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.

Consider partial maps ∑* → $\mathbb R$ with a rational domain. We show that two families of such series are actually the same: the unambiguous rational series on the one hand, and the max-plus and min-plus rational series on the other hand. The decidability of equality was known to hold in both families with different proofs, so the above unifies the picture. We give an effective procedure to build an unambiguous automaton from a max-plus automaton and a min-plus one that recognize the same series.

The MATRIX PACKING DOWN problem asks to find a row permutation of a given (0,1)-matrix in such a way that the total sum of the first non-zero column indexes is maximized. We study the computational complexity of this problem. We prove that the MATRIX PACKING DOWN problem is NP-complete even when restricted to zero trace symmetric (0,1)-matrices or to (0,1)-matrices with at most two 1's per column. Also, as intermediate results, we introduce several new simple graph layout problems which are proved to be NP-complete.

We present a model in which, due to the quantum nature of the signals controlling the implementation time of successive unitary computational steps, physical irreversibility appears in the execution of a logically reversible computation.

We characterize conjugation classes of Christoffel words (equivalently of standard words) by the number of factors. We give several geometric proofs of classical results on these words and sturmian words.

We introduce doubly-ranked (DR) monoids in order to study picturecodes. We show that a DR-monoid is free iff it is pictoriallystable. This allows us to associate with a set C of pictures apicture code B(C) which is the basis of the least DR-monoidincluding C.A weak version of the defect theorem for pictures is established.A characterization of picture codes through picture series isalso given.

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.

In the infinite Post Correspondence Problem an instance (h,g)consists of two morphisms h and g, and the problem is todetermine whether or not there exists an infinite word ωsuch that h(ω) = g(ω). This problem was shown to beundecidable by Ruohonen (1985) in general. Recently Blondel and Canterini (Theory Comput. Syst.36(2003) 231–245) showed that this problem is undecidable for domainalphabets of size 105. Here we give a proof that the infinite PostCorrespondence Problem is undecidable for instances where themorphisms have domains of 9 letters. The proof uses a recentresult of Matiyasevich and Sénizergues and a modification of aresult of Claus.

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.

It is known that the class of factorizing codes, i.e.,codes satisfying the factorization conjectureformulated by Schützenberger, isclosed under two operations:the classicalcomposition of codes and substitutionof codes.A natural question which arisesis whethera finite set Oof operations existssuch that each factorizingcode can be obtained by usingthe operations inO and starting with prefix or suffix codes.O is named herea complete setof operations (for factorizing codes).We show that composition and substitutionare not enough in order to obtaina completeset. Indeed, we exhibit a factorizing code over a two-letter alphabet A = {a,b}, precisely a 3-code, which cannot beobtained by decomposition or substitution.

Increasing integer sequences include many instances of interesting sequences and combinatorial structures, ranging from tournaments to addition chains, from permutations to sequences having the Goldbach property that any integer greater than 1 can be obtained as the sum of two elements in the sequence. The paper introduces and compares several of these classes of sequences, discussing recurrence relations, enumerative problems and questions concerning shortest sequences.

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 present a novel eye localization method which can be used in face recognition applications. It is based on two SVM classifiers which localize the eyes at different resolution levels exploiting the Haar wavelet representation of the images. We present an extensive analysis of its performance on images of very different public databases, showing very good results.

In secret sharing, different access structures have different difficulty degrees for acceding to the secret. We give a numerical measure of how easy or how difficult is to recover the secret, depending only on the structure itself and not on the particular scheme used for realizing it. We derive some consequences.

It is shown that small fragments of the first-order theory of the subword order, the (partial) lexicographic path ordering on words, the homomorphism preorder, and the infix order are undecidable. This is in contrast to the decidability of the monadic second-order theory of the prefix order [M.O. Rabin, Trans. Amer. Math. Soc., 1969] and of the theory of the total lexicographic path ordering [P. Narendran and M. Rusinowitch, Lect. Notes Artificial Intelligence, 2000] and, in case of the subword and the lexicographic path order, improves upon a result by Comon & Treinen [H. Comon and R. Treinen, Lect. Notes Comp. Sci., 1994]. Our proofs rely on the undecidability of the positive ∑1-theory of $(\mathbb N,+,\cdot)$ [Y. Matiyasevich, Hilbert's Tenth Problem, 1993] and on Treinen's technique [R. Treinen, J. Symbolic Comput., 1992] that allows to reduce Post's correspondence problem to logical theories.

Arithmetical complexity of a sequence is the number of words of length n that can be extracted from it according to arithmetic progressions. We study uniformly recurrent words of low arithmetical complexity and describe the family of such words having lowest complexity.

For both the edge deletion heuristic and the maximum-degree greedy heuristic, we studythe problem of recognizing those graphs for which that heuristic can approximate the size of a minimum vertex cover within a constant factor of r, where r is a fixed rational number. Our main results are that these problems are complete for the class of problems solvable via parallel access to NP. To achieve these main results, we also show that the restriction of the vertex cover problem to those graphs for which either of these heuristics can find an optimal solution remains NP-hard.

We compare various computational complexity classes defined within the framework of membrane systems, a distributed parallel computing device which is inspired from the functioning of the cell, with usual computational complexity classes for Turing machines. In particular, we focus our attention on the comparison among complexity classes for membrane systems with active membranes (where new membranes can be created by division of existing membranes) and the classes PSPACE, EXP, and EXPSPACE.

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.