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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
Numeration systems, linear recurrences, and regular sets
 Inform. and Comput
, 1994
"... A numeration system based on a strictly increasing sequence of positive integers u0 = 1, u1, u2,... expresses a nonnegative integer n as a sum n = � i j=0 ajuj. In this case we say the string aiai−1 · · · a1a0 is a representation for n. If gcd(u0, u1,...) = g, then every sufficiently large mult ..."
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Cited by 35 (4 self)
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A numeration system based on a strictly increasing sequence of positive integers u0 = 1, u1, u2,... expresses a nonnegative integer n as a sum n = � i j=0 ajuj. In this case we say the string aiai−1 · · · a1a0 is a representation for n. If gcd(u0, u1,...) = g, then every sufficiently large multiple of g has some representation. If the lexicographic ordering on the representations is the same as the usual ordering of the integers, we say the numeration system is orderpreserving. In particular, if u0 = 1, then the greedy representation, obtained via the greedy algorithm, is orderpreserving. We prove that, subject to some technical assumptions, if the set of all representations in an orderpreserving numeration system is regular, then the sequence u = (uj)j≥0 satisfies a linear recurrence. The converse, however, is not true. The proof uses two lemmas about regular sets that may be of independent interest. The first shows that if L is regular, then the set of lexicographically greatest strings of every length in L is also regular. The second shows that the number of strings of length n in a regular language L is bounded by a constant (independent of n) iff L is the finite union of sets of the form xy ∗ z. 1
L systems
 In G. Rozenberg and A. Salomaa (Eds.), Handbook of Formal Languages
, 1997
"... 1.1 Parallel rewriting L systems are parallel rewriting systems which were originally introduced in 1968 to model the development of multicellular organisms, [L1]. The basic ideas gave rise to an abundance of languagetheoretic problems, both mathematically ..."
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Cited by 20 (3 self)
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1.1 Parallel rewriting L systems are parallel rewriting systems which were originally introduced in 1968 to model the development of multicellular organisms, [L1]. The basic ideas gave rise to an abundance of languagetheoretic problems, both mathematically
On vanishing coefficients of algebraic power series over fields of positive characteristic
, 2012
"... ..."
On the Periodicity of Morphic Words
"... Abstract. Given a morphism h prolongable on a and an integer p, we present an algorithm that calculates which letters occur infinitely often in congruent positions modulo p in the infinite word h ω (a). As a corollary, we show that it is decidable whether a morphic word is ultimately pperiodic. Mor ..."
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Abstract. Given a morphism h prolongable on a and an integer p, we present an algorithm that calculates which letters occur infinitely often in congruent positions modulo p in the infinite word h ω (a). As a corollary, we show that it is decidable whether a morphic word is ultimately pperiodic. Moreover, using our algorithm we can find the smallest similarity relation such that the morphic word is ultimately relationally pperiodic. The problem of deciding whether an automatic sequence is ultimately weakly Rperiodic is also shown to be decidable. Key words: automatic sequence, decidability, morphic word, periodicity, similarity relation 1
Periodicity, Repetitions, and Orbits of an Automatic Sequence
, 2009
"... We revisit a technique of S. Lehr on automata and use it to prove old and new results in a simple way. We give a very simple proof of the 1986 theorem of Honkala that it is decidable whether a given kautomatic sequence is ultimately periodic. We prove that it is decidable whether a given kautomati ..."
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We revisit a technique of S. Lehr on automata and use it to prove old and new results in a simple way. We give a very simple proof of the 1986 theorem of Honkala that it is decidable whether a given kautomatic sequence is ultimately periodic. We prove that it is decidable whether a given kautomatic sequence is overlapfree (or squarefree, or cubefree, etc.) We prove that the lexicographically least sequence in the orbit closure of a kautomatic sequence is kautomatic, and use this last result to show that several related quantities, such as the critical exponent, irrationality measure, and recurrence quotient for Sturmian words with slope α, have automatic continued fraction expansions if α does.