The first-order theory of binary overlap-free words is decidable

Abstract 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 previously obtained about this class through tedious case-based proofs can now be proved “automatically,” using a decision procedure, and new claims can be proved or disproved simply by restating them as logical formulas.