Aleksi Saarela

Problems in between words and abelian words: k-abelian avoidability

We consider a recently defined notion of k-abelian equivalence of words in connection with avoidability problems. This equivalence relation, for a fixed natural number k, takes into account the numbers of occurrences of the different factors of length k and the prefix and the suffix of length k-1. We search for the smallest alphabet in which k-abelian squares and cubes can be avoided, respectively. For 2-abelian squares this is four-as in the case of abelian words, while for 2-abelian cubes we have only strong evidence that the size is two-as it is in the case of words. However, we are able to prove this optimal value only for 8-abelian cubes.

Local squares, periodicity and finite automata

We consider the general problem when local regularity implies the global one in the setting where local regularity means the existence of a square of certain length in every position of an infinite word. The square can occur as centered or to the left or to the right from each position. In each case there are three variants of the problem depending on whether the square is that of words, that of abelian words or, as an in between case, that of so called k-abelian words. The above nine variants of the problem are completely solved, and some open problems are addressed in the k-abelian case. Finally, an amazing unavoidability result for 2-abelian squares is obtained.

5-Abelian cubes are avoidable on binary alphabets ∗ ∗∗

A k -abelian cube is a word uvw , where the factors u , v , and w are either pairwise equal, or have the same multiplicities for every one of their factors of length at most k . Previously it has been shown that k -abelian cubes are avoidable over a binary alphabet for k ≥ 8. Here it is proved that this holds for k ≥ 5.

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 p...

3-Abelian Cubes Are Avoidable on Binary Alphabets

A k-abelian cube is a word uvw, where u, v, w have the same factors of length at most k with the same multiplicities. Previously it has been known that k-abelian cubes are avoidable over a binary alphabet for k ≥ 5. Here it is proved that this holds for k ≥ 3.

On growth and fluctuation of k-abelian complexity☆

An 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.

Degrees of Transducibility

Our objects of study are infinite sequences and how they can be transformed into each other. As transformational devices, we focus here on Turing Machines, sequential finite state transducers and Mealy Machines. For each of these choices, the resulting transducibility relation \(\ge \) is a preorder on the set of infinite sequences. This preorder induces equivalence classes, called degrees, and a partial order on the degrees.

Systems of Word Equations, Polynomials and Linear Algebra: A New Approach

Abstract We develop a new tool, namely polynomial and linear algebraic methods, for studying systems of word equations. We illustrate its usefulness by giving essentially simpler proofs of several hard problems. At the same time we prove extensions of these results. Finally, we obtain the first nontrivial upper bounds for the fundamental problem of the maximal size of independent systems. These bounds depend quadratically on the size of the shortest equation. No methods of having such bounds have been known before.

On maximal chains of systems of word equations

We consider systems of word equations and their solution sets. We discuss some fascinating properties of those, namely the size of a maximal independent set of word equations, and proper chains of solution sets of those. We recall the basic results, extend some known results and formulate several fundamental problems of the topic.

On Growth and Fluctuation of k -Abelian Complexity

An 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.