# Complementation of rational sets on countable scattered linear orderings

@article{Rispal2005ComplementationOR, title={Complementation of rational sets on countable scattered linear orderings}, author={C. Rispal and Olivier Carton}, journal={Int. J. Found. Comput. Sci.}, year={2005}, volume={16}, pages={767-786} }

In a preceding paper (Bruyere and Carton, automata on linear orderings, MFCS'01), automata have been introduced for words indexed by linear orderings. These automata are a generalization of automata for finite, infinite, bi-infinite and even transfinite words studied by Buchi. Kleene's theorem has been generalized to these words. We prove that rational sets of words on countable scattered linear orderings are closed under complementation using an algebraic approach.

#### 28 Citations

A Kleene Theorem for Languages of Words Indexed by Linear Orderings

- Mathematics, Computer Science
- Developments in Language Theory
- 2005

In a preceding paper, Bruyere and the second author introduced automata, as well as rational expressions, which allow to deal with words indexed by linear orderings, and this paper extends this result to languages of Words indexed by all linear ordering. Expand

Automata on linear orderings

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It is proved that for countable scattered linear orderings, the two notions of finite automata and rational expressions are equivalent, which extends Kleene's theorem. Expand

Logic and Bounded-Width Rational Languages of Posets over Countable Scattered Linear Orderings

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It is proved that a language of labelled N-free posets is series-rational if and only if it is recognizable by a finite depth-nilpotent algebra and bounded-width and monadic second-order definable. Expand

Regular Languages of Words over Countable Linear Orderings

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- ICALP
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An algebraic model for recognizing languages of words indexed by countable linear orderings that is effectively equivalent to definability in monadic second-order (MSO) logic is developed. Expand

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- Computer Science, Mathematics
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The various acceptance modes of automata, and two algebraic proofs of McNaughton's theorem on the equivalence between Buchi and Muller automata are given. Expand

Algebraic Characterization of FO for Scattered Linear Orderings

- Mathematics, Computer Science
- CSL
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We prove that for the class of sets of words indexed by countable scattered linear orderings, there is an equivalence between definability in first-order logic, star-free expressions with marked… Expand

AN ALGEBRAIC APPROACH TO MSO-DEFINABILITY ON COUNTABLE LINEAR ORDERINGS

- Computer Science, Mathematics
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An algebraic notion of recognizability for languages of words indexed by countable linear orderings is developed and it is proved that this notion is effectively equivalent to definability in monadic second-order (MSO) logic. Expand

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- Computer Science
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The purpose of this book is to describe the algebraic approach to regular languages in a way that covers these extensions to structures beyond words, e.g. trees or graphs. Expand

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- Mathematics, Computer Science
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This paper gives decidable algebraic characterizations of several sublogics of monadic second- order logic over countable linear orderings, such as first-order logic, first-orders logic on cuts, weak monadicSecond-order Logic with cuts, as well as fragments of monads in which sets have to be well ordered or scattered. Expand

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It is extended that the star-free sets of finite words are exactly the languages recognized by finite aperiodic semigroups, and the variety theorem of Eilenberg for finite words is extended: there is a one-to-one correspondence between varieties of languages of words on countable scattered linear orderings and pseudo-varieties of algebras. Expand

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