Privileged Words and Sturmian Words

This dissertation has two almost unrelated themes: privileged words and Sturmian words. Privileged words are a new class of words introduced recently. A word is privileged if it is a complete first return to a shorter privileged word, the shortest privileged words being letters and the empty word. Here we give and prove almost all results on privileged words known to date. On the other hand, the study of Sturmian words is a well-established topic in combinatorics on words. In this dissertation, we focus on questions concerning repetitions in Sturmian words, reproving old results and giving new ones, and on establishing completely new research directions. The study of privileged words presented in this dissertation aims to derive their basic properties and to answer basic questions regarding them. We explore a connection between privileged words and palindromes and seek out answers to questions on context-freeness, computability, and enumeration. It turns out that the language of privileged words is not context-free, but privileged words are recognizable by a linear-time algorithm. A lower bound on the number of binary privileged words of given length is proven. The main interest, however, lies in the privileged complexity functions of the Thue-Morse word and Sturmian words. We derive recurrences for computing the privileged complexity function of the Thue-Morse word, and we prove that Sturmian words are characterized by their privileged complexity function. As a slightly separate topic, we give an overview of a certain method of automated theorem-proving and show how it can be applied to study privileged factors of automatic words. The second part of this dissertation is devoted to Sturmian words. We extensively exploit the interpretation of Sturmian words as irrational rotation words. The essential tools are continued fractions and elementary, but powerful, results of Diophantine approximation theory. With these tools at our disposal, we reprove old results on powers occurring in Sturmian words with emphasis on the fractional index of a Sturmian word. Further, we consider abelian powers and abelian repetitions and characterize the maximum exponents of abelian powers with given period occurring in a Sturmian word in terms of the continued fraction expansion of its slope. We define the notion of abelian critical exponent for Sturmian words and explore its connection to the Lagrange spectrum of irrational numbers. The results obtained are often specialized for the Fibonacci word; for instance, we show that the minimum abelian period of a factor of the Fibonacci word is a Fibonacci number. In addition, we propose a completely new research topic: the square root map. We prove that the square root map preserves the language of any Sturmian word. Moreover, we construct a family of non-Sturmian optimal squareful words whose language the square root map also preserves.This construction yields examples of aperiodic infinite words whose square roots are periodic.

Translation – The fifth language skill? A comparison of the role of translation in Finnish lukio and the International Baccalaureate Diploma Programme

This study attempts to answer the question “Should translation be considered a fifth language skill?” by examining and comparing the use of translation as a language learning and assessment method in the national Finnish lukio curriculum and the curriculum of the International Baccalaureate Diploma Programme (IBDP). Furthermore, the students’ ability to translate and their opinions on the usefulness of translation in language learning will be examined. The students’ opinions were gathered through a questionnaire that was given to 156 students studying in either lukio or the IBDP in Turku and Rovaniemi. I present and compare the role of translation in selected language teaching and learning methods and approaches, and discuss the effectiveness of translation as a language learning method and an assessment method. The theoretical discussion provides the basis for examining the role of translation as a language learning method and an assessment method in the curricula and final examinations of both education programs. The analysis of the two curricula indicated that there is a significant difference in the use of translation, as translation is used as a language learning method and as an assessment method in lukio, but is not used in either form in the IB. The data obtained through the questionnaire indicated that there is a difference in the level of language competence between the lukio and IB students and suggested that the curriculum in which the student studies has some effect on his/her cognitive use of translation, ability to translate and opinions concerning the usefulness of translation in language learning. The results indicated that both groups of students used translation, along with their mother tongue, as a cognitive language learning method, and, contrary to the expectations set by the analysis of the two curricula, the IB students performed better in the translation exercises than lukio students. Both groups of students agreed that translation is a useful language learning method, and indicated that the most common dictionaries they use are bilingual Internet dictionaries. The results suggest that translation is a specific skill that requires teaching and practice, and that perhaps the translation exercises used in lukio should be developed from translating individual words and phrases to translating cultural elements. In addition, the results suggest that perhaps the IB curriculum should include the use of translation exercises (e.g., communicative translation exercises) in order to help students learn to mediate between languages and cultures rather than learn languages in isolation from each other.