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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 68 (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
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...
Discrete Planes, Z²Actions, JacobiPerron Algorithm and Substitutions
 ANN. INST. FOURIER (GRENOBLE
, 2001
"... We introduce twodimensional substitutions generating twodimensional sequences related to discrete approximations of irrational planes. These twodimensional substitutions are produced by the classical JacobiPerron continued fraction algorithm, by the way of induction of a Z²action by rotations ..."
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We introduce twodimensional substitutions generating twodimensional sequences related to discrete approximations of irrational planes. These twodimensional substitutions are produced by the classical JacobiPerron continued fraction algorithm, by the way of induction of a Z²action by rotations on the circle. This gives a new geometric interpretation of the JacobiPerron algorithm, as a map operating on the parameter space of Z²actions by rotations.