Tomi Kärki

Overlap-freeness in infinite partial words

We prove that there exist infinitely many infinite overlap-free binary partial words containing at least one hole. Moreover, we show that these words cannot contain more than one hole and the only hole must occur either in the first or in the second position. We define that a partial word is k-overlap-free if it does not contain a factor of the form xyxyx where the length of x is at least k. We prove that there exist infinitely many 2-overlap-free binary partial words containing an infinite number of holes.

Relational codes of words

We consider words, i.e. strings over a finite alphabet together with a similarity relation induced by a compatibility relation on letters. This notion generalizes that of partial words. The theory of codes on combinatorics on words is revisited by defining (R,S)-codes for arbitrary similarity relations R and S. We describe an algorithm to test whether or not a finite set of words is an (R,S)-code. Coding properties of finite sets of words are explored by finding maximal and minimal relations with respect to relational codes.

ON THE NUMBER OF SQUARES IN PARTIAL WORDS

In combinatorics on words, factors of the form ww, i.e., squares can be studiedfrom two perspectives. On one hand, one may try to avoid squares by constructingsquare-free words. A classical example of an infinite square-free word over a3-letter alphabet is obtained from the famous Thue-Morse word[1] using a certainmapping; see [14, 15]. On the other hand, one may try to maximize the numberof square factors in a word. The theorem of Fraenkel and Simpson states that aword of length n contains always less than 2n distinct squares [6]. A very shortproof for this and an improved upper bound 2n−Θ(logn)was given by Ilie in [9]and [10]. However, based on the numerical evidence, the conjectured bound is n.In this paper we consider squares in partial words, which are words with “donot know”-symbols ⋄ called holes. Here a square is a factor of the form ww

Multidimensional generalized automatic sequences and shape-symmetric morphic words

An infinite word is S-automatic if, for all n>=0, its (n+1)th letter is the output of a deterministic automaton fed with the representation of n in the numeration system S. In this paper, we consider an analogous definition in a multidimensional setting and study its relation to the shape-symmetric infinite words introduced by Arnaud Maes. More precisely, for d>=1, we show that a multidimensional infinite word x:N^d->@S over a finite alphabet @S is S-automatic for some abstract numeration system S built on a regular language containing the empty word if and only if x is the image by a coding of a shape-symmetric infinite word.

The theorem of Fine and Wilf for relational periods

We consider relational periods, where the relation is a compatibility relation on words induced by a relation on letters. We prove a variant of the theorem of Fine and Wilf for a (pure) period and a relational period.

On the periodicity of morphic words

Given a morphism h prolongable on a and an integer p, we present an algorithm that calculates which letters occur infinitely often in congruent positions modulo p in the infinite word hω(a). As a corollary, we show that it is decidable whether a morphic word is ultimately p-periodic. Moreover, using our algorithm we can find the smallest similarity relation such that the morphic word is ultimately relationally p-periodic. The problem of deciding whether an automatic sequence is ultimately weakly R-periodic is also shown to be decidable.

Relationally Periodic Sequences and Subword Complexity

By the famous theorem of Morse and Hedlund, a word is ultimately periodic if and only if it has bounded subword complexity, i.e., for sufficiently large n, the number of factors of length nis constant. In this paper we consider relational periods and relationally periodic sequences, where the relation is a similarity relation on words induced by a compatibility relation on letters. We investigate what would be a suitable definition for a relational subword complexity function such that it would imply a Morse and Hedlund-like theorem for relationally periodic words. We consider strong and weak relational periods and two candidates for subword complexity functions.

Repetition-freeness with Cyclic Relations and Chain Relations

A similarity relation R is a relation on words of equal length induced by a symmetric and reflexive relation on letters. Such a relation is called cyclic if the graph of the relation on letters is a cycle. A chain relation is obtained from a cyclic relation by removing one symmetric relation from the cycle. A word uv is an R-square if u and v are in relation R. The avoidability index of R-squares is the size of the minimal alphabet such that there exists an R-square-free infinite word having infinitely many occurrences of each letter of the alphabet. We prove that the avoidability index of R-squares is 7 in the case of cyclic relations and 6 in the case of chain relations. We also consider R-overlaps and show that they are 5-avoidable with cyclic relations and 4-avoidable with chain relations.

A new proof for the decidability of D0L ultimate periodicity

We give a new proof for the decidability of the D0L ultimate periodicity problem based on the decidability of p-periodicity of morphic words adapted to the approach of Harju and Linna.

COMPATIBILITY RELATIONS ON CODES AND FREE MONOIDS

A compatibility relation on letters induces a reflexive and symmetric relation on words of equal length. We consider these word relations with respect to the theory of variable length codes and free monoids. We define an (R, S)-code and an (R, S)-free monoid for arbitrary word relations R and S. Modified Sardinas-Patterson algorithm is presented for testing whether finite sets of words are (R,S)-codes. Coding capabilities of relational codes are measured algorithmically by finding minimal and maximal relations. We generalize the stability criterion of Schiitzenberger and Tilsons closure result for (R, S)-free monoids. The (R, S)-free hull of a set of words is introduced and we show how it can be computed. We prove a defect theorem for (R, S)-free hulls. In addition, a defect theorem of partial words is proved as a corollary.