Undecidability of Dyadic First-Order Logic in Coq

Johannes Hostert, Andrej Dudenhefner, Dominik Kirst

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

2 Scopus citations


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.

Original languageEnglish
Title of host publication13th International Conference on Interactive Theorem Proving, ITP 2022
EditorsJune Andronick, Leonardo de Moura
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772525
StatePublished - 1 Aug 2022
Externally publishedYes
Event13th International Conference on Interactive Theorem Proving, ITP 2022 - Haifa, Israel
Duration: 7 Aug 202210 Aug 2022

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
ISSN (Print)1868-8969


Conference13th International Conference on Interactive Theorem Proving, ITP 2022


  • Coq
  • first-order logic
  • synthetic computability
  • undecidability

ASJC Scopus subject areas

  • Software


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