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

TL;DR

This paper formalizes finite first-order satisfiability within constructive dependent type theory, proving undecidability with binary relations and decidability for monadic logic, all mechanized in Coq.

Contribution

It provides a constructive formalization of Trakhtenbrot's theorem and related results in Coq, including reductions and decidability classifications for FSAT.

Findings

FSAT is undecidable with at least binary relations

FSAT is decidable for monadic first-order logic

Mechanized proofs in Coq library

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. 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.