18 research outputs found
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
On a generalization of Abelian equivalence and complexity of infinite words
In this paper we introduce and study a family of complexity functions of
infinite words indexed by k \in \ints ^+ \cup {+\infty}. Let k \in \ints ^+
\cup {+\infty} and be a finite non-empty set. Two finite words and
in are said to be -Abelian equivalent if for all of length
less than or equal to the number of occurrences of in is equal to
the number of occurrences of in This defines a family of equivalence
relations on bridging the gap between the usual notion of
Abelian equivalence (when ) and equality (when We show that
the number of -Abelian equivalence classes of words of length grows
polynomially, although the degree is exponential in Given an infinite word
\omega \in A^\nats, we consider the associated complexity function \mathcal
{P}^{(k)}_\omega :\nats \rightarrow \nats which counts the number of
-Abelian equivalence classes of factors of of length We show
that the complexity function is intimately linked with
periodicity. More precisely we define an auxiliary function q^k: \nats
\rightarrow \nats and show that if for
some k \in \ints ^+ \cup {+\infty} and the is ultimately
periodic. Moreover if is aperiodic, then if and only if is Sturmian. We also
study -Abelian complexity in connection with repetitions in words. Using
Szemer\'edi's theorem, we show that if has bounded -Abelian
complexity, then for every D\subset \nats with positive upper density and for
every positive integer there exists a -Abelian power occurring in
at some position $j\in D.
Fine and Wilf's theorem for k-abelian periods
Two words u and v are k-abelian equivalent if they contain the same number of occurrences of each factor of length at most k. This leads to a hierarchy of equivalence relations on words which lie properly in between the equality and abelian equality. The goal of this paper is to analyze Fine and Wilf's periodicity theorem with respect to these equivalence relations. Fine and Wilf's theorem tells exactly how long a word with two periods p and q can be without having the greatest common divisor of p and q as a period. Recently, the same question has been studied for abelian periods. In this paper we show that for k-abelian periods the situation is similar to the abelian case: In general, there is no bound for the lengths of such words, but the values of the parameters p, q and k for which the length is bounded can be characterized. In the latter case we provide nontrivial upper and lower bounds for the maximal lengths of such words. In some cases (e.g., for k = 2) we found the maximal length precisely.</p
On a generalization of Abelian equivalence and complexity of infinite words
In this paper we introduce and study a family of complexity functions of infinite words indexed by k in Z^+ U {+infinity}. Let k in Z^+ U {+infinity} and A be a finite non-empty set. Two finite words u and v in A* are said to be k-Abelian equivalent if for all x in A* of length less than or equal to k, the number of occurrences of x in u is equal to the number of occurrences of x in v. This defines a family of equivalence relations sim_k on A*, bridging the gap between the usual notion of Abelian equivalence (when k = 1) and equality (when k = +infinity). We show that the number of k-Abelian equivalence classes of words of length n grows polynomially, although the degree is exponential in k. Given an infinite word omega in A^N, we consider the associated complexity function P^(k)_omega : N -> N which counts the number of k-Abelian equivalence classes of factors of omega of length n. We show that the complexity function P_k is intimately linked with periodicity. More precisely we define an auxiliary function q^k : N -> N and show that if P^(k)_omega(n) < q^k(n) for some k in Z^+ U {+infinity} and n >= 0, then omega is ultimately periodic. Moreover if omega is aperiodic, then P^(k)_omega(n) = q^k(n) if and only if omega is Sturmian. We also study k-Abelian complexity in connection with repetitions in words. Using Szemeredi's theorem, we show that if omega has bounded k-Abelian complexity, then for every D subset of N with positive upper density and for every positive integer N, there exists a k-Abelian N-power occurring in omega at some position j in D
Matrix equations and Hilbert's tenth problem
We show a reduction of Hilbert's tenth problem to the solvability of
the matrix equation Xi1
1 Xi2
2 Xik
k = Z over non-commuting integral
matrices, where Z is the zero matrix, thus proving that the solvability
of the equation is undecidable. This is in contrast to the case whereby
the matrix semigroup is commutative in which the solvability of the
same equation was shown to be decidable in general.
The restricted problem where k = 2 for commutative matrices is
known as the \A-B-C Problem" and we show that this problem is
decidable even for a pair of non-commutative matrices over an algebraic
number field
Noneffective Regularity of Equality Languages and Bounded Delay Morphisms
special issue dedicated to the second edition of the conference AutoMathA: from Mathematics to ApplicationsWe give an instance of a class of morphisms for which it is easy to prove that their equality set is regular, but its emptiness is still undecidable. The class is that of bounded delay 2 morphisms
On k-abelian palindromes
International audienceA word is called a palindrome if it is equal to its reversal. In the paper we consider a k-abelian modification of this notion. Two words are called k-abelian equivalent if they contain the same number of occurrences of each factor of length at most k. We say that a word is a k-abelian palindrome if it is k-abelian equivalent to its reversal. A question we deal with is the following: how many distinct palindromes can a word contain? It is well known that a word of length n can contain at most n + 1 distinct palindromes as its factors; such words are called rich. On the other hand, there exist infinite words containing only finitely many distinct palindromes as their factors; such words are called poor. We show that in the k-abelian case there exist infinite words containing finitely many distinct k-abelian palindromic factors. For rich words we show that there exist finite words of length ncontaining Theta(n(2)) distinct k-abelian palindromes as their factors
On growth and fluctuation of k-abelian complexity
International audienceAn extension of abelian complexity, so called k-abelian complexity, has been considered recently in a number of articles. This paper considers two particular aspects of this extension: First, how much the complexity can increase when moving from a level k to the next one. Second, how much the complexity of a given word can fluctuate. For both questions we give optimal solutions