On Relativisation and Complexity Gap

We study the proof complexity of Taut, the class of Second-Order Existential (SO∃) logical sentences which fail in all finite models. The Complexity-Gap theorem for Tree-like Resolution says that the shortest Tree-like Resolution refutation of any such sentence Φ is either fully exponential, \(2^{\Omega \left(n\right)}\), or polynomial, \(n^{O\left(1\right)}\), where n is the size of the finite model. Moreover, there is a very simple model-theoretics criteria which separates the two cases: the exponential lower bound holds if and only if Φ holds in some infinite model.