Successor-invariant first-order logic on finite structures

Abstract

AbstractWe consider successor-invariant first-order logic (FO + succ)inv, consisting of sentences Φ involving an “auxiliary” binary relation S such that (, S1) ⊨ Φ ⇔ (, S2) ⊨ Φ for all finite structures and successor relations S1, S2 on . A successor-invariant sentence Φ has a well-defined semantics on finite structures with no given successor relation: one simply evaluates Φ on (, 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].