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Documents authored by Forster, Yannick

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Invited Talk
DOI: 10.4230/LIPIcs.CSL.2025.3
Synthetic Mathematics for the Mechanisation of Computability Theory and Logic (Invited Talk)

Authors: Yannick Forster

Published in: LIPIcs, Volume 326, 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)

Abstract
Mathematical practice in most areas of mathematics is based on the assumption that proofs could be made fully formal in a chosen foundation in principle. This assumption is backed by partial attempts at formalisation and by full mechanisation of various areas of mathematics in various proof assistants and various foundations. Areas that have been largely neglected for computer-assisted and machine-checked proofs are computability theory and logic: Fundamental results like Gödel’s second incompleteness theorem in its stronger forms due to Kleene and Rosser, Löb’s theorem, Post’s theorem connecting the arithmetical hierarchy and Turing jumps, or the Friedberg-Muĉnik theorem solving Post’s problem have not or only very recently been re-produced in proof assistants. This is due to the fact that making these arguments formal is several orders of magnitude more involved than formalising other areas of mathematics, due to the amount of invisible mathematics (a term coined by Andrej Bauer) involved. In computability theory, invisible arguments occur mainly behind proofs that a certain intuitively sketched procedure is computable in - citing Emil Post - "forbidding, diverse and alien formalisms in which this [...] work of Gödel, Church, Turing, Kleene, Rosser [...] is embodied.". For instance, there have been various approaches of formalising Turing machines, all to the ultimate dissatisfaction of the respective authors, and none going further than constructing a universal machine and proving the halting problem undecidable. Professional computability theorist and teachers of computability theory thus rely on the informal Church Turing thesis to carry out their work and only argue the computability of described algorithms informally. For computability theory, a way out was proposed in the 1980s by Fred Richman and developed during the last decade by Andrej Bauer: Synthetic computability theory, where one assumes axioms in a constructive foundation which essentially identify all (constructively definable) functions with computable functions. A drawback of the approach is that assuming such an axiom on top of the axiom of countable choice - which is routinely assumed in this branch of constructive mathematics and computable analysis - is that the law of excluded middle, i.e. classical logic, becomes invalid. Computability theory is however, as all mainstream branches of mathematics, making routine use of the axiom of excluded middle. In the case of logic, the invisible mathematics usually is either centered around encoding formulas and proofs as numbers using Gödel or similar encodings or about provability arguments that certain results can be proved in restricted proof systems such as Peano arithmetic. In several settings, synthetic computability arguments can be employed to mechanise these proofs. We observe that a slight foundational shift rectifies the situation: By basing synthetic computability theory in the Calculus of Inductive Constructions, the type theory underlying amongst others the Coq and Lean proof assistants, where countable choice is independent and thus not provable, axioms for synthetic computability are compatible with the law of excluded middle. This insight can be used to finally mechanise computability theory and logic, in an elegant, concise way where invisible arguments stay invisible: with Felix Jahn I have mechanised arguments related to many-one and truth-table reduction theory (published at CSL '23), Dominik Kirst and Benjamin Peters have presented Gödel’s first incompleteness theorem in this style (at CSL '23), and in collaboration with Dominik Kirst and Niklas Mück I have given a proof of Post’s hierarchy theorem (at CSL '24). In this invited talk, I will give a broader overview of this line of research investigating a mechanised synthetic approach to logic and computability theory. In particular, I will discuss a Coq library of undecidability proofs, results in the theory of reducibility degrees, constructive reverse analysis of theorems, as well as generalised incompleteness results such as Löb’s theorem.
Cite as

Yannick Forster. Synthetic Mathematics for the Mechanisation of Computability Theory and Logic (Invited Talk). In 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 326, pp. 3:1-3:2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{forster:LIPIcs.CSL.2025.3,
  author =	{Forster, Yannick},
  title =	{{Synthetic Mathematics for the Mechanisation of Computability Theory and Logic}},
  booktitle =	{33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)},
  pages =	{3:1--3:2},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-362-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{326},
  editor =	{Endrullis, J\"{o}rg and Schmitz, Sylvain},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2025.3},
  URN =		{urn:nbn:de:0030-drops-227603},
  doi =		{10.4230/LIPIcs.CSL.2025.3},
  annote =	{Keywords: Synthetic mathematics, computability theory, logic}
}
Document
DOI: 10.4230/LIPIcs.CSL.2024.29
The Kleene-Post and Post’s Theorem in the Calculus of Inductive Constructions

Authors: Yannick Forster, Dominik Kirst, and Niklas Mück

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

Abstract
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.
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.CSL.2023.21
Constructive and Synthetic Reducibility Degrees: Post’s Problem for Many-One and Truth-Table Reducibility in Coq

Authors: Yannick Forster and Felix Jahn

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

Abstract
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.
Cite as

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.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}
}
Document
DOI: 10.4230/LIPIcs.ITP.2022.12
Synthetic Kolmogorov Complexity in Coq

Authors: Yannick Forster, Fabian Kunze, and Nils Lauermann

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

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.
Cite as

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.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
DOI: 10.4230/LIPIcs.ITP.2021.19
A Mechanised Proof of the Time Invariance Thesis for the Weak Call-By-Value λ-Calculus

Authors: Yannick Forster, Fabian Kunze, Gert Smolka, and Maximilian Wuttke

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

Abstract
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.
Cite as

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.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
DOI: 10.4230/LIPIcs.CSL.2021.21
Church’s Thesis and Related Axioms in Coq’s Type Theory

Authors: Yannick Forster

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

Abstract
"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.
Cite as

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.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}
}
Document
DOI: 10.4230/LIPIcs.ITP.2019.17
A Certifying Extraction with Time Bounds from Coq to Call-By-Value Lambda Calculus

Authors: Yannick Forster and Fabian Kunze

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

Abstract
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.
Cite as

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.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}
}
Document
DOI: 10.4230/LIPIcs.FSCD.2019.27
Hilbert’s Tenth Problem in Coq

Authors: Dominique Larchey-Wendling and Yannick Forster

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

Abstract
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.
Cite as

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.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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