Automata and Differentiable Words

TL;DR

This paper constructs automata recognizing k-differentiable and infinitely differentiable words, providing a new framework and classification for C-infinity-words, and demonstrating polynomial bounds on repetitions within these words.

Contribution

It introduces a novel automaton construction for differentiable words and extends it to infinite differentiability, along with a new framework and classification for C-infinity-words.

Findings

Automaton recognizing k-differentiable words constructed

Automaton extended to C-infinity-words

Polynomial bounds on repetitions in C-infinity-words proved

Abstract

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\infinity-words, i.e., words differentiable arbitrary many times. We thus obtain an infinite automaton for representing the set of C\infinity-words. We derive a classification of C\infinity-words induced by the structure of the automaton. Then, we introduce a new framework for dealing with \infinity-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\infinity-word admits a repetition in C\infinity whose length is polynomially bounded.