Trakhtenbrot’s Theorem in Coq

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.

Undecidability, incompleteness, and completeness of second-order logic in Coq

“…Therefore sparing tedious encodings in a concrete model of computation such as Turing machines, the mechanisation of positive properties like enumerability of deduction systems and negative properties like undecidability of validity becomes feasible. In previous work, this approach has already been employed for synthetic undecidability [8,21] and synthetic incompleteness [20] of FOL, and with this paper, we illustrate that the same methods scale to the case of SOL.…”

“…Our mechanisation of SOL is based on the existing FOL developments in [9,20,21]. In particular, we define signatures Σ as type classes allowing for the appropriate signature to be inferred based on context.…”

“…Mechanised first-and higher-order logic. While we are not aware of any previous mechanisations of SOL, there have been many formalisations concerned with first-or higherorder logics in various theorem provers like Isabelle/HOL [13,14,25], Mizar [3], Lean [40], and Coq [2,9,17,21]. Our work on SOL is inspired by the Coq formalisation of FOL developed in [9] and [21].…”

“…While we are not aware of any previous mechanisations of SOL, there have been many formalisations concerned with first-or higherorder logics in various theorem provers like Isabelle/HOL [13,14,25], Mizar [3], Lean [40], and Coq [2,9,17,21]. Our work on SOL is inspired by the Coq formalisation of FOL developed in [9] and [21]. We adopted their major design decisions of using a syntax with de Bruijn binders parametrised over an arbitrary signature.…”