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Documents authored by Larchey-Wendling, Dominique

Document
DOI: 10.4230/LIPIcs.ITP.2023.21
Proof Pearl: Faithful Computation and Extraction of μ-Recursive Algorithms in Coq

Authors: Dominique Larchey-Wendling and Jean-François Monin

Published in: LIPIcs, Volume 268, 14th International Conference on Interactive Theorem Proving (ITP 2023)

Abstract
Basing on an original Coq implementation of unbounded linear search for partially decidable predicates, we study the computational contents of μ-recursive functions via their syntactic representation, and a correct by construction Coq interpreter for this abstract syntax. When this interpreter is extracted, we claim the resulting OCaml code to be the natural combination of the implementation of the μ-recursive schemes of composition, primitive recursion and unbounded minimization of partial (i.e., possibly non-terminating) functions. At the level of the fully specified Coq terms, this implies the representation of higher-order functions of which some of the arguments are themselves partial functions. We handle this issue using some techniques coming from the Braga method. Hence we get a faithful embedding of μ-recursive algorithms into Coq preserving not only their extensional meaning but also their intended computational behavior. We put a strong focus on the quality of the Coq artifact which is both self contained and with a line of code count of less than 1k in total.
Cite as

Dominique Larchey-Wendling and Jean-François Monin. Proof Pearl: Faithful Computation and Extraction of μ-Recursive Algorithms in Coq. In 14th International Conference on Interactive Theorem Proving (ITP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 268, pp. 21:1-21:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{larcheywendling_et_al:LIPIcs.ITP.2023.21,
  author =	{Larchey-Wendling, Dominique and Monin, Jean-Fran\c{c}ois},
  title =	{{Proof Pearl: Faithful Computation and Extraction of \mu-Recursive Algorithms in Coq}},
  booktitle =	{14th International Conference on Interactive Theorem Proving (ITP 2023)},
  pages =	{21:1--21:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-284-6},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{268},
  editor =	{Naumowicz, Adam and Thiemann, Ren\'{e}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2023.21},
  URN =		{urn:nbn:de:0030-drops-183963},
  doi =		{10.4230/LIPIcs.ITP.2023.21},
  annote =	{Keywords: Unbounded linear search, \mu-recursive functions, computational contents, Coq, extraction, OCaml}
}
Document
DOI: 10.4230/LIPIcs.FSCD.2021.18
Synthetic Undecidability of MSELL via FRACTRAN Mechanised in Coq

Authors: Dominique Larchey-Wendling

Published in: LIPIcs, Volume 195, 6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021)

Abstract
We present an alternate undecidability proof for entailment in (intuitionistic) multiplicative sub-exponential linear logic (MSELL). We contribute the result and its mechanised proof to the Coq library of synthetic undecidability. The result crucially relies on the undecidability of the halting problem for two counters Minsky machines, which we also hand out to the library. As a seed of undecidability, we start from FRACTRAN halting which we (many-one) reduce to Minsky machines termination by implementing Euclidean division using two counters only. We then give an alternate presentation of those two counters machines as sequent rules, where computation is performed by proof-search, and halting reduced to provability. We use this system called non-deterministic two counters Minsky machines to describe and compare both the legacy reduction to linear logic, and the more recent reduction to MSELL. In contrast with that former MSELL undecidability proof, our correctness argument for the reduction uses trivial phase semantics in place of a focused calculus.
Cite as

Dominique Larchey-Wendling. Synthetic Undecidability of MSELL via FRACTRAN Mechanised in Coq. In 6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 195, pp. 18:1-18:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{larcheywendling:LIPIcs.FSCD.2021.18,
  author =	{Larchey-Wendling, Dominique},
  title =	{{Synthetic Undecidability of MSELL via FRACTRAN Mechanised in Coq}},
  booktitle =	{6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021)},
  pages =	{18:1--18:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-191-7},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{195},
  editor =	{Kobayashi, Naoki},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2021.18},
  URN =		{urn:nbn:de:0030-drops-142568},
  doi =		{10.4230/LIPIcs.FSCD.2021.18},
  annote =	{Keywords: Undecidability, computability theory, many-one reduction, Minsky machines, Fractran, sub-exponential linear logic, Coq}
}
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-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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