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Logic and precognizable sets of integers
 Bull. Belg. Math. Soc
, 1994
"... We survey the properties of sets of integers recognizable by automata when they are written in pary expansions. We focus on Cobham’s theorem which characterizes the sets recognizable in different bases p and on its generalization to N m due to Semenov. We detail the remarkable proof recently given ..."
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Cited by 92 (4 self)
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We survey the properties of sets of integers recognizable by automata when they are written in pary expansions. We focus on Cobham’s theorem which characterizes the sets recognizable in different bases p and on its generalization to N m due to Semenov. We detail the remarkable proof recently given by Muchnik for the theorem of CobhamSemenov, the original proof being published in Russian. 1
On iterating linear transformations over recognizable sets of integers
 Theoretical Computer Science
"... It has been known for a long time that the sets of integer vectors that are recognizable by finitestate automata are those that can be defined in an extension of Presburger arithmetic. In this paper, we address the problem of deciding whether the closure of a linear transformation preserves the re ..."
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Cited by 20 (2 self)
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It has been known for a long time that the sets of integer vectors that are recognizable by finitestate automata are those that can be defined in an extension of Presburger arithmetic. In this paper, we address the problem of deciding whether the closure of a linear transformation preserves the recognizable nature of sets of integer vectors. We solve this problem by introducing an original extension of the concept of recognizability to sets of vectors with complex components. This generalization allows to obtain a simple necessary and sufficient condition over linear transformations, in terms of the eigenvalues of the transformation matrix. We then show that these eigenvalues do not need to be computed explicitly in order to evaluate the condition, and we give a full decision procedure based on simple integer arithmetic. The proof of this result is constructive, and can be turned into an algorithm for applying the closure of a linear transformation that satisfies the condition to a finitestate representation of a set. Finally, we show that the necessary and sufficient condition that we have obtained can straightforwardly be turned into a sufficient condition for linear transformations with linear guards. Key words: automata, iterations, Presburger arithmetic, recognizable sets of integers
Automatabased presentations of infinite structures
, 2009
"... The model theory of finite structures is intimately connected to various fields ..."
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Cited by 6 (0 self)
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The model theory of finite structures is intimately connected to various fields
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 secondorder 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 in ..."
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Cited by 5 (2 self)
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We prove that every rational language of words indexed by linear orderings is definable in monadic secondorder 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 boundedwidth rational languages of posets over countable scattered linear orderings
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Automata and Numeration Systems
"... This article is a short survey on the following problem: given a set X ` ..."
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