# Trakhtenbrot's Theorem in Coq: Finite Model Theory through the Constructive Lens

arXiv: https://arxiv.org/abs/2104.14445 · HTML: https://mlsanity.com/paper/2104.14445/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 37 references that are in the map.
- First posted on arXiv 29 Apr 2021.

## Builds on (most cited first, 4 of 4)

- [Verification of PCP-Related Computational Reductions in Coq](https://mlsanity.com/paper/1711.07023/) (2017)
- [A certifying extraction with time bounds from Coq to call-by-value λ-calculus](https://mlsanity.com/paper/1904.11818/) (2019)
- [Trakhtenbrot's Theorem in Coq, A Constructive Approach to Finite Model Theory](https://mlsanity.com/paper/2004.07390/) (2020)
- [Completeness Theorems for First-Order Logic Analysed in Constructive Type Theory (Extended Version)](https://mlsanity.com/paper/2006.04399/) (2020)

## Abstract (arXiv)

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. To showcase an application of Trakhtenbrot's theorem, we continue our reduction chain with a many-one reduction from FSAT to separation logic. All our results are mechanised in the framework of a growing Coq library of synthetic undecidability proofs.