467 research outputs found
Overlap-Free Words and Generalizations
The study of combinatorics on words dates back at least to the beginning of the 20th century and the work of Axel Thue. Thue was the first to give an example of an infinite word over a three letter alphabet that contains no squares (identical adjacent blocks) xx. This result was eventually used to solve some longstanding open problems in algebra and has remarkable connections to other areas of mathematics and computer science as well.
This thesis will consider several different generalizations of Thue's work. In particular we shall study the properties of infinite words avoiding various types of repetitions.
In Chapter 1 we introduce the theory of combinatorics on words. We present the basic definitions and give an historical survey of the area.
In Chapter 2 we consider the work of Thue in more detail. We present various well-known properties of the Thue-Morse word and give some generalizations. We examine Fife's characterization of the infinite overlap-free words and give a simpler proof of this result. We also present some applications to transcendental number theory, generalizing a classical result of Mahler.
In Chapter 3 we generalize a result of Seebold by showing that the only infinite 7/3-power-free binary words that can be obtained by iterating a morphism are the Thue-Morse word and its complement.
In Chapter 4 we continue our study of overlap-free and 7/3-power-free words. We discuss the squares that can appear as subwords of these words. We also show that it is possible to construct infinite 7/3-power-free binary words containing infinitely many overlaps.
In Chapter 5 we consider certain questions of language theory. In particular, we examine the context-freeness of the set of words containing overlaps. We show that over a three-letter alphabet, this set is not context-free, and over a two-letter alphabet, we show that this set cannot be unambiguously context-free.
In Chapter 6 we construct infinite words over a four-letter alphabet that avoid squares in any arithmetic progression of odd difference. Our constructions are based on properties of the paperfolding words. We use these infinite words to construct non-repetitive tilings of the integer lattice.
In Chapter 7 we consider approximate squares rather than squares. We give constructions of infinite words that avoid such approximate squares.
In Chapter 8 we conclude the work and present some open problems
Deleting and Testing Forbidden Patterns in Multi-Dimensional Arrays
Understanding the local behaviour of structured multi-dimensional data is a
fundamental problem in various areas of computer science. As the amount of data
is often huge, it is desirable to obtain sublinear time algorithms, and
specifically property testers, to understand local properties of the data.
We focus on the natural local problem of testing pattern freeness: given a
large -dimensional array and a fixed -dimensional pattern over a
finite alphabet, we say that is -free if it does not contain a copy of
the forbidden pattern as a consecutive subarray. The distance of to
-freeness is the fraction of entries of that need to be modified to make
it -free. For any and any large enough pattern over
any alphabet, other than a very small set of exceptional patterns, we design a
tolerant tester that distinguishes between the case that the distance is at
least and the case that it is at most , with query
complexity and running time , where and
depend only on .
To analyze the testers we establish several combinatorial results, including
the following -dimensional modification lemma, which might be of independent
interest: for any large enough pattern over any alphabet (excluding a small
set of exceptional patterns for the binary case), and any array containing
a copy of , one can delete this copy by modifying one of its locations
without creating new -copies in .
Our results address an open question of Fischer and Newman, who asked whether
there exist efficient testers for properties related to tight substructures in
multi-dimensional structured data. They serve as a first step towards a general
understanding of local properties of multi-dimensional arrays, as any such
property can be characterized by a fixed family of forbidden patterns
A generalization of Thue freeness for partial words
This paper approaches the combinatorial problem of Thue freeness for partial words. Partial words are sequences over a finite alphabet that may contain a number of ?holes?. First, we give an infinite word over a three-letter alphabet which avoids squares of length greater than two even after we replace an infinite number of positions with holes. Then, we give an infinite word over an eight-letter alphabet that avoids longer squares even after an arbitrary selection of its positions are replaced with holes, and show that the alphabet size is optimal. We find similar results for overlap-free partial words
Repetitions in partial words
El objeto de esta tesis está representado por las repeticiones de palabras parciales, palabras que, además de las letras regulares, pueden tener un número de sÃmbolos desconocidos,llamados sÃmbolos "agujeros" o "no sé qué". Más concretamente, se presenta y se resuelve una extensión de la noción de repetición establecida por Axel Thue. Investigamos las palabras parciales con un número infinito de agujeros que cumplen estas propiedades y, también las palabras parciales que conservan las propiedades después de la inserción de un número arbitrario de agujeros, posiblemente infinito. Luego, hacemos un recuento del número máximo de 2-repeticiones distintas compatibles con los factores de una palabra parcial. Se demuestra que el problema en el caso general es difÃcil, y estudiamos el problema en el caso de un agujero. Al final, se estudian algunas propiedades de las palabras parciales sin fronteras y primitivas (palabras sin repeticiones) y se da una caracterización del lenguaje de palabras parciales con una factorización crÃtica
Specular sets
We introduce the notion of specular sets which are subsets of groups called
here specular and which form a natural generalization of free groups. These
sets are an abstract generalization of the natural codings of linear
involutions. We prove several results concerning the subgroups generated by
return words and by maximal bifix codes in these sets.Comment: arXiv admin note: substantial text overlap with arXiv:1405.352
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