# The Dot-Depth Hierarchy, 45 Years Later

@inproceedings{Pin2017TheDH, title={The Dot-Depth Hierarchy, 45 Years Later}, author={Jean-{\'E}ric Pin}, booktitle={The Role of Theory in Computer Science}, year={2017} }

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…

## 28 Citations

### The complexity of separation for levels in concatenation hierarchies

- Computer ScienceFSTTCS
- 2018

It is built on results to show that when the alphabet is fixed, there are polynomial time algorithms for both levels of the famous Straubing-Th\'erien hierarchy.

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

- MathematicsSemigroup Forum
- 2019

This is a tribute to the Brazilian mathematician Imre Simon, with emphasis on three results that had a considerable influence on the development of automata and semigroup theory. Imre Simon, a…

### Separation for dot-depth two

- Computer Science2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
- 2017

The dot-depth hierarchy of Brzozowski and Cohen is a classification of all first-order definable languages: each level contains languages that can be defined with a prescribed number of quantifier blocks.

### The Covering Problem

- Computer ScienceLog. Methods Comput. Sci.
- 2018

The main contribution of this paper is a suitable candidate to play this role: the Covering Problem, which admits an elementary set theoretic formulation, similar to membership, and develops a mathematical framework and a methodology tailored to the investigation of this problem.

### Separation and covering for group based concatenation hierarchies

- Computer Science2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
- 2019

Using a generic approach, it is shown that for any concatenation hierarchy, if separation is decidable for the basis, then it isdecidable as well for levels 1/2, 1 and 3/2 (the authors actually solve a more general problem called covering).

### The Power of Programs over Monoids in DA

- MathematicsMFCS
- 2017

It is proved that the class known as DA satisfies tameness and hence that the regular languages recognized by programs over monoids in DA are precisely those recognizable in the classical sense by morphisms from QDA.

### The Covering Problem: A Unified Approach for Investigating the Expressive Power of Logics

- Computer ScienceMFCS
- 2016

The main contribution of this paper is a suitable candidate to play this role: the Covering Problem, which admits an elementary set theoretic formulation, similar to membership, and develops a mathematical framework as well as a methodology tailored to the investigation of this problem.

### The Power of Programs over Monoids in

- Computer ScienceLATA
- 2020

It is shown that those programs actually can recognise all languages from a class of restricted dot-depth one languages, using a non-trivial trick, and conjecture that this class suffices to characterise the regular languages recognised by programs over monoids in .

### Group separation strikes back

- Mathematics, Computer ScienceArXiv
- 2022

It is proved that covering, which generalizes separation, is decidable and also that covering is also decidable for two strict sub- classes: languages recognized by commutative groups, and modulo languages .

### The Regular Languages of First-Order Logic with One Alternation

- MathematicsLICS
- 2022

The regular languages with a neutral letter expressible in first-order logic with one alternation are characterized. Specifically, it is shown that if an arbitrary Σ2 formula defines a regular…

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