A semigroup characterization of dot-depth one languages

Knast, Robert

doi:10.1051/ita/1983170403211

Cited by 86 publications

(69 citation statements)

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“…The syntactic characterization of these languages was settled by Knast [24,25]. More generally, one can show that the languages of dot-depth n form a +-variety of languages.…”

Section: Back To the Early Attemptsmentioning

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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“…The syntactic characterization of these languages was settled by Knast [24,25]. More generally, one can show that the languages of dot-depth n form a +-variety of languages.…”

Section: Back To the Early Attemptsmentioning

confidence: 99%

Finite Semigroups and Recognizable Languages: An Introduction

Pin¹

1995

Semigroups, Formal Languages and Groups

View full text Add to dashboard Cite

show abstract

“…We conjecture that the variety of semigroups (resp. monoids) corresponding to BPol V is B 1 M V. The conjecture was proved to be true if V is the trivial variety of monoids, the trivial variety of semigroups or the variety of monoids consisting of all groups [32,10,12]. Note also that every language of BPol V is recognized by a semigroup of B 1 M V. Finally, it is proved in [21] that the identities of B 1 M V are (x ω py ω qx ω ) ω x ω py ω sx ω (x ω ry ω sx ω ) ω = (x ω py ω qx ω ) ω (x ω ry ω sx ω ) ω for all x, y, p, q, r, s ∈Â * for some finite alphabet A such that V satisfies x 2 = x = y = p = q = r = s.…”

Section: The Sequential Calculusmentioning

confidence: 99%

Polynomial closure and unambiguous product

Weil

1997

Theory of Computing Systems

123

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“…We conjecture that the variety of semigroups (resp. monoids) corresponding to BPol V is B 1 M V. The conjecture was proved to be true if V is the trivial variety of monoids, the trivial variety of semigroups or the variety of monoids consisting of all groups [32,10,12]. Note also that every language of BPol V is recognized by a semigroup of for all x, y, p, q, r, s ∈Â * for some finite alphabet A such that V satisfies x 2 = x = y = p = q = r = s.…”

Section: The Sequential Calculusmentioning

confidence: 99%

Polynomial closure and unambiguous product

Weil

1995

Automata, Languages and Programming

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A semigroup characterization of dot-depth one languages

Cited by 86 publications

References 4 publications

Finite Semigroups and Recognizable Languages: An Introduction

Finite Semigroups and Recognizable Languages: An Introduction

Polynomial closure and unambiguous product

Polynomial closure and unambiguous product

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