This Special Issue of RAIRO, Theoretical Informatics and Applications is devoted to full versions of selected papers from the workshop Word Avoidability, Complexity and Morphisms which took place in Turku (Finland) on July 2004, as a satellite event of the conference ICALP'2004.
The topics of this one day workshop concern particular aspects of Combinatorics on Words: string pattern avoidability, complexities of finite or infinite words, and free monoids morphims.
The scientific program of the worshop consisted of two invited lectures given by A.E. Frid and J. Shallit as well as presentations of 8 contributed papers selected by the Program Committee from a total of 12 submissions.
The Program Committee consisted of : A. Carpi (Perugia), J. Cassaigne (Marseille), J. Currie (Winnipeg), I. Petre (Turku), G. Richomme (Amiens), A. M. Shur (Ekaterinburg), and P. Séébold (Montpellier). 
This special issue contains the full text of five of the papers presented at the workshop. All of them were normally refereed and revised.
I would like to thank everybody who contributed to the workshop and this issue including invited speakers, members of the program Committee, ICALP'2004's organization, authors and all referees of papers submitted to the worshop and this special issue. I am indebtful to J. Kärhumaki and J. Berstel for thinking of me to organize this workshop. Special thanks to P. Séébold for all his advices. Finally I am grateful to C. Choffrut, the editor-chief of the RAIRO-TIA for accepting to publish this special issue.

Let I be a finite set of words and $\Rightarrow_I^*$ be the derivation relation generated by the set of productions {ε → u | u ∈ I}. Let $L_I^{\epsilon}$ be the set of words u such that $\epsilon \Rightarrow_I^* u$. We prove that the set I is unavoidable if and only if the relation $\Rightarrow_I^*$ is a well quasi-order on the set $L_I^{\epsilon}$. This result generalizes a theorem of [Ehrenfeucht et al.,Theor. Comput. Sci.27 (1983) 311–332]. Further generalizations are investigated.

We present an algorithm which produces, in some cases, infinite words avoiding both large fractional repetitions and a given set of finite words. We use this method to show that all the ternary patterns whose avoidability index was left open in Cassaigne's thesis are 2-avoidable. We also prove that there exist exponentially many $\frac{7}{4}^+$-free ternary words and $\frac{7}{5}^+$-free 4-ary words. Finally we give small morphisms for binary words containing only the squares 2, 12 and (01)² and for binary words avoiding large squares and fractional repetitions.

The arithmetical complexity of infinite words, defined by Avgustinovich, Fon-Der-Flaass and the author in 2000, is the number of words of length n which occur in the arithmetical subsequences of the infinite word. This is one of the modifications of the classical function of subword complexity, which is equal to the number of factors of the infinite word of length n. In this paper, we show that the orders of growth of the arithmetical complexity can behave as many sub-polynomial functions. More precisely, for each sequence u of subword complexity ƒu(n) and for each prime p ≥ 3 we build a Toeplitz word on the same alphabet whose arithmetical complexity is $a(n)=\Theta(n f_u(\lceil \log_p n \rceil))$.

We divide infinite sequences of subword complexity 2n+1 into four subclasses with respect to left and right special elements and examine the structure of the subclasses with the help of Rauzy graphs. Let k ≥ 2 be an integer. If the expansion in base k of a number is an Arnoux-Rauzy word, then it belongs to Subclass I and the number is known to be transcendental. We prove the transcendence of numbers with expansions in the subclasses II and III.

We consider the position and number of occurrences of squares in the Thue-Morse sequence, and show that the corresponding sequences are 2-regular. We also prove that changing any finite but nonzero number of bits in the Thue-Morse sequence creates an overlap, and any linear subsequence of the Thue-Morse sequence (except those corresponding to decimation by a power of 2) contains an overlap.

The paper treats the question whether there always exists a minimal nondeterministic finite automaton of n states whose equivalent minimal deterministic finite automaton has α states for any integers n and α with n ≤ α ≤ 2n. Partial answers to this question were given by Iwama, Kambayashi, and Takaki (2000) and by Iwama, Matsuura, and Paterson (2003). In the present paper, the question is completely solved by presenting appropriate automata for all values of n and α. However, in order to give an explicit construction of the automata, we increase the input alphabet to exponential sizes. Then we prove that 2n letters would be sufficient but we describe the related automata only implicitly. In the last section, we investigate the above question for automata over binary and unary alphabets.

We investigate the complexity of several problems concerning Las Vegas finite automata. Our results are as follows. (1) The membership problem for Las Vegas finite automata is in NL. (2) The nonemptiness and inequivalence problems for Las Vegas finite automata are NL-complete. (3) Constructing for a given Las Vegas finite automaton a minimum state deterministic finite automaton is in NP. These results provide partial answers to some open problems posed by Hromkovič and Schnitger [Theoret. Comput. Sci.262 (2001) 1–24)].

Among Sturmian words, some of them are morphic, i.e. fixed point of a non-identical morphism on words. Berstel and Séébold (1993) have shown that if a characteristic Sturmian word is morphic, then it can be extended by the left with one or two letters in such a way that it remains morphic and Sturmian. Yasutomi (1997) has proved that these were the sole possible additions and that, if we cut the first letters of such a word, it didn't remain morphic. In this paper, we give an elementary and combinatorial proof of this result.