135,469 research outputs found
Least Periods of Factors of Infinite Words
We show that any positive integer is the least period of a factor of the Thue-Morse word. We also characterize the set of least periods of factors of a Sturmian word. In particular, the corresponding set for the Fibonacci word is the set of Fibonacci numbers. As a byproduct of our results, we give several new proofs and tightenings of well-known properties of Sturmian words.Work of the first author supported by a Discovery Grant from NSERC. Work of the second author supported by the Finnish Academy under grant 8206039.https://www.rairo-ita.org/articles/ita/abs/2009/01/ita08003/ita08003.htm
Cyclic Complexity of Words
We introduce and study a complexity function on words called
\emph{cyclic complexity}, which counts the number of conjugacy classes of
factors of length of an infinite word We extend the well-known
Morse-Hedlund theorem to the setting of cyclic complexity by showing that a
word is ultimately periodic if and only if it has bounded cyclic complexity.
Unlike most complexity functions, cyclic complexity distinguishes between
Sturmian words of different slopes. We prove that if is a Sturmian word and
is a word having the same cyclic complexity of then up to renaming
letters, and have the same set of factors. In particular, is also
Sturmian of slope equal to that of Since for some
implies is periodic, it is natural to consider the quantity
We show that if is a Sturmian word,
then We prove however that this is
not a characterization of Sturmian words by exhibiting a restricted class of
Toeplitz words, including the period-doubling word, which also verify this same
condition on the limit infimum. In contrast we show that, for the Thue-Morse
word , Comment: To appear in Journal of Combinatorial Theory, Series
Transition Property For Cube-Free Words
We study cube-free words over arbitrary non-unary finite alphabets and prove
the following structural property: for every pair of -ary cube-free
words, if can be infinitely extended to the right and can be infinitely
extended to the left respecting the cube-freeness property, then there exists a
"transition" word over the same alphabet such that is cube free. The
crucial case is the case of the binary alphabet, analyzed in the central part
of the paper.
The obtained "transition property", together with the developed technique,
allowed us to solve cube-free versions of three old open problems by Restivo
and Salemi. Besides, it has some further implications for combinatorics on
words; e.g., it implies the existence of infinite cube-free words of very big
subword (factor) complexity.Comment: 14 pages, 5 figure
Representations of Circular Words
In this article we give two different ways of representations of circular
words. Representations with tuples are intended as a compact notation, while
representations with trees give a way to easily process all conjugates of a
word. The latter form can also be used as a graphical representation of
periodic properties of finite (in some cases, infinite) words. We also define
iterative representations which can be seen as an encoding utilizing the
flexible properties of circular words. Every word over the two letter alphabet
can be constructed starting from ab by applying the fractional power and the
cyclic shift operators one after the other, iteratively.Comment: In Proceedings AFL 2014, arXiv:1405.527
On Generating Binary Words Palindromically
We regard a finite word up to word isomorphism as an
equivalence relation on where is equivalent to if
and only if Some finite words (in particular all binary words) are
generated by "{\it palindromic}" relations of the form for some
choice of and That is to say,
some finite words are uniquely determined up to word isomorphism by the
position and length of some of its palindromic factors. In this paper we study
the function defined as the least number of palindromic relations
required to generate We show that every aperiodic infinite word must
contain a factor with and that some infinite words have
the property that for each factor of We obtain a
complete classification of such words on a binary alphabet (which includes the
well known class of Sturmian words). In contrast for the Thue-Morse word, we
show that the function is unbounded
On the Sets of Real Numbers Recognized by Finite Automata in Multiple Bases
This article studies the expressive power of finite automata recognizing sets
of real numbers encoded in positional notation. We consider Muller automata as
well as the restricted class of weak deterministic automata, used as symbolic
set representations in actual applications. In previous work, it has been
established that the sets of numbers that are recognizable by weak
deterministic automata in two bases that do not share the same set of prime
factors are exactly those that are definable in the first order additive theory
of real and integer numbers. This result extends Cobham's theorem, which
characterizes the sets of integer numbers that are recognizable by finite
automata in multiple bases.
In this article, we first generalize this result to multiplicatively
independent bases, which brings it closer to the original statement of Cobham's
theorem. Then, we study the sets of reals recognizable by Muller automata in
two bases. We show with a counterexample that, in this setting, Cobham's
theorem does not generalize to multiplicatively independent bases. Finally, we
prove that the sets of reals that are recognizable by Muller automata in two
bases that do not share the same set of prime factors are exactly those
definable in the first order additive theory of real and integer numbers. These
sets are thus also recognizable by weak deterministic automata. This result
leads to a precise characterization of the sets of real numbers that are
recognizable in multiple bases, and provides a theoretical justification to the
use of weak automata as symbolic representations of sets.Comment: 17 page
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