It is well-known that every first-order property on words is expressible
using at most three variables. The subclass of properties expressible with
only two variables is also quite interesting and well-studied. We prove
precise structure theorems that characterize the exact expressive power of
first-order logic with two variables on words. ... more >>>

We present a trichotomy theorem for the quantum query complexity of regular languages. Every regular language has quantum query complexity $\Theta(1)$, $\tilde{\Theta}(\sqrt n)$, or $\Theta(n)$. The extreme uniformity of regular languages prevents them from taking any other asymptotic complexity. This is in contrast to even the context-free languages, which we ... more >>>