28 research outputs found
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
On the aperiodic avoidability of binary patterns with variables and reversals
In this work we present a characterisation of the avoidability of all unary and binary patterns, that do not only contain variables but also reversals of their instances, with respect to aperiodic infinite words. These types of patterns were studied recently in either more general or particular cases
Contextual partial commutations
We consider the monoid T with the presentation which is "close" to trace monoids. We prove two different types of results. First, we give a combinatorial description of the lexicographically minimum and maximum representatives of their congruence classes in the free monoid {a; b}* and solve the classical equations, such as commutation and conjugacy in T. Then we study the closure properties of the two subfamilies of the rational subsets of T whose lexicographically minimum and maximum cross-sections respectively, are rational in {a; b}*. © 2010 Discrete Mathematics and Theoretical Computer Science
A note on Thue games
In this work we improve on a result from [1]. In particular,
we investigate the situation where a word is constructed jointly by two
players who alternately append letters to the end of an existing word.
One of the players (Ann) tries to avoid (non-trivial) repetitions, while
the other one (Ben) tries to enforce them. We show a construction that
is closer to the lower bound showed in [2] using entropy compression, and
building on the probabilistic arguments based on a version of the Lov´asz
Local Lemma from [3]. We provide an explicit strategy for Ann to avoid
(non-trivial) repetitions over a 7-letter alphabet
Binary Patterns in Binary Cube-Free Words: Avoidability and Growth
The avoidability of binary patterns by binary cube-free words is investigated
and the exact bound between unavoidable and avoidable patterns is found. All
avoidable patterns are shown to be D0L-avoidable. For avoidable patterns, the
growth rates of the avoiding languages are studied. All such languages, except
for the overlap-free language, are proved to have exponential growth. The exact
growth rates of languages avoiding minimal avoidable patterns are approximated
through computer-assisted upper bounds. Finally, a new example of a
pattern-avoiding language of polynomial growth is given.Comment: 18 pages, 2 tables; submitted to RAIRO TIA (Special issue of Mons
Days 2012
Contextual partial commutations
We consider the monoid T with the presentation which is "close" to trace monoids. We prove two different types of results. First, we give a combinatorial description of the lexicographically minimum and maximum representatives of their congruence classes in the free monoid {a; b}* and solve the classical equations, such as commutation and conjugacy in T. Then we study the closure properties of the two subfamilies of the rational subsets of T whose lexicographically minimum and maximum cross-sections respectively, are rational in {a; b}*. © 2010 Discrete Mathematics and Theoretical Computer Science
Pattern Matching with Variables: Fast Algorithms and New Hardness Results
A pattern (i. e., a string of variables and terminals) maps to a word, if this is obtained by uniformly replacing the variables by terminal words; deciding this is NP-complete. We present efficient
algorithmsfootnote{The computational model we use is the standard unit-cost RAM with logarithmic word size. Also, all logarithms appearing in our time complexity evaluations are in base 2.} that solve this problem for restricted classes of patterns. Furthermore, we show that it is NP-complete to decide, for a given number k and a word w, whether w can be factorised into k distinct factors; this shows that the injective version (i.e., different variables are replaced by different words) of the above matching problem is NP-complete even for very restricted cases
Finding Pseudo-repetitions
Pseudo-repetitions are a natural generalization of the classical notion of repetitions in sequences. We solve fundamental algorithmic questions on pseudo-repetitions by application of insightful combinatorial results on words. More precisely, we efficiently decide whether a word is a pseudo-repetition and find all the pseudo-repetitive factors of a word
Pattern matching with variables: Efficient algorithms and complexity results
A pattern α (i. e., a string of variables and terminals) matches a word w, if w can be obtained by uniformly replacing the variables of α by terminal words. The respective matching problem, i. e., deciding whether or not a given pattern matches a given word, is generally NP-complete, but can be solved in polynomial-time for restricted classes of patterns. We present efficient algorithms for the matching problem with respect to patterns with a bounded number of repeated variables and patterns with a structural restriction on the order of variables. Furthermore, we show that it is NP-complete to decide, for a given number k and a word w, whether w can be factorised into k distinct factors. As an immediate consequence of this hardness result, the injective version (i. e., different variables are replaced by different words) of the matching problem is NP-complete even for very restricted clases of patterns