On the context-freeness of the set of words containing overlaps
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It is shown that 2+-repetition, i.e. a word of the form uvuvu where u is a letter and v is a word, is the smallest repetition which can be avoided in infinite words over binary alphabet. Such binary words avoiding pattern uvuvu, finite or infinite, are called as 2+-free words and those words are the main topic of this work. It is shown here that 2+-free words over binary alphabet can be presented as words built from special kind of blocks, called Morse-blocks, with some rules. In particular, the given presentation by these blocks is unique for 2+-free words long enough. Moreover, it is also shown that the language generated by this presentation can be described by some automaton. In fact, the corresponding presentation in blocks for finite 2+-free words can be generated by a modified version of cross-product of two gsm''s. On the other hand, for infinite 2+-free words, one gsm is enough. Also, a little more accurate bounds for the count of 2+-free words of a given length are given here.