Open Problems About Regular Languages, 35 Years Later

Abstract

In December 1979, Janusz A. Brzozowski presented a selection of six open problems about regular languages and mentioned two other problems in the conclusion of his article. These problems have been the source of some of the greatest breakthroughs in automata theory over the past 35 years. This survey article summarizes the state of the art on these questions and the hopes for the next 35 years. Thirty-five years ago, at a conference in Santa Barbara in December 1979, Janusz A. Brzozowski [8] presented a selection of six open problems about regular languages and mentioned two other topics in the conclusion of his article. These six open problems were, in order, star height, restricted star height, group complexity, star removal, regularity of non-counting classes and optimality of prefix codes. The two other topics were the limitedness problem and the dotdepth hierarchy. These problems proved to be very influential in the development of automata theory and were the source of critical breakthroughs. The aim of this paper is to survey these results, to describe their impact on current research and to outline some hopes for the next thirty-five years. Due to the lack of space, the dot-depth hierarchy is treated in a separate article [61].

Generic Results for Concatenation Hierarchies

“…there exists a regular language of generalized star-height greater than 1. In other terms, membership is open for the class of languages of generalized star-height 1 (see [21] for a historical presentation).…”

The influence of Imre Simon’s work in the theory of automata, languages and semigroups

“…Simon was led to his factorisation forest theorem in connection with his research on three topics: Brown's theorem on locally finite semigroups, the limitedness problem for the tropical semiring [34,49,51,56] and the star-height problem. Despite their importance, I will not develop further these topics, which are already covered in the historical survey [40].…”

The Dot-Depth Hierarchy, 45 Years Later

“…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.…”

“…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.…”

“…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.…”