259 research outputs found
Abelian Primitive Words
We investigate Abelian primitive words, which are words that are not Abelian
powers. We show that unlike classical primitive words, the set of Abelian
primitive words is not context-free. We can determine whether a word is Abelian
primitive in linear time. Also different from classical primitive words, we
find that a word may have more than one Abelian root. We also consider
enumeration problems and the relation to the theory of codes
Undecidability and Finite Automata
Using a novel rewriting problem, we show that several natural decision
problems about finite automata are undecidable (i.e., recursively unsolvable).
In contrast, we also prove three related problems are decidable. We apply one
result to prove the undecidability of a related problem about k-automatic sets
of rational numbers
Avoiding and Enforcing Repetitive Structures in Words
The focus of this thesis is on the study of repetitive structures in words, a central topic in the area of combinatorics on words. The results presented in the thesis at hand are meant to extend and enrich the existing theory concerning the appearance and absence of such structures. In the first part we examine whether these structures necessarily appear in infinite words over a finite alphabet. The repetitive structures we are concerned with involve functional dependencies between the parts that are repeated. In particular, we study avoidability questions of patterns whose repetitive structure is disguised by the application of a permutation. This novel setting exhibits the surprising behaviour that avoidable patterns may become unavoidable in larger alphabets. The second and major part of this thesis deals with equations on words that enforce a certain repetitive structure involving involutions in their solution set. Czeizler et al. (2009) introduced a generalised version of the classical equations u` Æ vmwn that were studied by Lyndon and Schützenberger. We solve the last two remaining and most challenging cases and thereby complete the classification of these equations in terms of the repetitive structures appearing in the admitted solutions. In the final part we investigate the influence of the shuffle operation on words avoiding ordinary repetitions. We construct finite and infinite square-free words that can be shuffled with themselves in a way that preserves squarefreeness. We also show that the repetitive structure obtained by shuffling a word with itself is avoidable in infinite words
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