Logic and Rational Languages of Words Indexed by Linear Orderings

Abstract

Abstract. 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 countable scattered linear orderings, but false in the general case. As a corollary we prove that the inclusion problem for rational languages of words indexed by countable linear orderings is decidable.

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

“…It is known from [2] that a language of scattered and countable linear structures is linear-rational if and only if it is in MSO [<]. When non-scattered linear structures are taken into account, a shuffle operation [4] must be added to linear-rational expressions to keep the equivalence with automata.…”

“…Later, Bruyère and Carton [5] introduced automata and equivalent rational expressions for words whose shape is a countable and scattered linear ordering. These automata and rational expressions were recently connected to MSO[<] by Bedon, Bès, Carton and Rispal [2]. Adding the notion of parallelism to the notion of sequentiality in words, Lodaya and Weil [14] defined automata, rational expressions and finite algebra for languages of particular finite partially ordered sets, obtained using the letters and closure under sequential and parallel composition.…”

Regular Languages of Words over Countable Linear Orderings

“…In [RC05], ◇-automata are proven to be closed under complementation with respect to countable and scattered orderings (a linear order is scattered if it is nowhere dense, namely, if none of its sub-orders is isomorphic to the rational line). More precisely, ◇-automata have the same expressive power as MSO logic over countable and scattered orderings [BBCR10]. However, it was already noticed in [BBCR10] that ◇-automata are strictly weaker than MSO logic over countable (possibly nonscattered) linear orderings: indeed, the closure under complementation fails as there is an automaton that accepts all words with non-scattered domains, whereas there is none for scattered words.…”

“…More precisely, ◇-automata have the same expressive power as MSO logic over countable and scattered orderings [BBCR10]. However, it was already noticed in [BBCR10] that ◇-automata are strictly weaker than MSO logic over countable (possibly nonscattered) linear orderings: indeed, the closure under complementation fails as there is an automaton that accepts all words with non-scattered domains, whereas there is none for scattered words.…”

On Context-Free Languages of Scattered Words

“…In the last decade, the theory of automata on well-ordered words has been extended to automata on all countable words, including scattered and dense words. In [2,3,12], both operational and logical characterizations of the class of languages of countable words recognized by finite automata were obtained.…”