177 research outputs found
Language complexity of rotations and Sturmian sequences
AbstractGiven a rotation of the circle, we study the complexity of formal languages that are generated by the itineraries of interval covers. These languages are regular iff the rotation is rational. In the case of irrational rotations, our study reduces to that of the language complexity of the corresponding Sturmian sequences. We show that for a large class of irrationals, including e, all quadratic numbers and more generally all Hurwitz numbers, the corresponding languages can be recognized by a nondeterministic Turing machine in linear time (in other words, belongs to NLIN)
On the Structure of Bispecial Sturmian Words
A balanced word is one in which any two factors of the same length contain
the same number of each letter of the alphabet up to one. Finite binary
balanced words are called Sturmian words. A Sturmian word is bispecial if it
can be extended to the left and to the right with both letters remaining a
Sturmian word. There is a deep relation between bispecial Sturmian words and
Christoffel words, that are the digital approximations of Euclidean segments in
the plane. In 1997, J. Berstel and A. de Luca proved that \emph{palindromic}
bispecial Sturmian words are precisely the maximal internal factors of
\emph{primitive} Christoffel words. We extend this result by showing that
bispecial Sturmian words are precisely the maximal internal factors of
\emph{all} Christoffel words. Our characterization allows us to give an
enumerative formula for bispecial Sturmian words. We also investigate the
minimal forbidden words for the language of Sturmian words.Comment: arXiv admin note: substantial text overlap with arXiv:1204.167
Coding rotations on intervals
We show that the coding of rotation by on intervals with
rationally independent lengths can be recoded over Sturmian words of angle
More precisely, for a given an universal automaton is constructed such
that the edge indexed by the vector of values of the th letter on each
Sturmian word gives the value of the th letter of the coding of rotation.Comment: LIAFA repor
On the number of return words in infinite words with complexity 2n+1
In this article, we count the number of return words in some infinite words
with complexity 2n+1. We also consider some infinite words given by codings of
rotation and interval exchange transformations on k intervals. We prove that
the number of return words over a given word w for these infinite words is
exactly k.Comment: see also http://liafa.jussieu.fr/~vuillon/articles.htm
The number of binary rotation words
We consider binary rotation words generated by partitions of the unit circle
to two intervals and give a precise formula for the number of such words of
length n. We also give the precise asymptotics for it, which happens to be
O(n^4). The result continues the line initiated by the formula for the number
of all Sturmian words obtained by Lipatov in 1982, then independently by
Berenstein, Kanal, Lavine and Olson in 1987, Mignosi in 1991, and then with
another technique by Berstel and Pocchiola in 1993.Comment: Submitted to RAIRO IT
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