Trakhtenbrot's Theorem in Coq, A Constructive Approach to Finite Model Theory

TL;DR

This paper formalizes finite first-order satisfiability within constructive type theory using Coq, proving undecidability for signatures with relations and decidability for monadic logic, advancing mechanized finite model theory.

Contribution

It provides a constructive, mechanized classification of FSAT based on signature complexity, including a Coq formalization of Trakhtenbrot's theorem and decidability results.

Findings

FSAT is undecidable for signatures with at least binary relations.

FSAT is decidable for monadic first-order logic.

Mechanized proofs are developed in Coq, supporting synthetic undecidability.

Abstract

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.