A dichotomy in classifying quantifiers for finite models

Abstract

We consider a family 𝔲 of finite universes. The second order existential quantifier Q_ℜ, means for each U∈ 𝔲 quantifying over a set of n(ℜ)-place relations isomorphic to a given relation. We define a natural partial order on such quantifiers called interpretability. We show that for every Q_ℜ, either Q_ℜ is interpretable by quantifying over subsets of U and one to one functions on U both of bounded order, or the logic L(Q_ℜ) (first order logic plus the quantifier Q_ℜ) is undecidable.