Testing avoidability on sets of partial words is hard

We prove that the problem of deciding whether a finite set of partial words is unavoidable is NP-hard for any alphabet of size larger than or equal to two, which is in contrast with the well-known feasability results for unavoidability of a set of full words. We raise some related questions on avoidability of sets of partial words

Unavoidable Sets of Partial Words

The notion of an unavoidable set of words appears frequently in the fields of mathematics and theoretical computer science, in particular with its connection to the study of combinatorics on words. The theory of unavoidable sets has seen extensive study over the past twenty years. In this paper we extend the definition of unavoidable sets of words to unavoidable sets of partial words. Partial words, or finite sequences that may contain a number of ?do not know? symbols or ?holes,? appear naturally in several areas of current interest such as molecular biology, data communication, and DNA computing. We demonstrate the utility of the notion of unavoidability of sets of partial words by making use of it to identify several new classes of unavoidable sets of full words. Along the way we begin work on classifying the unavoidable sets of partial words of small cardinality. We pose a conjecture, and show that affirmative proof of this conjecture gives a sufficient condition for classifying all the unavoidable sets of partial words of size two. We give a result which makes the conjecture easy to verify for a significant number of cases. We characterize many forms of unavoidable sets of partial words of size three over a binary alphabet, and completely characterize such sets over a ternary alphabet. Finally, we extend our results to unavoidable sets of partial words of size k over a k-letter alphabet

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 símbolos desconocidos,llamados símbolos "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 difícil, 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 crítica

Two Element Unavoidable Sets Of Partial Words

The notion of an unavoidable set of words appears frequently in the fields of mathematics and theoretical computer science, in particular with its connection to the study of combinatorics on words. The theory of unavoidable sets has seen extensive study over the past twenty years. In this paper we extend the definition of unavoidable sets of words to unavoidable sets of partial words. Partial words, or finite sequences that may contain a number of do not know symbols or holes, appear in natural ways in several areas of current interest such as molecular biology, data communication, DNA computing, etc. We demonstrate the utility of the notion of unavoidability on partial words by making use of it to identify several new classes of unavoidable sets of full words. Along the way we begin work on classifying the unavoidable sets of partial words of small cardinality. We pose a conjecture, and show that affirmative proof of this conjecture gives a sufficient condition for classifying all the unavoidable sets of partial words of size two. Lastly we give a result which makes the conjecture easy to verify for a significant number of cases. © Springer-Verlag Berlin Heidelberg 2007