63,238 research outputs found
Binary Patterns in Binary Cube-Free Words: Avoidability and Growth
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
In combinatorics on words, a word over an alphabet is said to
avoid a pattern over an alphabet of variables if there is no
factor of such that where is a
non-erasing morphism. A pattern is said to be -avoidable if there exists
an infinite word over a -letter alphabet that avoids . 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 -avoidable, and if it is -avoidable, we
determine whether it is avoided by exponentially many binary words
Tower-type bounds for unavoidable patterns in words
A word is said to contain the pattern if there is a way to substitute
a nonempty word for each letter in so that the resulting word is a subword
of . Bean, Ehrenfeucht and McNulty and, independently, Zimin characterised
the patterns which are unavoidable, in the sense that any sufficiently long
word over a fixed alphabet contains . 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 and . We
study the quantitative aspects of this theorem, obtaining essentially tight
tower-type bounds for the function , the least integer such that any
word of length over an alphabet of size contains . When , the first non-trivial case, we determine up to a constant factor,
showing that .Comment: 17 page
Random subshifts of finite type
Let be an irreducible shift of finite type (SFT) of positive entropy, and
let be its set of words of length . Define a random subset
of by independently choosing each word from with some
probability . Let be the (random) SFT built from the set
. For each and tending to infinity, we compute
the limit of the likelihood that is empty, as well as the limiting
distribution of entropy for . For near 1 and tending
to infinity, we show that the likelihood that 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
- …