3 Search Results for "Baillon, Martin"


Document
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-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}
}
Document
Inductive Continuity via Brouwer Trees

Authors: Liron Cohen, Bruno da Rocha Paiva, Vincent Rahli, and Ayberk Tosun

Published in: LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)


Abstract
Continuity is a key principle of intuitionistic logic that is generally accepted by constructivists but is inconsistent with classical logic. Most commonly, continuity states that a function from the Baire space to numbers, only needs approximations of the points in the Baire space to compute. More recently, another formulation of the continuity principle was put forward. It states that for any function F from the Baire space to numbers, there exists a (dialogue) tree that contains the values of F at its leaves and such that the modulus of F at each point of the Baire space is given by the length of the corresponding branch in the tree. In this paper we provide the first internalization of this "inductive" continuity principle within a computational setting. Concretely, we present a class of intuitionistic theories that validate this formulation of continuity thanks to computations that construct such dialogue trees internally to the theories using effectful computations. We further demonstrate that this inductive continuity principle implies other forms of continuity principles.

Cite as

Liron Cohen, Bruno da Rocha Paiva, Vincent Rahli, and Ayberk Tosun. Inductive Continuity via Brouwer Trees. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 37:1-37:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{cohen_et_al:LIPIcs.MFCS.2023.37,
  author =	{Cohen, Liron and da Rocha Paiva, Bruno and Rahli, Vincent and Tosun, Ayberk},
  title =	{{Inductive Continuity via Brouwer Trees}},
  booktitle =	{48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)},
  pages =	{37:1--37:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-292-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{272},
  editor =	{Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.37},
  URN =		{urn:nbn:de:0030-drops-185718},
  doi =		{10.4230/LIPIcs.MFCS.2023.37},
  annote =	{Keywords: Continuity, Dialogue trees, Stateful computations, Intuitionistic Logic, Extensional Type Theory, Constructive Type Theory, Realizability, Theorem proving, Agda}
}
Document
Gardening with the Pythia A Model of Continuity in a Dependent Setting

Authors: Martin Baillon, Assia Mahboubi, and Pierre-Marie Pédrot

Published in: LIPIcs, Volume 216, 30th EACSL Annual Conference on Computer Science Logic (CSL 2022)


Abstract
We generalize to a rich dependent type theory a proof originally developed by Escardó that all System 𝚃 functionals are continuous. It relies on the definition of a syntactic model of Baclofen Type Theory, a type theory where dependent elimination must be strict, into the Calculus of Inductive Constructions. The model is given by three translations: the axiom translation, that adds an oracle to the context; the branching translation, based on the dialogue monad, turning every type into a tree; and finally, a layer of algebraic binary parametricity, binding together the two translations. In the resulting type theory, every function f : (ℕ → ℕ) → ℕ is externally continuous.

Cite as

Martin Baillon, Assia Mahboubi, and Pierre-Marie Pédrot. Gardening with the Pythia A Model of Continuity in a Dependent Setting. In 30th EACSL Annual Conference on Computer Science Logic (CSL 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 216, pp. 5:1-5:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{baillon_et_al:LIPIcs.CSL.2022.5,
  author =	{Baillon, Martin and Mahboubi, Assia and P\'{e}drot, Pierre-Marie},
  title =	{{Gardening with the Pythia A Model of Continuity in a Dependent Setting}},
  booktitle =	{30th EACSL Annual Conference on Computer Science Logic (CSL 2022)},
  pages =	{5:1--5:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-218-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{216},
  editor =	{Manea, Florin and Simpson, Alex},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2022.5},
  URN =		{urn:nbn:de:0030-drops-157256},
  doi =		{10.4230/LIPIcs.CSL.2022.5},
  annote =	{Keywords: Type theory, continuity, syntactic model}
}
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