Constructing Semantic Automata for Quantifier Iterations Sarah McWhirter, Jakub Szymanik Abstract: We expand on recent work extending the semantic automata model of quantifier verification to iteration. We demonstrate a simple and intuitive method to construct a minimal iteration DFA from two DFA recognizing monadic regular quantifier languages and prove that deterministic CFL are also closed under quantifier iteration. We also touch upon the relation between computational results and linguistic discussion of Frege boundary.