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Hilbert’s Tenth Problem in Coq

Authors Dominique Larchey-Wendling , Yannick Forster

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Dominique Larchey-Wendling
  • Université de Lorraine, CNRS, LORIA, Vandœuvre-lès-Nancy, France
Yannick Forster
  • Saarland University, Saarland Informatics Campus, Saarbrücken, Germany


We would like to thank Gert Smolka, Dominik Kirst and Simon Spies for helpful discussion regarding the presentation.

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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)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Models of computation
  • Theory of computation → Type theory
  • Hilbert’s tenth problem
  • Diophantine equations
  • undecidability
  • computability theory
  • reduction
  • Minsky machines
  • Fractran
  • Coq
  • type theory


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