A Kleene Theorem for Languages of Words Indexed by Linear Orderings [chapter]

Alexis Bès, Olivier Carton
2005 Lecture Notes in Computer Science  
In a preceding paper, Bruyère and Carton introduced automata, as well as rational expressions, which allow to deal with words indexed by linear orderings. A Kleene-like theorem was proved for words indexed by countable scattered linear orderings. In this paper we extend this result to languages of words indexed by all linear orderings. Words and rational expressions Given a finite alphabet A, a word (a j ) j∈J is a function from J to A which maps any element j of J to a letter a j of A. We say
more » ... hat J is the length |x| of the word x. For instance, the empty word ε is indexed by the empty linear ordering J = ∅. Usual finite words are the words indexed by finite orderings J = {1, 2, . . . , n}, n ≥ 0. A word of length J = ω is usually called an ω-word or an infinite word. A word of length ζ = −ω +ω is a sequence . . . a −2 a −1 a 0 a 1 a 2 . . . of letters which is usually called a bi-infinite word. The sum operation on linear orderings leads to a notion of product of words as follows. Let J and K j for j ∈ J, be linear orderings. Let x j = (a k,j ) k∈Kj be a word of length K j , for any j ∈ J. The product j∈J x j is the word z of
doi:10.1007/11505877_14 fatcat:lpwwn7tidzg3zjrlmo7vkgnjoy