 In van Benthem’s semantic automata setting  (section 9),  Steinert-Threlkeld and Icard (2013) proved that the Frege boundary  (section 16)  is robust in the sense that if two Conserv and Ext type \({\langle}1,1{\rangle}\) quantifiers are recognizable by finite (or push-down) automata, then so is their iteration. Moreover, Steinert-Threlkeld (2016) showed that for large classes of type \({\langle}1,1,2{\rangle}\) quantifiers, it is decidable whether they are iterations of type \({\langle}1,1{\rangle}\) quantifiers or not. A recent presentation of both theoretical and empirical results around the cognitive aspects of quantifier recognition is Szymanik (2016).