This talk is the first in a series of lecture intended to describe the main ideas and the use of automata limitedness and related questions.

The goal of the first session is to present the famous (restricted) star-height problem, and show the first steps toward its resolution. The question is the following: given a regular language L and a non-negative integer k, can we decide whether L can bee represented as a regular expression of star-height (i.e., nesting of Kleene stars) at most k. This problem has been open for 25 years unit Hashiguchi provided a proof in a series of four paper, between 81 and 88. This proof is notoriously difficult, and the presentation here will be based on the ideas in the more modern, and much simpler, proof of Kirsten (2005).

The idea behind both these proofs is to reduce the original problem to a question of existence of bounds for a function computed by some specific forms of automata (distance automata, nested-distance desert automata, B-automata, …). This idea goes much beyond the scope of the star-height problem, and I will try to convey this idea through several examples.