Successor-invariant first-order logic on finite structures
We consider successor-invariant first-order logic (FO + succ) inv, consisting of sentences Φ involving an "auxiliary" binary relation S such that (Θ, S1) |= Φ ⇔ (Θ, S2) |= Φ for all finite structures Θ and successor relations S1, S2 on Θ. A successor-invariant sentence Φ has a well-defined semantics on finite structures Θ with no given successor relation: one simply evaluates Φ on (Θ, 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  and epsilon-invariant logic . © 2007, Association for Symbolic Logic.
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