Transition graphs and the star-height of regular events.

Eggan, L. C.

doi:10.1307/mmj/1028998975

Cited by 130 publications

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“…Theorem 3(1) is known as Eggan's Theorem [7] and proved in its present form by Cohen [6]. Theorem 3(3) is due to McNaughton [25].…”

Section: Lemma 2 ([625]) If a Graph H Is A Minor Of A Graph G Thenmentioning

confidence: 94%

Succinctness of Regular Expressions with Interleaving, Intersection and Counting

Gelade

Lecture Notes in Computer Science

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a b s t r a c tIn this paper, we study the succinctness of regular expressions (REs) extended with interleaving, intersection and counting operators. We show that in a translation from REs with interleaving to standard regular expressions a double exponential size increase cannot be avoided. We also consider the complexity of translations to finite automata. We give a tight exponential lower bound on the translation of REs with intersection to NFAs, and, for each of the three classes of REs, we show that in a translation to a DFA a double exponential size increase cannot be avoided. Together with known results, this gives a complete picture of the complexity of translating REs extended with interleaving, intersection or counting into (standard) regular expressions, NFAs, and DFAs.

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“…Theorem 3(1) is known as Eggan's Theorem [7] and proved in its present form by Cohen [6]. Theorem 3(3) is due to McNaughton [25].…”

Section: Lemma 2 ([625]) If a Graph H Is A Minor Of A Graph G Thenmentioning

confidence: 94%

Succinctness of Regular Expressions with Interleaving, Intersection and Counting

Gelade

Lecture Notes in Computer Science

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“…It is easy to show that (1,2) in the definition of k . In order to verify (3), we have to show for every π ∈ X, ∆(π) > K. This follows from Corollary 1(2) and the length assumption on w.…”

Section: Lemma 6 [6] Let W ∈ Smentioning

confidence: 99%

Distance Desert Automata and the Star Height One Problem

Kirsten

2004

Lecture Notes in Computer Science

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Abstract. We show that it is decidable in time complexity 2whether the language accepted by an n-state non-deterministic automaton is of star height one, which is the first ever complexity result for the star height one problem. To achieve this, we introduce distance desert automata as a joint generalization of distance automata and desert automata, and show the decidability of its limitedness problem by solving the underlying Burnside problem.

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“…The star height of a recognizable language is obtained by considering rational expressions instead of extended rational expressions [15].…”

Section: Early Attempts To Classify Recognizable Languagesmentioning

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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Transition graphs and the star-height of regular events.

Cited by 130 publications

References 0 publications

Succinctness of Regular Expressions with Interleaving, Intersection and Counting

Succinctness of Regular Expressions with Interleaving, Intersection and Counting

Distance Desert Automata and the Star Height One Problem

Finite Semigroups and Recognizable Languages: An Introduction

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