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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 66 (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
A characterization of substitutive sequences using return words. Discrete Mathematics
, 1998
"... We prove that a sequence is primitive substitutive if and only if the set of its derived sequences is finite; we defined these sequences here. ..."
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Cited by 59 (7 self)
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We prove that a sequence is primitive substitutive if and only if the set of its derived sequences is finite; we defined these sequences here.
A generalization of Cobham’s theorem
 Theory Comput. Syst
, 1998
"... Abstract If a nonperiodic sequence X is the image by a morphism of a fixed point of both a primitive substitution σ and a primitive substitution τ, then the dominant eigenvalues of the matrices of σ and of τ are multiplicatively dependent. This is the way we propose to generalize Cobham’s Theorem. ..."
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Cited by 11 (3 self)
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Abstract If a nonperiodic sequence X is the image by a morphism of a fixed point of both a primitive substitution σ and a primitive substitution τ, then the dominant eigenvalues of the matrices of σ and of τ are multiplicatively dependent. This is the way we propose to generalize Cobham’s Theorem. 1
Bertrand Numeration Systems and Recognizability
, 1995
"... . There exist various wellknown characterizations of sets of numbers recognizable by a finite automaton, when they are represented in some integer base p 2. We show how to modify these characterizations, when integer bases p are replaced by linear numeration systems whose characteristic polynomial ..."
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. There exist various wellknown characterizations of sets of numbers recognizable by a finite automaton, when they are represented in some integer base p 2. We show how to modify these characterizations, when integer bases p are replaced by linear numeration systems whose characteristic polynomial is the minimal polynomial of a Pisot number. We also prove some related interesting properties. 1 Introduction Since the work of [9], sets of integers recognizable by finite automata have been studied in numerous papers. One of the jewels in this topic is the famous Cobham's theorem [11]: the only sets of numbers recognizable by finite automata, independently of the base of representation, are those which are ultimately periodic. Other studies are concerned with computation models equivalent to finite automata in the recognition of sets of integers. The proposed models use firstorder logical formulae [9], uniform substitutions [12], algebraic formal series [10]. We refer the reader to the...