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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 70 (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 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 13 (4 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
Generalization of automatic sequences for numeration systems on a regular language
, 1999
"... Let L be an infinite regular language on a totally ordered alphabet (Σ, <). Feeding a finite deterministic automaton (with output) with the words of L enumerated lexicographically with respect to < leads to an infinite sequence over the output alphabet of the automaton. This process generalize ..."
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Cited by 11 (5 self)
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Let L be an infinite regular language on a totally ordered alphabet (Σ, <). Feeding a finite deterministic automaton (with output) with the words of L enumerated lexicographically with respect to < leads to an infinite sequence over the output alphabet of the automaton. This process generalizes the concept of kautomatic sequence for abstract numeration systems on a regular language (instead of systems in base k). Here, I study the first properties of these sequences and their relations with numeration systems.
Regular Maps in Generalized Number Systems
, 2000
"... This paper extends some results of Allouche and Shallit for qregular sequences to numeration systems in algebraic number fields and to linear numeration systems. We also construct automata that perform addition and multiplication by a fixed number. 1 Introduction A sequence is called qautomat ..."
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Cited by 2 (1 self)
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This paper extends some results of Allouche and Shallit for qregular sequences to numeration systems in algebraic number fields and to linear numeration systems. We also construct automata that perform addition and multiplication by a fixed number. 1 Introduction A sequence is called qautomatic if its nth term can be generated by a finite state machine from the qary digits of n. The concept of automatic sequences was introduced in 1969 and 1972 by Cobham [8, 9]. In 1979 Christol [6] (see also Christol, Kamae, Mend`es France and Rauzy [7]) discovered a nice arithmetic property of automatic sequences: a sequence with values in a finite field of characteristic p is pautomatic if and only if the corresponding power series is algebraic over the field of rational functions over this finite field. A brief survey on this subject is given in [2], see also [10]. Some generalizations of this concept were studied in [27, 23, 24, 3], see also the survey [1]. An automatic sequence has to t...
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...