Recurrence and Frequencies

A word w is called repetitive if it contains two consecutive equal factors ; otherwise w is nonrepetitive. Thus the word abacacb is repetitive, and abcacbabcbac is nonrepetitive. There is no nonrepetitive word of length 4 over a two letter alphabet ; on the contrary, there exist infinite nonrepetitive words over a three letter alphabet. Most of the explicitly known infinite nonrepetitive words are constructed by iteration of a morphism. In this paper, we show that it is decidable whether an infinite word over a three letter alphabet obtained by iterating a morphism is nonrepetitive. We also investigate nonrepetitive morphisms, i.e. morphisms preserving nonrepetitive words, and we show that it is decidable whether a morphism (over an arbitrary finite alphabet) is nonrepetitive.

Multidimensional generalized automatic sequences and shape-symmetric morphic words

On square-free arithmetic progressions in infinite words

In this paper we undertake the general study of the Abelian complexity of an infinite word on a finite alphabet. We investigate both similarities and differences between the Abelian complexity and the usual subword complexity. While the Thue-Morse minimal subshift is neither characterized by its Abelian complexity nor by its subword complexity alone, we show that the subshift is completely characterized by the two complexity functions together. We give an affirmative answer to an old question of G. Rauzy by exhibiting a class of words whose Abelian complexity is everywhere equal to 3. We also investigate links between Abelian complexity and the existence of Abelian powers. Using van der Waerden's Theorem, we show that any minimal subshift having bounded Abelian complexity contains Abelian k-powers for every positive integer k. In the case of Sturmian words we prove something stronger: For every Sturmian word w and positive integer k, each sufficiently long factor of w begins in an Abelian k-power.

Quasiperiods of biinfinite words

Automated reasoning about systems with infinite domains requires an extension of regular automata to infinite alphabets. Existing formalisms of such automata cope with the infiniteness of the alphabet by adding to the automaton a set of registers or pebbles, or by attributing the alphabet by labels from an auxiliary finite alphabet that is read by an intermediate transducer. These formalisms involve a complicated mechanism on top of the transition function of automata over finite alphabets and are therefore difficult to understand and to work with. We introduce and study variable finite automata over infinite alphabets (VFA). VFA form a natural and simple extension of regular (and ω-regular) automata, in which the alphabet consists of letters as well as variables that range over the infinite alphabet domain. Thus, VFAs have the same structure as regular automata, only that some of the transitions are labeled by variables. We compare VFA with existing formalisms, and study their closure properties and classical decision problems. We consider the settings of both finite and infinite words. In addition, we identify and study the deterministic fragment of VFA. We show that while this fragment is sufficiently strong to express many interesting properties, it is closed under union, intersection, and complementation, and its nonemptiness and containment problems are decidable. Finally, we describe a determinization process for a determinizable subset of VFA.

The aim of this chapter is to present some recent research work in the combinatorial aspects of the theory of semigroups which are of great interest both for Algebra and Theoretical Computer Science. This research mainly concerns that part of combinatorics of finite and infinite words over a finite alphabet, which is usually called the theory of “unavoidable” regularities. The book by Lothaire [77] can be considered an excellent introduction to this theory.

We consider several problems of infinite words over a finite alphabet. In particular, we describe a few automata-theoretic methods to define infinite words. Properties of infinite words studied in more details are repetition-freeness, periodicity and different kinds of complexity issues. Examples are used to illustrate the power of infinite words in many applications, as well as illustrations of problems from different areas of mathematics.

We study infinite words u over an alphabet $\mathcal{A}$ satisfying the property $\mathcal{P} :~\mathcal{P}(n)+ \mathcal{P}(n+1) = 1+ \#\mathcal{A} {\rm any} n \in \mathbb{N}$ , where $\mathcal{P}(n)$ denotes the number of palindromic factors of length n occurring in the language of u . We study also infinite words satisfying a stronger property $\mathcal{PE}$ : every palindrome of u has exactly one palindromic extension in u . For binary words, the properties $\mathcal{P}$ and $\mathcal{PE}$ coincide and these properties characterize Sturmian words, i.e., words with the complexity C(n) = n + 1 for any $n \in \mathbb{N}$ . In this paper, we focus on ternary infinite words with the language closed under reversal. For such words u , we prove that if C(n) = 2n + 1 for any $n \in \mathbb{N}$ , then u satisfies the property $\mathcal{P}$ and moreover u is rich in palindromes. Also a sufficient condition for the property $\mathcal{PE}$ is given. We construct a word demonstrating that $\mathcal{P}$ on a ternary alphabet does not imply $\mathcal{PE}$ .

Let r and s be multiplicatively independent positive integers. We establish that the r-ary expansion and the s-ary expansion of an irrational real number, viewed as infinite words on {0,1,...,r-1} and {0,1,...,s-1}, respectively, cannot have simultaneously a low block complexity. In particular, they cannot be both Sturmian words.

Prefix normal words are binary words that have no factor with more 1s than the prefix of the same length. Finite prefix normal words were introduced in [Fici and Liptak, DLT 2011]. In this paper, we study infinite prefix normal words and explore their relationship to some known classes of infinite binary words. In particular, we establish a connection between prefix normal words and Sturmian words, between prefix normal words and abelian complexity, and between prefix normality and lexicographic order.

In this paper we give a broad unified framework via group actions for constructing complexity functions of infinite words x = x 0 x 1 x 2 ⋯ ∈ A N x=x_0x_1x_2\cdots \in \mathbb {A}^{\mathbb {N}} with values in a finite set A . \mathbb {A}. Factor complexity, Abelian complexity and cyclic complexity are all particular cases of this general construction. We consider infinite sequences of permutation groups ω = ( G n ) n ≥ 1 \omega = (G_n)_{n\geq 1} with each G n ⊆ S n . G_n \subseteq S_n. Associated with every such sequence is a complexity function p ω , x : N → N p_{\omega ,x}:\mathbb {N} \rightarrow \mathbb {N} which counts, for each length n , n, the number of equivalence classes of factors of x x of length n n under the action of G n G_n on A n \mathbb {A}^n given by g ∗ ( u 1 u 2 ⋯ u n ) = u g − 1 ( 1 ) u g − 1 ( 2 ) ⋯ u g − 1 ( n ) . g* (u_1u_2\cdots u_n)=u_{g^{-1}(1)}u_{g^{-1}(2)}\cdots u_{g^{-1}(n)}. Each choice of ω = ( G n ) n ≥ 1 \omega =(G_n)_{n\geq 1} defines a unique complexity function which reflects a different combinatorial property of a given infinite word. For instance, an infinite word x x has bounded Abelian complexity if and only if x x is k k -balanced for some positive integer k , k, while bounded cyclic complexity is equivalent to x x being ultimately periodic. A celebrated result of G.A. Hedlund and M. Morse states that every aperiodic infinite word x ∈ A N x\in \mathbb {A}^{\mathbb {N}} contains at least n + 1 n+1 distinct factors of each length n . n. Moreover x ∈ A N x\in \mathbb {A}^{\mathbb {N}} has exactly n + 1 n+1 distinct factors of each length n n if and only if x x is a Sturmian word, i.e., binary, aperiodic and balanced. We prove that this characterisation of aperiodicity and Sturmian words extends to this general framework.

In our paper [A square root map on Sturmian words, Electron. J. Combin. 24.1 (2017)], we introduced a symbolic square root map. Every optimal squareful infinite word s contains exactly six minimal squares and can be written as a product of these squares: \(s = X_1^2 X_2^2 \cdots \). The square root \(\sqrt{s}\) of s is the infinite word \(X_1 X_2 \cdots \) obtained by deleting half of each square. We proved that the square root map preserves the languages of Sturmian words (which are optimal squareful words). The dynamics of the square root map on a Sturmian subshift are well understood. In our earlier work, we introduced another type of subshift of optimal squareful words which together with the square root map form a dynamical system. In this paper, we study these dynamical systems in more detail and compare their properties to the Sturmian case. The main results are characterizations of periodic points and the limit set. The results show that while there is some similarity it is possible for the square root map to exhibit quite different behavior compared to the Sturmian case.

Standard Sturmian infinite words have a curious property, discovered by G. Rauzy. If in such a word we delete all occurrences of each letter, except every pth one, then we get the same infinite word. This property and several generalizations are studied here. In the last part a short and self-contained theory of Sturmian words, using only combinatorial arguments, is presented.

Temporal logics, first-order logics, and automata over data words have recently attracted considerable attention. A data word is a word over a finite alphabet, together with a datum (an element of an infinite domain) at each position. Examples include timed words and XML documents. To refer to the data, temporal logics are extended with the freeze quantifier, first-order logics with predicates over the data domain, and automata with registers or pebbles. We investigate relative expressiveness and complexity of standard decision problems for LTL with the freeze quantifier (LTLdarr), 2-variable first-order logic (FO2) over data words, and register automata. The only predicate available on data is equality. Previously undiscovered connections among those formalisms, and to counter automata with incrementing errors, enable us to answer several questions left open in recent literature. We show that the future-time fragment of LTLdarr which corresponds to FO2 over finite data words can be extended considerably while preserving decidability, but at the expense of non-primitive recursive complexity, and that most of further extensions are undecidable. We also prove that surprisingly, over infinite data words, LTLdarr without the 'until' operator, as well as nonemptiness of one-way universal register automata, are undecidable even when there is only 1 register