On Infinite Prefix Normal Words

We present a new class of binary words: the prefix normal words. They are defined by the property that for any given length k, no factor of length k has more a’s than the prefix of the same length. These words arise in the context of indexing for jumbled pattern matching (a.k.a. permutation matching or Parikh vector matching), where the aim is to decide whether a string has a factor with a given multiplicity of characters, i.e., with a given Parikh vector. Using prefix normal words, we give the first non-trivial characterization of binary words having the same set of Parikh vectors of factors. We prove that the language of prefix normal words is not context-free and is strictly contained in the language of pre-necklaces, which are prefixes of powers of Lyndon words. We discuss further properties and state open problems.

On generating binary words palindromically

G. Fici proved that a finite word has a minimal suffix automaton if and only if all its left special factors occur as prefixes. He called LSP all finite and infinite words having this latter property. We characterize here infinite LSP words in terms of S-adicity. More precisely we provide a finite set of morphisms S and an automaton \(\mathcal{A}\) such that an infinite word is LSP if and only if it is S-adic and all its directive words are recognizable by \(\mathcal{A}\).

Numerical characteristics of polynomial identities of left nilpotent algebras are examined. Previously, we came up with a construction which, given an infinite binary word, allowed us to build a two-step left nilpotent algebra with specified properties of the codimension sequence. However, the class of the infinite words used was confined to periodic words and Sturm words. Here the previously proposed approach is generalized to a considerably more general case. It is proved that for any algebra constructed given a binary word with subexponential function of combinatorial complexity, there exists a PI-exponent, and its precise value is computed.

We consider two-variable first-order logic FO2 and its quantifier alternation hierarchies over both finite and infinite words. Our main results are forbidden patterns for deterministic automata (finite words) and for Carton-Michel automata (infinite words). In order to give concise patterns, we allow the use of subwords on paths in finite graphs. This concept is formalized as subword-patterns. For certain types of subword-patterns there exists a non-deterministic logspace algorithm to decide their presence or absence in a given automaton. In particular, this leads to NL algorithms for deciding the levels of the FO2 quantifier alternation hierarchies. This applies to both full and half levels, each over finite and infinite words. Moreover, we show that these problems are NL-hard and, hence, NL-complete.

This paper is concerned with the language inclusion problem for timed automata: given timed automata ${\mathcal A}$ and ${\mathcal B}$, is every word accepted by ${\mathcal B}$ also accepted by ${\mathcal A}$? Alur and Dill [3] showed that the language inclusion problem is decidable if ${\mathcal A}$ has no clocks and undecidable if ${\mathcal A}$ has two clocks (with no restriction on ${\mathcal B}$). However, the status of the problem when ${\mathcal A}$ has one clock is not determined by [3]. In this paper we close this gap for timed automata over infinite words by showing that the one-clock language inclusion problem is undecidable. For timed automata over finite words, building on our earlier paper [20], we show that the one-clock language inclusion problem is decidable with non-primitive recursive complexity. This reveals a surprising divergence between the theory of timed automata over finite words and over infinite words. Finally, we show that if e-transitions or non-singular postconditions are allowed, then the one-clock language inclusion problem is undecidable over both finite and infinite words.

Over finite words, languages of dot-depth one are expressively complete for alternation-free first-order logic. This fragment is also known as the Boolean closure of existential first-order logic. Here, the atomic formulas comprise order, successor, minimum, and maximum predicates. Knast (1983) has shown that it is decidable whether a language has dot-depth one. We extend Knast's result to infinite words. In particular, we describe the class of languages definable in alternation-free first-order logic over infinite words, and we give an effective characterization of this fragment. This characterization has two components. The first component is identical to Knast's algebraic property for finite words and the second component is a topological property, namely being a Boolean combination of Cantor sets. As an intermediate step we consider finite and infinite words simultaneously. We then obtain the results for infinite words as well as for finite words as special cases. In particular, we give a new proof of Knast's Theorem on languages of dot-depth one over finite 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.

An infinite word is an infinite Lyndon word if it is smaller, with respect to the lexicographic order, than all its proper suffixes, or equivalently if it has infinitely many finite Lyndon words as prefixes. A characterization of binary endomorphisms generating Lyndon infinite words is provided.

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.

We obtain the following results about the avoidance of ternary formulas. Up to renaming of the letters, the only infinite ternary words avoiding the formula $ABCAB.ABCBA.ACB.BAC$ (resp. $ABCA.BCAB.BCB.CBA$) have the same set of recurrent factors as the fixed point of $0\mapsto 012$, $1\mapsto 02$, $2\mapsto 1$. The formula $ABAC.BACA.ABCA$ is avoided by polynomially many binary words and there exists arbitrarily many infinite binary words with different sets of recurrent factors that avoid it. If every variable of a ternary formula appears at least twice in the same fragment, then the formula is $3$-avoidable. The pattern $ABACADABCA$ is unavoidable for the class of $C_4$-minor-free graphs with maximum degree~$3$. This disproves a conjecture of Grytczuk. The formula $ABCA.ACBA$, or equivalently the palindromic pattern $ABCADACBA$, has avoidability index $4$.

We introduce and study a new complexity function in combinatorics on words, which takes into account the smallest second occurrence time of a factor of an infinite word. We characterize the eventually periodic words and the Sturmian words by means of this function. Then, we establish a new result on repetitions in Sturmian words and show that it is best possible. Let b ≥ 2 b \ge 2 be an integer. We deduce a lower bound for the irrationality exponent of real numbers whose sequence of b b -ary digits is a Sturmian sequence over { 0 , 1 , … , b − 1 } \{0, 1, \ldots , b-1\} and we prove that this lower bound is best possible. As an application, we derive some information on the b b -ary expansion of log ⁡ ( 1 + 1 a ) \log (1 + \frac {1}{a}) for any integer a ≥ 34 a \ge 34 .

Automata over infinite words provide a powerful framework to solve various decision problems. However, the mechanized reasoning with restricted classes of automata over infinite words is often simpler and more efficient. For instance, weak deterministic Büchi automata (wdbas) can be handled algorithmically almost as efficient as deterministic automata over finite words. In this paper, we show how and when the standard powerset construction for automata over finite words can be used to determinize automata over infinite words. An instance is the class of automata that accept wdba-recognizable languages. Furthermore, we present applications of this new determinization construction. Namely, we apply it to improve the automata-based approach for the mixed first-order linear arithmetic over the reals and the integers, and we utilize it to accelerate finite state model checking. We report on experimental results for these two applications.

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 $\mathcal{PE}$ .

Polynomial identities and codimension growth of nonassociative algebras over a field of characteristic zero are considered. A new approach is proposed for constructing nonassociative algebras starting from a given infinite binary word. The sequence of codimensions of such an algebra is closely connected with the combinatorial complexity of the defining word. These constructions give new examples of algebras with abnormal codimension growth. The first important achievement of the given approach is that the algebras under study are finitely generated. The second one is that the asymptotic behavior of codimension sequences is widely different from all previous examples.