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.

The avoidability of binary patterns by binary cube-free words is investigated and the exact bound between unavoidable and avoidable patterns is found. All avoidable patterns are shown to be D0L-avoidable. For avoidable patterns, the growth rates of the avoiding languages are studied. All such languages, except for the overlap-free language, are proved to have exponential growth. The exact growth rates of languages avoiding minimal avoidable patterns are approximated through computer-assisted upper bounds. Finally, a new example of a pattern-avoiding language of polynomial growth is given.

We consider the position and number of occurrences of squares in the Thue-Morse sequence, and show that the corresponding sequences are 2-regular. We also prove that changing any finite but nonzero number of bits in the Thue-Morse sequence creates an overlap, and any linear subsequence of the Thue-Morse sequence (except those corresponding to decimation by a power of 2) contains an overlap.