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

Dominik Kirst (1) , Dominique Larchey-Wendling (2)

1 Saarland University [Saarbrücken]
2 TYPES - Logic, Proof Theory and Programming

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.

Domaines: Logique en informatique [cs.LO]; Informatique et langage [cs.CL]; Mathématique discrète [cs.DM]; Langage de programmation [cs.PL]; Théorie et langage formel [cs.FL]; Génie logiciel [cs.SE]; Logique [math.LO]

Dominik Kirst, Dominique Larchey-Wendling. Trakhtenbrot’s Theorem in Coq: A Constructive Approach to Finite Model Theory. 10th International Joint Conference on Automtated Reasoning, IJCAR 2020, Jul 2020, Paris, France. pp.79-96, ⟨10.1007/978-3-030-51054-1_5⟩. ⟨hal-02944203⟩