66 research outputs found
Shortest Repetition-Free Words Accepted by Automata
We consider the following problem: given that a finite automaton of
states accepts at least one -power-free (resp., overlap-free) word, what is
the length of the shortest such word accepted? We give upper and lower bounds
which, unfortunately, are widely separated.Comment: 12 pages, conference pape
Computing maximal-exponent factors in an overlap-free word
The exponent of a string is the quotient of its length over its smallest period. The exponent and the period of a string can be computed in time proportional to the string length. We design an algorithm to compute the maximal exponent of all factors of an overlap-free string. Our algorithm runs in linear time on a fixed-size alphabet, while a naive solution of the question would run in cubic time. The solution for non overlap-free strings derives from algorithms to compute all maximal repetitions, also called runs, occurring in the string.
We also show there is a linear number of occurrences of maximal-exponent factors in an overlap-free string. Their maximal number lies between 0.66n and 2.25n in a string of length n. The algorithm can additionally locate all of them in linear time
Avoiding Abelian powers in binary words with bounded Abelian complexity
The notion of Abelian complexity of infinite words was recently used by the
three last authors to investigate various Abelian properties of words. In
particular, using van der Waerden's theorem, they proved that if a word avoids
Abelian -powers for some integer , then its Abelian complexity is
unbounded. This suggests the following question: How frequently do Abelian
-powers occur in a word having bounded Abelian complexity? In particular,
does every uniformly recurrent word having bounded Abelian complexity begin in
an Abelian -power? While this is true for various classes of uniformly
recurrent words, including for example the class of all Sturmian words, in this
paper we show the existence of uniformly recurrent binary words, having bounded
Abelian complexity, which admit an infinite number of suffixes which do not
begin in an Abelian square. We also show that the shift orbit closure of any
infinite binary overlap-free word contains a word which avoids Abelian cubes in
the beginning. We also consider the effect of morphisms on Abelian complexity
and show that the morphic image of a word having bounded Abelian complexity has
bounded Abelian complexity. Finally, we give an open problem on avoidability of
Abelian squares in infinite binary words and show that it is equivalent to a
well-known open problem of Pirillo-Varricchio and Halbeisen-Hungerb\"uhler.Comment: 16 pages, submitte
Almost overlap-free words and the word problem for the free Burnside semigroup satisfying x^2=x^3
In this paper we investigate the word problem of the free Burnside semigroup
satisfying x^2=x^3 and having two generators. Elements of this semigroup are
classes of equivalent words. A natural way to solve the word problem is to
select a unique "canonical" representative for each equivalence class. We prove
that overlap-free words and so-called almost overlap-free words (this notion is
some generalization of the notion of overlap-free words) can serve as canonical
representatives for corresponding equivalence classes. We show that such a word
in a given class, if any, can be efficiently found. As a result, we construct a
linear-time algorithm that partially solves the word problem for the semigroup
under consideration.Comment: 33 pages, submitted to Internat. J. of Algebra and Compu
Binary words containing infinitely many overlaps
We characterize the squares occurring in infinite overlap-free binary words
and construct various alpha power-free binary words containing infinitely many
overlaps.Comment: 9 page
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
Polynomial versus Exponential Growth in Repetition-Free Binary Words
It is known that the number of overlap-free binary words of length n grows
polynomially, while the number of cubefree binary words grows exponentially. We
show that the dividing line between polynomial and exponential growth is 7/3.
More precisely, there are only polynomially many binary words of length n that
avoid 7/3-powers, but there are exponentially many binary words of length n
that avoid (7/3+)-powers. This answers an open question of Kobayashi from 1986.Comment: 12 page
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