# Complementation of rational sets on countable scattered linear orderings

@article{Rispal2004ComplementationOR, title={Complementation of rational sets on countable scattered linear orderings}, author={Chloe Rispal and Olivier Carton}, journal={Int. J. Found. Comput. Sci.}, year={2004}, 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 ordering are closed under complementation using an algebraic approach.

## 34 Citations

### A Kleene Theorem for Languages of Words Indexed by Linear Orderings

- MathematicsInt. J. Found. Comput. Sci.
- 2006

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.

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

- Mathematics, Computer ScienceLFCS
- 2009

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.

### Logic and Rational Languages of Words Indexed by Linear Orderings

- MathematicsCSR
- 2008

We prove that every rational language of words indexed by linear orderings is definable in monadic second-order logic. We also show that the converse is true for the class of languages indexed by…

### Logic and Rational Languages of Words Indexed by Linear Orderings

- MathematicsTheory of Computing Systems
- 2009

We prove that every rational language of words indexed by linear orderings is definable in monadic second-order logic. We also show that the converse is true for the class of languages indexed by…

### Factorisation forests for infinite words application to countable scattered linear orderings

- Mathematics
- 2007

The theorem of factorisation forests shows the existence of nested factorisations -- a la Ramsey -- for finite words. This theorem has important applications in semigroup theory, and beyond.
We…

### Tree Automata and Automata on Linear Orderings

- MathematicsRAIRO Theor. Informatics Appl.
- 2009

It is shown that the inclusion problem is decidable for rational languages of words indexed by scattered countable linear orderings and a reduction to the Monadic Second Order theory of the infinite binary tree is leaned on.

### Factorization forests for infinite words and applications to countable scattered linear orderings

- Mathematics, Computer ScienceTheor. Comput. Sci.
- 2010

### Regular Languages of Words over Countable Linear Orderings

- MathematicsICALP
- 2011

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.

### Automata and semigroups recognizing infinite words

- Mathematics, Computer ScienceLogic and Automata
- 2008

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

### Algebraic Characterization of FO for Scattered Linear Orderings

- MathematicsCSL
- 2011

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…

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