Hierarchies of aperiodic languages

Brzozowski, Janusz

doi:10.1051/ita/197610r200331

Cited by 39 publications

(28 citation statements)

References 14 publications

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“…Therefore, in the rest of the paper we consider languages as subsets of A + . For an excellent and gênerai présentation of problems related to this paper the reader is referred to Brzozowski's survey paper [1] or the above mentioned monograph of Eilenberg [4].…”

Section: Any Subset Ofmentioning

confidence: 99%

A semigroup characterization of dot-depth one languages

Knast¹

1983

RAIRO. Inform. théor.

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Section: Any Subset Ofmentioning

confidence: 99%

A semigroup characterization of dot-depth one languages

Knast¹

1983

RAIRO. Inform. théor.

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“…As product is often denoted by a dot, Brzozowski defined the "dot-depth" of languages of the free semigroup [5]. Later on, Thérien (implicitly) and Straubing (explicitly) introduced a similar notion (often called the concatenation level in the literature) for the languages of the free monoid.…”

Section: Theorem 52 There Is An Algorithm To Determine the Star Heigmentioning

confidence: 99%

Finite Semigroups and Recognizable Languages: An Introduction

Pin¹

1995

Semigroups, Formal Languages and Groups

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This paper is an attempt to share with a larger audience some modern developments in the theory of finite automata. It is written for the mathematician who has a background in semigroup theory but knows next to nothing on automata and languages. No proofs are given, but the main results are illustrated by several examples and counterexamples.What is the topic of this theory? It deals with languages, automata and semigroups, although recent developments have shown interesting connections with model theory in logic, symbolic dynamics and topology. Historically, in their attempt to formalize natural languages, linguists such as Chomsky gave a mathematical definition of natural concepts such as words, languages or grammars: given a finite set A, a word on A is simply an element of the free monoid on A, and a language is a set of words. But since scientists are fond of classifications of all sorts, language theory didn't escape to this mania. Chomsky established a first hierarchy, based on his formal grammars. In this paper, we are interested in the recognizable languages, which form the lower level of the Chomsky hierarchy. A recognizable language can be described in terms of finite automata while, for the higher levels, more powerful machines, ranging from pushdown automata to Turing machines, are required. For this reason, problems on finite automata are often under-estimated, according to the vague -but totally erroneous -feeling that "if a problem has been reduced to a question about finite automata, then it should be easy to solve".Kleene's theorem [23] is usually considered as the foundation of the theory. It shows that the class of recognizable languages (i.e. recognized by finite automata), coincides with the class of rational languages, which are given by rational expressions. Rational expressions can be thought of as a generalization of polynomials involving three operations: union (which plays the role of addition), product and star operation. An important corollary of Kleene's theorem is that rational languages are closed under complement. In the sixties, several classification schemes for the rational languages were proposed, based on the * From 1st Sept 1993, LITP, Université Paris VI, Tour 55-65, 4 Place Jussieu, 75252 Paris Cedex 05, FRANCE.

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“…Three concatenation hierarchies of rational languages have been considered since the 1970's: the dot-depth hierarchy introduced by Brzozowski [10], the Straubing-Thérien hierarchy [31,32], and the group hierarchy presented in [18]. All these hierarchies may be regarded as being indexed by half integers, that is numbers of the form n or n+1/2 where n is a non-negative integer, and where each level is obtained in the following way: the languages of level n + 1/2 are finite unions of products of the form L 0 a 1 L 1 .…”

Section: Introductionmentioning

confidence: 99%

The globals of pseudovarieties of ordered semigroups containingB₂and an application to a problem proposed by Pin

Almeida

Escada

2005

RAIRO-Theor. Inf. Appl.

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Given a basis of pseudoidentities for a pseudovariety of ordered semigroups containing the 5-element aperiodic Brandt semigroup B 2 , under the natural order, it is shown that the same basis, over the most general graph over which it can be read, defines the global. This is used to show that the global of the level 3/2 of the refinement of Straubing-Thérien's concatenation hierarchy introduced by Pin and Weil has infinite vertex rank.

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Hierarchies of aperiodic languages

Cited by 39 publications

References 14 publications

A semigroup characterization of dot-depth one languages

A semigroup characterization of dot-depth one languages

Finite Semigroups and Recognizable Languages: An Introduction

The globals of pseudovarieties of ordered semigroups containingB₂and an application to a problem proposed by Pin

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Hierarchies of aperiodic languages

Cited by 39 publications

References 14 publications

A semigroup characterization of dot-depth one languages

A semigroup characterization of dot-depth one languages

Finite Semigroups and Recognizable Languages: An Introduction

The globals of pseudovarieties of ordered semigroups containingB2and an application to a problem proposed by Pin

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The globals of pseudovarieties of ordered semigroups containingB₂and an application to a problem proposed by Pin