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Undecidability of Dyadic First-Order Logic in Coq

Authors Johannes Hostert , Andrej Dudenhefner , Dominik Kirst

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Author Details

Johannes Hostert
  • Universität des Saarlandes, Saarland Informatics Campus, Saarbrücken, Germany
Andrej Dudenhefner
  • Universität des Saarlandes, Saarland Informatics Campus, Saarbrücken, Germany
Dominik Kirst
  • Universität des Saarlandes, Saarland Informatics Campus, Saarbrücken, Germany

Acknowledgements

We thank Yannick Forster for valuable feedback on drafts of this paper.

Cite AsGet BibTex

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)
https://doi.org/10.4230/LIPIcs.ITP.2022.19

Abstract

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Constructive mathematics
  • Theory of computation → Type theory
  • Theory of computation → Logic and verification
Keywords
  • undecidability
  • synthetic computability
  • first-order logic
  • Coq

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