Trakhtenbrot’s Theorem in Coq: A Constructive Approach to Finite Model Theory

Dominik Kirst, Dominique Larchey-Wendling

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

6 Scopus citations


We study finite first-order satisfiability (FSAT) in the constructive setting of dependent type theory. Employing synthetic accounts of enumerability and decidability, we give a full classification of FSAT depending on the first-order signature of non-logical symbols. On the one hand, our development focuses on Trakhtenbrot’s theorem, stating that FSAT is undecidable as soon as the signature contains an at least binary relation symbol. Our proof proceeds by a many-one reduction chain starting from the Post correspondence problem. On the other hand, we establish the decidability of FSAT for monadic first-order logic, i.e. where the signature only contains at most unary function and relation symbols, as well as the enumerability of FSAT for arbitrary enumerable signatures. All our results are mechanised in the framework of a growing Coq library of synthetic undecidability proofs.

Original languageEnglish
Title of host publicationAutomated Reasoning - 10th International Joint Conference, IJCAR 2020, Proceedings
EditorsNicolas Peltier, Viorica Sofronie-Stokkermans
Number of pages18
ISBN (Print)9783030510534
StatePublished - 1 Jan 2020
Externally publishedYes
Event10th International Joint Conference on Automated Reasoning, IJCAR 2020 - Virtual, Online
Duration: 1 Jul 20204 Jul 2020

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume12167 LNAI
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference10th International Joint Conference on Automated Reasoning, IJCAR 2020
CityVirtual, Online

ASJC Scopus subject areas

  • Theoretical Computer Science
  • General Computer Science


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