Successor-invariant first-order logic on finite structures

Abstract

We consider successor-invariant first-order logic (FO + succ)_inv, consisting of sentences Φ involving an “auxiliary” binary relation S such that (𝔞,S₁) ⊨ Φ {⋔} (𝔞,S₂) ⊨ Φ for all finite structures 𝔞 and successor relations S₁,S₂ on 𝔞. A successor-invariant sentence Φ has a well-defined semantics on finite structures 𝔞 with no given successor relation: one simply evaluates Φ on (𝔞,S) for an arbitrary choice of successor relation S. In this article, we prove that (FO + succ)_inv is more expressive on finite structures than first-order logic without a successor relation. This extends similar results for order-invariant logic [8] and epsilon-invariant logic [10].