The dot-depth hierarchy, 45 years later Dedicated to Janusz A. Brzozowski for his 80th birthday

In 1970, R. S. Cohen and Janusz A. Brzozowski introduced a hierarchy of star-free languages called the dot-depth hierarchy. This hierarchy and its generalisations, together with the problems attached to them, had a long-lasting influence on the development of automata theory. This survey article reports on the numerous results and conjectures attached to this hierarchy. This paper is a follow-up of the survey article Open problems about regular languages, 35 years later [57]. The dot-depth hierarchy, also known as Brzozowski hierarchy, is a hierarchy of star-free languages first introduced by Cohen and Brzozowski [25] in 1971. It immediately gave rise to many interesting questions and an account of the early results can be found in Brzozowski’s survey [20] from 1976. 1 Terminology, notation and background Most of the terminology used in this paper was introduced in [57]. We just complete these definitions by giving the ordered versions of the notions of syntactic monoid and variety of finite monoids. 1.1 Syntactic order and positive varieties An ordered monoid is a monoid equipped with an order 6 compatible with the multiplication: x 6 y implies zx 6 zy and xz 6 yz. The syntactic preorder of a language L of A is the relation 6L defined on A by u 6L v if and only if, for every x, y ∈ A ,