Automata and differentiable words

We exhibit the construction of a deterministic automaton that, given k > 0, recognizes the (regular) language of k-differentiable words. Our approach follows a scheme of Crochemore et al. based on minimal forbidden words. We extend this construction to the case of C∞-words, i.e., words differentiable arbitrary many times. We thus obtain an infinite automaton for representing the set of C∞-words. We derive a classification of C∞-words induced by the structure of the automaton. Then, we introduce a new framework for dealing with C∞-words, based on a three letter alphabet. This allows us to define a compacted version of the automaton, that we use to prove that every C∞-word admits a repetition whose length is polynomially bounded.