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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 64 (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
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 10 (4 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.
Automata and Numeration Systems
"... This article is a short survey on the following problem: given a set X ` ..."
An analogue of the ThueMorse sequence
"... We consider the finite binary words Z(n), n ∈ N, defined by the following selfsimilar process: Z(0): = 0, Z(1): = 01, and Z(n + 1): = Z(n) · Z(n − 1), where the dot · denotes word concatenation, and w the word obtained from w by exchanging the zeros and the ones. Denote by Z(∞) = 01110100... the l ..."
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We consider the finite binary words Z(n), n ∈ N, defined by the following selfsimilar process: Z(0): = 0, Z(1): = 01, and Z(n + 1): = Z(n) · Z(n − 1), where the dot · denotes word concatenation, and w the word obtained from w by exchanging the zeros and the ones. Denote by Z(∞) = 01110100... the limiting word of this process, and by z(n) the n’th bit of this word. This sequence z is an analogue of the ThueMorse sequence. We show that a theorem of Bacher and Chapman relating the latter to a “Sierpiński matrix ” has a natural analogue involving z. The semiinfinite selfsimilar matrix which plays the role of the Sierpiński matrix here is the zeta matrix of the poset of finite subsets of N without two consecutive elements, ordered by inclusion. We observe that this zeta matrix is nothing but the exponential of the incidence matrix of the Hasse diagram of this poset. We prove that the corresponding Möbius matrix has a simple expression in terms of the zeta matrix and the sequence z. 1