Vectorization hierarchies of some graph quantifiers

Abstract.

We give a sufficient condition for the inexpressibility of the k-th extended vectorization of a generalized quantifier $\sf Q$ in ${\rm FO}({\vec Q}_k)$, the extension of first-order logic by all k-ary quantifiers. The condition is based on a model construction which, given two ${\rm FO}({\vec Q}_1)$-equivalent models with certain additional structure, yields a pair of ${\rm FO}({\vec Q}_k)$-equivalent models. We also consider some applications of this condition to quantifiers that correspond to graph properties, such as connectivity and planarity.