The Alternation Hierarchy of First-Order Logic on Words is Decidable

We show that for any $i > 0$, it is decidable, given a regular language, whether it is expressible in the $\Sigma_i[<]$ fragment of first-order logic FO[<]. This settles a question open since 1971. Our main technical result relies on the notion of polynomial closure of a class of languages $\mathcal{V}$, that is, finite unions of languages of the form $L_0a_1L_1\cdots a_nL_n$ where each $a_i$ is a letter and each $L_i$ a language of $\mathcal{V}$. We show that if a class $\mathcal{V}$ of regular languages with some closure properties (namely, a positive variety) has a decidable separation problem, then so does its polynomial closure Pol($\mathcal{V}$). The resulting algorithm for Pol($\mathcal{V}$) has time complexity that is exponential in the time complexity for $\mathcal{V}$ and we propose a natural conjecture that would lead to a polynomial time blowup instead. Corollaries include the decidability of half levels of the dot-depth hierarchy and the group-based concatenation hierarchy.