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

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