An infinite permutation α is a linear ordering of N. We study propertiesof infinite permutations analogous to those of infinite words, and show some resemblancesand some differences between permutations and words. In this paper, we try to extend topermutations the notion of automaticity. As we shall show, the standard definitions whichare equivalent in the case of words are not equivalent in the context of permutations. Weinvestigate the relationships between these definitions and prove that they constitute achain of inclusions. We also construct and study an automaton generating the Thue-Morsepermutation.

The fixed point submonoid of an endomorphism of a free product of a free monoid andcyclic groups is proved to be rational using automata-theoretic techniques. Maslakova’sresult on the computability of the fixed point subgroup of a free group automorphism isgeneralized to endomorphisms of free products of a free monoid and a free group which areautomorphisms of the maximal subgroup.

Multiwords are words in which a single symbol can be replaced by a nonempty set of symbols. They extend the notion of partial words. A word w is certain in a multiword M if it occurs in every word that can be obtained by selecting one single symbol among the symbols provided in each position of M. Motivated by a problem on incomplete databases, we investigate a variant of the pattern matching problem which is to decide whether a word w is certain in a multiword M. We study the language CERTAIN(w) of multiwords in which w is certain. We show that this regular language is aperiodic for three large families of words. We also show its aperiodicity in the case of partial words over an alphabet with at least three symbols.

For an extensive range of infinite words, and the associated symbolic dynamical systems, we compute, together with the usual language complexity function counting the finite words, the minimal and maximal complexity functions we get by replacing finite words by finite patterns, or words with holes.

We study the avoidance of Abelian powers of words and consider three reasonablegeneralizations of the notion of Abelian power to fractional powers. Our main goal is tofind an Abelian analogue of the repetition threshold, i.e., a numericalvalue separating k-avoidable and k-unavoidable Abelianpowers for each size k of the alphabet. We prove lower bounds for theAbelian repetition threshold for large alphabets and all definitions of Abelian fractionalpower. We develop a method estimating the exponential growth rate of Abelian-power-freelanguages. Using this method, we get non-trivial lower bounds for Abelian repetitionthreshold for small alphabets. We suggest that some of the obtained bounds are the exactvalues of Abelian repetition threshold. In addition, we provide upper bounds for thegrowth rates of some particular Abelian-power-free languages.

Two deterministic finite automata are almost equivalent if they disagree in acceptanceonly for finitely many inputs. An automaton A is hyper-minimized if noautomaton with fewer states is almost equivalent to A. A regular languageL is canonical if the minimal automaton accepting L ishyper-minimized. The asymptotic state complexitys∗(L) of a regular languageL is the number of states of a hyper-minimized automaton for a languagefinitely different from L. In this paper we show that: (1) the class ofcanonical regular languages is not closed under: intersection, union, concatenation,Kleene closure, difference, symmetric difference, reversal, homomorphism, and inversehomomorphism; (2) for any regular languages L1 andL2 the asymptotic state complexity of their sumL1 ∪ L2, intersectionL1 ∩ L2, differenceL1 − L2, and symmetricdifference L1 ⊕ L2 can be boundedbys∗(L1)·s∗(L2).This bound is tight in binary case and in unary case can be met in infinitely many cases.(3) For any regular language L the asymptotic state complexity of itsreversal LR can be bounded by2s∗(L). This bound is tightin binary case. (4) The asymptotic state complexity of Kleene closure and concatenationcannot be bounded. Namely, for every k ≥ 3, there exist languagesK, L, and M such thats∗(K) = s∗(L) = s∗(M) = 1ands∗(K∗) = s∗(L·M) = k.These are answers to open problems formulated by Badr et al.[RAIRO-Theor. Inf. Appl. 43 (2009) 69–94].

This paper presents a new lower bound for the recursive algorithm for solving parity games which is induced by the constructive proof of memoryless determinacy by Zielonka. We outline a family of games of linear size on which the algorithm requires exponential time.

An algorithm is corrected here that was presented as Theorem 2 in [Š. Holub, RAIRO-Theor. Inf. Appl. 40 (2006) 583–591]. It is designed to calculate the maximum length of a nontrivial word with a given set of periods.

Given non-negative weights wS on the k-subsets S of a km-element set V, we consider the sum of the products wS1 ⋅⋅⋅ wSm over all partitions V = S1 ∪ ⋅⋅⋅ ∪Sm into pairwise disjoint k-subsets Si. When the weights wS are positive and within a constant factor of each other, fixed in advance, we present a simple polynomial-time algorithm to approximate the sum within a polynomial in m factor. In the process, we obtain higher-dimensional versions of the van der Waerden and Bregman–Minc bounds for permanents. We also discuss applications to counting of perfect and nearly perfect matchings in hypergraphs.