63,238 research outputs found

    Binary Patterns in Binary Cube-Free Words: Avoidability and Growth

    Get PDF
    The avoidability of binary patterns by binary cube-free words is investigated and the exact bound between unavoidable and avoidable patterns is found. All avoidable patterns are shown to be D0L-avoidable. For avoidable patterns, the growth rates of the avoiding languages are studied. All such languages, except for the overlap-free language, are proved to have exponential growth. The exact growth rates of languages avoiding minimal avoidable patterns are approximated through computer-assisted upper bounds. Finally, a new example of a pattern-avoiding language of polynomial growth is given.Comment: 18 pages, 2 tables; submitted to RAIRO TIA (Special issue of Mons Days 2012

    Avoidability of formulas with two variables

    Full text link
    In combinatorics on words, a word ww over an alphabet Σ\Sigma is said to avoid a pattern pp over an alphabet Δ\Delta of variables if there is no factor ff of ww such that f=h(p)f=h(p) where h:Δ∗→Σ∗h:\Delta^*\to\Sigma^* is a non-erasing morphism. A pattern pp is said to be kk-avoidable if there exists an infinite word over a kk-letter alphabet that avoids pp. We consider the patterns such that at most two variables appear at least twice, or equivalently, the formulas with at most two variables. For each such formula, we determine whether it is 22-avoidable, and if it is 22-avoidable, we determine whether it is avoided by exponentially many binary words

    Tower-type bounds for unavoidable patterns in words

    Get PDF
    A word ww is said to contain the pattern PP if there is a way to substitute a nonempty word for each letter in PP so that the resulting word is a subword of ww. Bean, Ehrenfeucht and McNulty and, independently, Zimin characterised the patterns PP which are unavoidable, in the sense that any sufficiently long word over a fixed alphabet contains PP. Zimin's characterisation says that a pattern is unavoidable if and only if it is contained in a Zimin word, where the Zimin words are defined by Z1=x1Z_1 = x_1 and Zn=Zn−1xnZn−1Z_n=Z_{n-1} x_n Z_{n-1}. We study the quantitative aspects of this theorem, obtaining essentially tight tower-type bounds for the function f(n,q)f(n,q), the least integer such that any word of length f(n,q)f(n, q) over an alphabet of size qq contains ZnZ_n. When n=3n = 3, the first non-trivial case, we determine f(n,q)f(n,q) up to a constant factor, showing that f(3,q)=Θ(2qq!)f(3,q) = \Theta(2^q q!).Comment: 17 page

    Random subshifts of finite type

    Full text link
    Let XX be an irreducible shift of finite type (SFT) of positive entropy, and let Bn(X)B_n(X) be its set of words of length nn. Define a random subset ω\omega of Bn(X)B_n(X) by independently choosing each word from Bn(X)B_n(X) with some probability α\alpha. Let XωX_{\omega} be the (random) SFT built from the set ω\omega. For each 0≤α≤10\leq \alpha \leq1 and nn tending to infinity, we compute the limit of the likelihood that XωX_{\omega} is empty, as well as the limiting distribution of entropy for XωX_{\omega}. For α\alpha near 1 and nn tending to infinity, we show that the likelihood that XωX_{\omega} contains a unique irreducible component of positive entropy converges exponentially to 1. These results are obtained by studying certain sequences of random directed graphs. This version of "random SFT" differs significantly from a previous notion by the same name, which has appeared in the context of random dynamical systems and bundled dynamical systems.Comment: Published in at http://dx.doi.org/10.1214/10-AOP636 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org
    • …
    corecore