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In this paper we introduce and study a family of complexity functions of infinite words indexed by k ∈ Z + ∪ {+∞}. Let k ∈ Z + ∪ {+∞} 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 ∈ 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 ∼ k on A * , bridging the gap between the usual notion of Abelian equivalence (when k = 1) and equality (when k = +∞). 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 ω ∈ A N , we consider the associated complexity function P (k) ω : N → N which counts the number of k-Abelian equivalence classes of factors of ω 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) ω (n) < q k (n) for some k ∈ Z + ∪ {+∞} and n ≥ 0, the ω is ultimately periodic. Moreover if ω is aperiodic, then P (k) ω (n) = q k (n) if and only if ω is Sturmian. We also study k-Abelian complexity in connection with repetitions in words. Using Szemerédi's theorem, we show that if ω has bounded k-Abelian complexity, then for every D ⊂ N with positive upper density and for every positive integer N, there exists a k-Abelian N power occurring in ω at some position j ∈ D.

Classically, several properties and relations of words, such as "being a power of the same word", can be expressed by using word equations. This paper is devoted to a general study of the expressive power of word equations. As main results we prove theorems which allow us to show that certain properties of words are not expressible as components of solutions of word equations. In particular, "the primitiveness" and "the equal length" are such properties, as well as being "any word over a proper subalphabet". knowledge, no attempt to synthesis or of a systematization of this topic has been done. This was emphasized also in a recent survey [Choffrut and Karhumäki 1997], where some results of the field were collected.Classical relations on words that are characterized as solution sets of word equations are for instance, "two words X and Y are powers of the same word" if and only if they constitute a solution of the equation XY ϭ YX, and "two words X and Y are conjugates" if and only if they constitute a solution of the equation XZ ϭ ZY. In the first case, we need no extra variables, while in the second case an additional variable is needed, see Example 4. As above, we identify names of variables and particular solutions of an equation.Motivated by above, we say that a property of words-either a language L ʕ ⌺* or a k-ary relation ʕ (⌺*) k -is expressible by a word equation, if there exists an equation e with t Ն k variables over ⌺ such that: -L coincides with the values of a fixed component of all solutions of e, or -coincides with the values of k fixed components of all solutions of e.Obviously, languages are k-ary relations with k ϭ 1, but, due to the importance of this particular case, we have chosen to define those separately. We allow e to contain constants from ⌺. An important feature here is also that t can be larger than k, that is, additional variables are allowed. This increases essentially the expressive power of equations, and in particular makes it much easier to express certain properties by equations.As an illustration we recall the following: The union of solution sets of two equations can be expressed as a solution set of one equation, as was shown in Büchi and Senger [1986/1987] using 4 additional variables, and later improved to require only 2 additional ones by S. Grigorieff (personal communications), cf. also Choffrut and Karhumäki. Similarly, the inequality, that is the set of t-tuples of words which does not satisfy a given equation e with t variables, can be expressed as a union of the solution sets of finitely many equations, each of them using 3 extra variables, cf. for example, Choffrut and Karhumäki [1997], and consequently the inequality is expressible by one equation if additional variables are allowed.This way of expressing relations on words using word equations is very natural and resembles the way of expressing enumerable relations on integers by diophantine equations. However, the expressive power of our method is weaker. Namely, while diophantine equations can express all recursi...

We slightly improve the result of Klarner, Birget and Satterfield, showing that the freeness of finitely presented multiplicative semigroups of 3×3 matrices over ℕ is undecidable even for triangular matrices. This is achieved by proving a new variant of Post correspondence problem. We also consider the freeness problem for 2×2 matrices. On the one hand, we show that it cannot be proved undecidable using the above methods which work in higher dimensions, and, on the other hand, we give some evidence that its decidability, if so, is also a challenging problem.

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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 þ -powers. This answers an open question of Kobayashi from 1986. r 2004 Elsevier Inc. All rights reserved.

Given a finite set of matrices with integer entries, consider the question of determining whether the semigroup they generated 1) is free, 2) contains the identity matrix, 3) contains the null matrix or 4) is a group. Even for matrices of dimension 3, questions 1) and 3) are undecidable. For dimension 2, they are still open as far as we know. Here we prove that problems 2) and 4) are decidable by proving more generally that it is recursively decidable whether or not a given non singular matrix belongs to a given finitely generated semigroup.

Using a result of B.H. Neumann we extend Eilenberg's Equality Theorem to a general result which implies that the multiplicity equivalence problem of two (nondeterministic) multitape finite automata is decidable. As a corollary we solve a long standing open problem in automata theory, namely, the equivalence problem for multitape deterministic finite automata. The main theorem states that there is a finite test set for the multiplicity equivalence of finite automata over conservative monoids embeddable in a fully ordered group.