The First-Order Theory of Binary Overlap-Free Words is Decidable

We show that the first-order logical theory of the binary overlap-free words (and, more generally, the $\alpha$-free words for rational $\alpha$, $2<\alpha \le 7\mathrm{/}3$), is decidable. As a consequence, many results previously obtained about this class through tedious case- based proofs can now be proved “automatically”, using a decision procedure.