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29
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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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
Counting OverlapFree Binary Words
 Springer LNCS 665
, 1993
"... A word on a finite alphabet A is said to be overlapfree if it contains no factor of the form xuxux, where x is a letter and u a (possibly empty) word. In this paper we study the number un of overlapfree binary words of length n, which is known to be bounded by a polynomial in n. First, we describe ..."
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Cited by 22 (2 self)
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A word on a finite alphabet A is said to be overlapfree if it contains no factor of the form xuxux, where x is a letter and u a (possibly empty) word. In this paper we study the number un of overlapfree binary words of length n, which is known to be bounded by a polynomial in n. First, we describe a bijection between the set of overlapfree words and a rational language. This yields recurrence relations for un , which allow to compute un in logarithmic time. Then, we prove that the numbers ff = sup f r j n r = O (un) g and fi = inf f r j un = O (n r ) g are distinct, and we give an upper bound for ff and a lower bound for fi. Finally, we compute an asymptotically tight bound to the number of overlapfree words of length less than n. 1 Introduction In general, the problem of evaluating the number un of words of length n in the language U consisting of words on some finite alphabet A with no factors in a certain set F is not easy. If F is finite, it amounts to counting words in...
Palindrome complexity
 To appear, Theoret. Comput. Sci
, 2002
"... We study the palindrome complexity of infinite sequences on finite alphabets, i.e., the number of palindromic factors (blocks) of given length occurring in a given sequence. We survey the known results and obtain new results for some sequences, in particular for Rote sequences and for fixed points o ..."
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Cited by 17 (2 self)
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We study the palindrome complexity of infinite sequences on finite alphabets, i.e., the number of palindromic factors (blocks) of given length occurring in a given sequence. We survey the known results and obtain new results for some sequences, in particular for Rote sequences and for fixed points of primitive morphisms of constant length belonging to “class P ” of HofKnillSimon. We also give an upper bound for the palindrome complexity of a sequence in terms of its (block)complexity. 1
Number of representations related to a linear recurrent basis
 Acta Arith
, 1999
"... Abstract. We consider a sequence (Gn) satisfying a stationary recurrence relation with a Perron dominant root φ of the corresponding polynomial. We fix q ≥ 2 and study the properties of the summation function Φ(x) = y<x f(y), where f(y) is defined as the number of representations of y in the basi ..."
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Cited by 15 (5 self)
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Abstract. We consider a sequence (Gn) satisfying a stationary recurrence relation with a Perron dominant root φ of the corresponding polynomial. We fix q ≥ 2 and study the properties of the summation function Φ(x) = y<x f(y), where f(y) is defined as the number of representations of y in the basis (Gn) with the digits 0, 1,..., q−1. The main result is the existence of a (unique) periodic function H with period 1 such that Φ(x) = xαH(logφ x) + o(x α), x→∞, where α = logφ q. It is proved that H is positive, continuous and a.e. differentiable. Besides, we reduce the problem of estimating the remainder term and Lipschitz exponent for H to finding a good exponent in the upper bound for f. A special attention is paid to the radix case with Gn = dn. 2 0. General results and notations. We first present a general statement on the number of representations related to a linear recurrent basis. Let r be an integer, r ≥ 1, and let a1, a2,..., ar be reals. We consider a sequence (Gk)k≥0 such that for
Transcendence of Formal Power Series With Rational Coefficients
, 1999
"... We give algebraic proofs of transcendence over Q(X) of formal power series with rational coefficients, by using inter alia reduction modulo prime numbers, and the Christol theorem. Applications to generating series of languages and combinatorial objects are given. Keywords: transcendental formal po ..."
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Cited by 12 (2 self)
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We give algebraic proofs of transcendence over Q(X) of formal power series with rational coefficients, by using inter alia reduction modulo prime numbers, and the Christol theorem. Applications to generating series of languages and combinatorial objects are given. Keywords: transcendental formal power series, binomial series, automatic sequences, pLucas sequences, ChomskySchutzenberger theorem. 1 Introduction Formal power series with integer coefficients often occur as generating series. Suppose that a set E contains exactly a n elements of "size" n for each integer n: the generating series of this set is the formal power series P n0 a n X n (this series belongs to Z[[X]] ae Q[[X]]). Properties of this formal power series reflect properties of its coefficients, and hence properties of the set E. Roughly speaking, algebraicity of the series over Q(X) means that the set has a strong structure. For example, the ChomskySchutzenberger theorem [16] asserts that the generating seri...
Regularity Properties of the Stern Enumeration of the Rationals
"... s(n) + s(n + 1). Stern showed in 1858 that gcd(s(n), s(n + 1)) = 1, and that every positive rational number a s(n) b occurs exactly once in the form s(n+1) for some n ≥ 1. We show that in a strong sense, the average value of these fractions is 3 2. We also show that for d ≥ 2, the pair (s(n), s(n+1 ..."
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Cited by 10 (2 self)
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s(n) + s(n + 1). Stern showed in 1858 that gcd(s(n), s(n + 1)) = 1, and that every positive rational number a s(n) b occurs exactly once in the form s(n+1) for some n ≥ 1. We show that in a strong sense, the average value of these fractions is 3 2. We also show that for d ≥ 2, the pair (s(n), s(n+1)) is uniformly distributed among all feasible pairs of congruence classes modulo d. More precise results are presented for d = 2 and 3. 1
Automatic Dirichlet series
, 2000
"... Dirichlet series whose coefficients are generated by finite automata define meromorphic functions on the whole complex plane. As consequences, a new proof of Cobham's theorem on the existence of logarithmic frequencies of symbols in automatic sequences is given, and certain infinite products ar ..."
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Cited by 5 (1 self)
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Dirichlet series whose coefficients are generated by finite automata define meromorphic functions on the whole complex plane. As consequences, a new proof of Cobham's theorem on the existence of logarithmic frequencies of symbols in automatic sequences is given, and certain infinite products are explicitly computed.
diatomic sequence 0,1,1,2,1,3,2,3,1,4
 American Math. Monthly
"... 1 Introduction. Stern’s diatomic sequence is a simply defined sequence with an amazing set of properties. Our goal is to present many of these properties—those that have most impressed the author. The diatomic sequence has a long history; perhaps ..."
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Cited by 4 (1 self)
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1 Introduction. Stern’s diatomic sequence is a simply defined sequence with an amazing set of properties. Our goal is to present many of these properties—those that have most impressed the author. The diatomic sequence has a long history; perhaps