84 research outputs found
Suffix conjugates for a class of morphic subshifts
Let A be a finite alphabet and f: A^* --> A^* be a morphism with an iterative
fixed point f^\omega(\alpha), where \alpha{} is in A. Consider the subshift (X,
T), where X is the shift orbit closure of f^\omega(\alpha) and T: X --> X is
the shift map. Let S be a finite alphabet that is in bijective correspondence
via a mapping c with the set of nonempty suffixes of the images f(a) for a in
A. Let calS be a subset S^N be the set of infinite words s = (s_n)_{n\geq 0}
such that \pi(s):= c(s_0)f(c(s_1)) f^2(c(s_2))... is in X. We show that if f is
primitive and f(A) is a suffix code, then there exists a mapping H: calS -->
calS such that (calS, H) is a topological dynamical system and \pi: (calS, H)
--> (X, T) is a conjugacy; we call (calS, H) the suffix conjugate of (X, T). In
the special case when f is the Fibonacci or the Thue-Morse morphism, we show
that the subshift (calS, T) is sofic, that is, the language of calS is regular
Words and forbidden factors
AbstractGiven a finite or infinite word v, we consider the set M(v) of minimal forbidden factors of v. We show that the set M(v) is of fundamental importance in determining the structure of the word v. In the case of a finite word w we consider two parameters that are related to the size of M(w): the first counts the minimal forbidden factors of w and the second gives the length of the longest minimal forbidden factor of w. We derive sharp upper and lower bounds for both parameters. We prove also that the second parameter is related to the minimal period of the word w. We are further interested to the algorithmic point of view. Indeed, we design linear time algorithm for the following two problems: (i) given w, construct the set M(w) and, conversely, (ii) given M(w), reconstruct the word w. In the case of an infinite word x, we consider the following two functions: gx that counts, for each n, the allowed factors of x of length n and fx that counts, for each n, the minimal forbidden factors of x of length n. We address the following general problem: what information about the structure of x can be derived from the pair (gx,fx)? We prove that these two functions characterize, up to the automorphism exchanging the two letters, the language of factors of each single infinite Sturmian word
On a Generalization of the 3x + 1 Problem
AbstractWe consider the following analogue of the 3x + 1 function, [formula], where ÎČ > 1 is real, and â â is the ceiling function (next largest integer). The case ÎČ = 32 is just the 3x + 1 function. We prove that for almost all ÎČ, TÎČ decreases iterates on average when 1 < ÎČ < 2 and increases iterates on average when ÎČ > 2. We find certain values of ÎČ where the analogue of the 3x + 1 conjecture has an affirmative answer and other values where it has a negative answer
A Characterization of Bispecial Sturmian Words
A finite Sturmian word w over the alphabet {a,b} is left special (resp. right
special) if aw and bw (resp. wa and wb) are both Sturmian words. A bispecial
Sturmian word is a Sturmian word that is both left and right special. We show
as a main result that bispecial Sturmian words are exactly the maximal internal
factors of Christoffel words, that are words coding the digital approximations
of segments in the Euclidean plane. This result is an extension of the known
relation between central words and primitive Christoffel words. Our
characterization allows us to give an enumerative formula for bispecial
Sturmian words. We also investigate the minimal forbidden words for the set of
Sturmian words.Comment: Accepted to MFCS 201
Words with the Maximum Number of Abelian Squares
An abelian square is the concatenation of two words that are anagrams of one
another. A word of length can contain distinct factors that
are abelian squares. We study infinite words such that the number of abelian
square factors of length grows quadratically with .Comment: To appear in the proceedings of WORDS 201
On Sturmian Graphs
In this paper we define Sturmian graphs and we prove that all of them have a certain ''''counting'''' property. We show deep connections between this counting property and two conjectures, by Moser and by Zaremba, on the continued fraction expansion of real numbers. These graphs turn out to be the underlying graphs of compact directed acyclic word graphs of central Sturmian words. In order to prove this result, we give a characterization of the maximal repeats of central Sturmian words. We show also that, in analogy with the case of Sturmian words, these graphs converge to infinite ones
Minimal Absent Words in Rooted and Unrooted Trees
We extend the theory of minimal absent words to (rooted and unrooted) trees, having edges labeled by letters from an alphabet of cardinality. We show that the set of minimal absent words of a rooted (resp. unrooted) tree T with n nodes has cardinality (resp.), and we show that these bounds are realized. Then, we exhibit algorithms to compute all minimal absent words in a rooted (resp. unrooted) tree in output-sensitive time (resp. assuming an integer alphabet of size polynomial in n
Optimal Computation of Avoided Words
The deviation of the observed frequency of a word from its expected
frequency in a given sequence is used to determine whether or not the word
is avoided. This concept is particularly useful in DNA linguistic analysis. The
value of the standard deviation of , denoted by , effectively
characterises the extent of a word by its edge contrast in the context in which
it occurs. A word of length is a -avoided word in if
, for a given threshold . Notice that such a word
may be completely absent from . Hence computing all such words na\"{\i}vely
can be a very time-consuming procedure, in particular for large . In this
article, we propose an -time and -space algorithm to compute all
-avoided words of length in a given sequence of length over a
fixed-sized alphabet. We also present a time-optimal -time and
-space algorithm to compute all -avoided words (of any
length) in a sequence of length over an alphabet of size .
Furthermore, we provide a tight asymptotic upper bound for the number of
-avoided words and the expected length of the longest one. We make
available an open-source implementation of our algorithm. Experimental results,
using both real and synthetic data, show the efficiency of our implementation
Minimal Forbidden Factors of Circular Words
Minimal forbidden factors are a useful tool for investigating properties of
words and languages. Two factorial languages are distinct if and only if they
have different (antifactorial) sets of minimal forbidden factors. There exist
algorithms for computing the minimal forbidden factors of a word, as well as of
a regular factorial language. Conversely, Crochemore et al. [IPL, 1998] gave an
algorithm that, given the trie recognizing a finite antifactorial language ,
computes a DFA recognizing the language whose set of minimal forbidden factors
is . In the same paper, they showed that the obtained DFA is minimal if the
input trie recognizes the minimal forbidden factors of a single word. We
generalize this result to the case of a circular word. We discuss several
combinatorial properties of the minimal forbidden factors of a circular word.
As a byproduct, we obtain a formal definition of the factor automaton of a
circular word. Finally, we investigate the case of minimal forbidden factors of
the circular Fibonacci words.Comment: To appear in Theoretical Computer Scienc
Repetitions in beta-integers
Classical crystals are solid materials containing arbitrarily long periodic
repetitions of a single motif. In this paper, we study the maximal possible
repetition of the same motif occurring in beta-integers -- one dimensional
models of quasicrystals. We are interested in beta-integers realizing only a
finite number of distinct distances between neighboring elements. In such a
case, the problem may be reformulated in terms of combinatorics on words as a
study of the index of infinite words coding beta-integers. We will solve a
particular case for beta being a quadratic non-simple Parry number.Comment: 11 page
- âŠ