100 research outputs found
Complementation of Rational sets on Scattered Linear Orderings of Finite Rank
International audienceIn a preceding paper [1], 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 show that deterministic automata do not have the same expressive power. Despite this negative result, we prove that rational sets of words of finite ranks are closed under complementation
Complementation of Rational Sets on Scattered Linear Orderings of Finite Rank
International audienceIn a preceding paper, automata have been introduced for words indexed by linear orderings. These automata are a generalization of automata for finite, infinite, bi-finite and even transfinite words studied by Buchi Kleene's theorem has been generalized to these words. We show that deterministic automata do not have the same expressive power. Despite this negative result, we prove that rational sets of words of finite ranks are closed under complementation
Complementation of Rational Sets on Countable Scattered Linear Orderings
In a preceding paper (Bruyère 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 Büchi. Kleene's theorem has been generalized to these words. We prove that rational sets of words on countable scattered linear orderings are closed under complementation using an algebraic approach
On factorisation forests
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. The purpose of this paper is to
illustrate the importance of this approach in the context of automata over
infinite words and trees. We extend the theorem of factorisation forest in two
directions: we show that it is still valid for any word indexed by a linear
ordering; and we show that it admits a deterministic variant for words indexed
by well-orderings. A byproduct of this work is also an improvement on the known
bounds for the original result. We apply the first variant for giving a
simplified proof of the closure under complementation of rational sets of words
indexed by countable scattered linear orderings. We apply the second variant in
the analysis of monadic second-order logic over trees, yielding new results on
monadic interpretations over trees. Consequences of it are new caracterisations
of prefix-recognizable structures and of the Caucal hierarchy.Comment: 27 page
An Algebraic Approach to Mso-Definability on Countable Linear Orderings
We develop an algebraic notion of recognizability for languages of words indexed by countable linear orderings. We prove that this notion is effectively equivalent to definability in monadic second-order (MSO) logic. We also provide three logical applications. First, we establish the first known collapse result for the quantifier alternation of MSO logic over countable linear orderings. Second, we solve an open problem posed by Gurevich and Rabinovich, concerning the MSO-definability of sets of rational numbers using the reals in the background. Third, we establish the MSO-definability of the set of yields induced by an MSO-definable set of trees, confirming a conjecture posed by Bruyère, Carton, and Sénizergues
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
Laver and set theory
In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.Accepted manuscrip
A non-regular language of infinite trees that is recognizable by a sort-wise finite algebra
-clones are multi-sorted structures that naturally emerge as algebras
for infinite trees, just as -semigroups are convenient algebras for
infinite words. In the algebraic theory of languages, one hopes that a language
is regular if and only if it is recognized by an algebra that is finite in some
simple sense. We show that, for infinite trees, the situation is not so simple:
there exists an -clone that is finite on every sort and finitely
generated, but recognizes a non-regular language
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