:0030-drops-162909}, doi = {10.4230/LIPIcs.FSCD.2022.9}, annote = {Keywords: first-order logic, Peano arithmetic, Tennenbaum’s theorem, constructive type theory, Church’s thesis, synthetic computability, Coq} }

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

We mechanise the undecidability of various first-order axiom systems in Coq, employing the synthetic approach to computability underlying the growing Coq Library of Undecidability Proofs. Concretely, we cover both semantic and deductive entailment in fragments of Peano arithmetic (PA) and Zermelo-Fraenkel set theory (ZF), with their undecidability established by many-one reductions from solvability of Diophantine equations, i.e. Hilbert’s tenth problem (H10), and the Post correspondence problem (PCP), respectively. In the synthetic setting based on the computability of all functions definable in a constructive foundation, such as Coq’s type theory, it suffices to define these reductions as meta-level functions with no need for further encoding in a formalised model of computation.
The concrete cases of PA and ZF are prepared by a general synthetic theory of undecidable axiomatisations, focusing on well-known connections to consistency and incompleteness. Specifically, our reductions rely on the existence of standard models, necessitating additional assumptions in the case of full ZF, and all axiomatic extensions still justified by such standard models are shown incomplete. As a by-product of the undecidability of ZF formulated using only membership and no equality symbol, we obtain the undecidability of first-order logic with a single binary relation.

Dominik Kirst and Marc Hermes. Synthetic Undecidability and Incompleteness of First-Order Axiom Systems in Coq. In 12th International Conference on Interactive Theorem Proving (ITP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 193, pp. 23:1-23:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{kirst_et_al:LIPIcs.ITP.2021.23, author = {Kirst, Dominik and Hermes, Marc}, title = {{Synthetic Undecidability and Incompleteness of First-Order Axiom Systems in Coq}}, booktitle = {12th International Conference on Interactive Theorem Proving (ITP 2021)}, pages = {23:1--23:20}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2021.23}, URN = {urn:nbn:de:0030-drops-139188}, doi = {10.4230/LIPIcs.ITP.2021.23}, annote = {Keywords: undecidability, synthetic computability, first-order logic, incompleteness, Peano arithmetic, ZF set theory, constructive type theory, Coq} }

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