Logic: Lecture 16, 06 October 2015 ---------------------------------- Formal definition of quantifier rank (Libkin, Definition 3.8) - Number of quantifiers can be exponential wrt quantifier rank ---------------------------------------------------------------------- Rank k-types (Libkin, Section 3.4) Lemma [Libkin, Lemma 3.13]: If L is finite, then up to logical equivalence, FO[k] over L contains only finitely many formulae in m free variables x1,...,xm. Definition of rank k m-type (Libkin Definition 3.14) - Every rank k m-type is defined by a single rank k formula - Every rank k formula can be expressed in terms of rank k m-types ======================================================================