On the growth rates of complexity of threshold languages

Shur, Arseny M.; Gorbunova, Irina A.

doi:10.1051/ita/2010012

Cited by 23 publications

(34 citation statements)

References 18 publications

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“…Initially this was part of our strategy. Unfortunately, our experience supports the conjecture in [17], that the number of these words grows approximately as 1.24 r (independently of n). -free words and satisfied the algebraic condition; this allowed us to verify the claim of [13] that the morphisms presented therein for 5 ≤ n ≤ 11 are shortest possible 'convenient morphisms'; the uniforms are all uniform, with lengths around 4n − 4 in each case.…”

Section: η | |π|supporting

confidence: 82%

A proof of Dejean’s conjecture

Currie

Rampersad

2011

Math. Comp.

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Section: η | |π|supporting

confidence: 82%

A proof of Dejean’s conjecture

Currie

Rampersad

2011

Math. Comp.

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“…It is not obvious that this is even possible: indeed it might be that RT (k) = RT (k) for k ≥ 3 (where the threshold RT (k), introduced by Kobayashi [13], is such that the growth is polynomial when RT (k) < α < RT (k), and not polynomial when α > RT (k); Lemma 1 states that RT (2) = 7/3) and there is no more polynomial growth. This is conjectured by Shur [20], and supported by some numerical evidence.…”

Section: Discussionsupporting

confidence: 61%

On the number of α-power-free binary words for 2<α≤7/3

Blondel

Cassaigne

Jungers

2009

Theoretical Computer Science

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a b s t r a c tWe study the number u α (n) of α-power-free binary words of length n, and the asymptotics of this number when n tends to infinity, for a fixed rational number α in (2, 7/3]. For any such α, we prove a structure result that allows us to describe constructively the sequence u α (n) as a 2-regular sequence. This provides an algorithm that computes the number u α (n) in logarithmic time, for fixed α. Then, generalizing recent results on 2 + -free words, we describe the asymptotic behaviour of u α (n) in terms of joint spectral quantities of a pair of matrices that one can efficiently construct, given a rational number α.For α = 7/3, we compute the automaton and give sharp estimates for the asymptotic behaviour of u α (n).

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“…. , 60 letters we observed in [12] that the sequence {α(T (3) k )} demonstrates fast convergence to the limit ≈1.242096777.…”

Section: Theorem 1 ( [12]) For Any Integer M ≥ 3 There Exists a Setmentioning

confidence: 76%

“…Any threshold language can be approximated from above by a series of regular languages consisting of words that locally satisfy the (k/(k−1)) + -freeness property. Namely, these words avoid all (k/(k−1)) + -powers w such that |w| − per(w) ≤ m, for some constant m. From our previous work [12], it is clear that the case m = 3 gives a lot of important structural information about the languages T k . Here we study this case in details, using cylindric representation that captures the properties common for considered words over all alphabets.…”

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confidence: 96%

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On Pansiot Words Avoiding 3-Repetitions

Gorbunova

Shur

2012

Int. J. Found. Comput. Sci.

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The recently confirmed Dejean's conjecture about the threshold between avoidable and unavoidable powers of words gave rise to interesting and challenging problems on the structure and growth of threshold words. Over any finite alphabet with k ≥ 5 letters, Pansiot words avoiding 3-repetitions form a regular language, which is a rather small superset of the set of all threshold words. Using cylindric and 2-dimensional words, we prove that, as k approaches infinity, the growth rates of complexity for these regular languages tend to the growth rate of complexity of some ternary 2-dimensional language. The numerical estimate of this growth rate is ≈1.2421.Powers, integral and fractional, are the simplest and most natural repetitions in words. Any repetition over an arbitrary fixed alphabet is characterized by the set of all words over this alphabet, avoiding this repetition. The main question concerning such a set is whether it is finite or infinite. For fractional powers, this question is answered by Dejean's conjecture [5], which is now proved in all cases by the efforts of different authors, see [2][3][4][8][9][10][11].Recall that the exponent of a word w is the ratio between its length and its minimal period: exp(w) = |w|/ per(w). If exp(w) = β > 1, then w is a fractional power (β -power). It is convenient to treat the notion of β -power as follows: a word w is a β -power if exp(w) ≥ β while (|w|−1)/ per(w) < β , and a β + -power if exp(w) > β while (|w|−1)/ per(w) ≤ β . As usual, β + is treated as a "number", covering β in the usual ≤ order. A word is called β -free (where β can be a number with plus as well) if it contains no β -powers as factors. A β -power is k-avoidable if the number of k-ary β -free words is infinite. Dejean's conjecture states that a β -power is k-avoidable if and only if β ≥ (7/4) + and k = 3, β ≥ (7/5) + and k = 4, or β ≥ (k/(k−1)) + and k = 2, k ≥ 5.The (k/(k−1)) + -free languages over k-letter alphabets, where k ≥ 5, are called threshold languages; we denote them by T k . We study structure and growth of these languages, aiming at the asymptotic properties as the size of the alphabet increases. Any threshold language can be approximated from above by a series of regular languages consisting of words that locally satisfy the (k/(k−1)) + -freeness property. Namely, these words avoid all (k/(k−1)) + -powers w such that |w| − per(w) ≤ m, for some constant m. From our previous work [12], it is clear that the case m = 3 gives a lot of important structural information about the languages T k . Here we study this case in details, using cylindric representation that captures the properties common for considered words over all alphabets. PreliminariesWe study finite words and two-sided infinite words (Z-words) over finite k-letter alphabets Σ k and over some special ternary alphabet introduced below. We also consider 2-dimensional words, which are just finite rectangular arrays of alphabetic symbols. Unlike to some commonly used models of 2-dimensional

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On the growth rates of complexity of threshold languages

Cited by 23 publications

References 18 publications

A proof of Dejean’s conjecture

A proof of Dejean’s conjecture

On the number of α-power-free binary words for 2<α≤7/3

On Pansiot Words Avoiding 3-Repetitions

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