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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… (More)

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.

We show that a special case of the Feferman-Vaught composition theorem gives rise to a natural notion of automata for finite words over an infinite alphabet, with good closure and decidability properties, as well as several logical characterizations. We also consider a slight extension of the Feferman-Vaught formalism which allows to express more relations… (More)

Let M = (A, <, P) where (A, <) is a linear ordering and P denotes a finite sequence of monadic predicates on A. We show that if A contains an interval of order type ω or −ω, and the monadic second-order theory of M is decidable, then there exists a non-trivial expansion M ′ of M by a monadic predicate such that the monadic second-order theory of M ′ is… (More)

We study expansions of the Weak Monadic Second Order theory of (N, <) by cardinality relations, which are predicates R(X1,. .. , Xn) whose truth value depends only on the cardinality of the sets X1,. .. , Xn. We first provide a (definable) criterion for definability of a cardinality relation in (N, <), and use it to prove that for every cardinality relation… (More)

Rationals and countable ordinals are important examples of structures with decidable monadic second-order theories. A chain is an expansion of a linear order by monadic predicates. We show that if the monadic second-order theory of a countable chain C is decidable then C has a non-trivial expansion with decidable monadic second-order theory.