Trakhtenbrot's Theorem in Coq: Finite Model Theory through the Constructive Lens
This work provides a constructive formalization of finite model theory results, which is incremental for the proof assistant community.
The paper tackles the problem of finite first-order satisfiability (FSAT) in constructive dependent type theory, proving Trakhtenbrot's theorem that FSAT is undecidable for signatures with at least binary relation symbols and establishing decidability for monadic first-order logic, with all results mechanized in Coq.
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.