Lower Bounds on Unambiguous Automata Complementation and Separation via Communication Complexity

We use results from communication complexity, both new and old ones, to prove lower bounds for problems on unambiguous finite automata (UFAs). We show: (1) Complementing UFAs with $n$ states requires in general at least $n^{\tilde{\Omega}(\log n)}$ states, improving on a bound by Raskin. (2) There are languages $L_n$ such that both $L_n$ and its complement are recognized by NFAs with $n$ states but any UFA that recognizes $L_n$ requires $n^{\Omega(\log n)}$ states, refuting a conjecture by Colcombet on separation.