Alexis Bès

Logic and Rational Languages of Words Indexed by Linear Orderings

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.

An Application of the Feferman-Vaught Theorem to Automata and Logics for Words over an Infinite Alphabet

We show that a special case of the Feferman-Vaught composition theorem givesnrise to a natural notion of automata for finite words over an infinitenalphabet, with good closure and decidability properties, as well as severalnlogical characterizations. We also consider a slight extension of thenFeferman-Vaught formalism which allows to express more relations betweenncomponent values (such as equality), and prove related decidability results.n From this result we get new classes of decidable logics for words over anninfinite alphabet.

Undecidable Extensions of Buchi Arithmetic and Cobham-Semenov Theorem

Let k and I be two multiplicatively independent integers, and let L C N be a I-recognizable set which is not definable in (N; +). We prove that the elementary theory of (N; +, Vk, L), where Vk (x) denotes the greatest power of k dividing x, is undecidable. This result leads to a new proof of the CobhamSemenov theorem. ?

An Extension of the Cobham-Semenov Theorem

Let θ, θ′ be two multiplicatively independent Pisot numbers, and let U , U ′ be two linear numeration systems whose characteristic polynomial is the minimal polynomial of θ and θ′, respectively. For every n ≥ 1, if A ⊆ ℕ n is U -and U ′ -recognizable then A is definable in 〈ℕ: + 〉.

A kleene theorem for languages of words indexed by linear orderings

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. 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.

Logic and rational languages of words indexed by linear orderings

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.

On Pascal triangles modulo a prime power

Abstract In the first part of the paper we study arithmetical properties of Pascal triangles modulo a prime power; the main result is the generalization of Lucas theorem. Then we investigate the structure 〈N; Bpx〉, where p is a prime, α is an integer greater than one, and B p x (x, y) = Rem (( x+y x ), p x ) ; it is shown that addition is first-order definable in this structure, and that its elementary theory is decidable.

Algebraic Characterization of FO for Scattered Linear Orderings

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 product, and recognizability by finite aperiodic semigroups which satisfy some additional equation.

Decidable Expansions of Labelled Linear Orderings

Consider a linear ordering equipped with a finite sequence of monadicnpredicates. If the ordering contains an interval of order type omega orn-omega, and the monadic second-order theory of the combined structure isndecidable, there exists a non-trivial expansion by a further monadic predicatenthat is still decidable.