Bi-infinite Words Research Articles

On bi-infinite and conjugate post correspondence problems

We study two modifications of the Post Correspondence Problem (PCP), namely (1) the bi-infinite version, where it is asked whether there exists a bi-infinite word such that two given morphisms agree on it, and (2) the conjugate version, where we require the images of a solution for two given morphisms are conjugates of each other. For the conjugate PCP we give an undecidability proof by reducing it to the word problem for a special type of semi-Thue systems and for the bi-infinite PCP we give a simple argument that it is in the class Σ20 of the arithmetical hierarchy.

Quasiperiods of biinfinite words

We study the notion of quasiperiodicity, in the sense of “coverability”, for biinfinite words. All previous work about quasiperiodicity focused on right infinite words, but the passage to the biinfinite case could help to prove stronger results about quasiperiods of Sturmian words. We demonstrate this by showing that all biinfinite Sturmian words have infinitely many quasiperiods, which is not quite (but almost) true in the right infinite case, and giving a characterization of those quasiperiods.The main difference between right infinite and the biinfinite words is that, in the latter case, we might have several quasiperiods of the same length. This is not possible with right infinite words because a quasiperiod has to be a prefix of the word. We study in depth the relations between quasiperiods of the same length in a given biinfinite quasiperiodic word. This study gives enough information to allow to determine the set of quasiperiods of an arbitrary word.

Variations of the Morse-Hedlund Theorem for k-Abelian Equivalence

In this paper we investigate local-to-global phenomena for a new family of complexity functions of infinite words indexed by k ≥ 0. Two finite words u and v are said to be k-abelian equivalent if for all words x of length less than or equal to k, the number of occurrences of x in u is equal to the number of occurrences of x in v. This defines a family of equivalence relations, bridging the gap between the usual notion of abelian equivalence (when k = 1) and equality (when k = ∞). Given an infinite word w, we consider the associated complexity function which counts the number of k-abelian equivalence classes of factors of w of length n. As a whole, these complexity functions have a number of common features: each gives a characterization of periodicity in the context of bi-infinite words, and each can be used to characterize Sturmian words in the framework of aperiodic one-sided infinite words. Nevertheless, they also exhibit a number of striking differences, the study of which is one of the main topics of our paper.

Local rule distributions, language complexity and non-uniform cellular automata

This paper investigates a variant of cellular automata, namely ν-CA. Indeed, ν-CA are cellular automata which can have different local rules at each site of their lattice. The assignment of local rules to sites of the lattice completely characterizes ν-CA. In this paper, sets of assignments sharing some interesting properties are associated with languages of bi-infinite words. The complexity classes of these languages are investigated providing an initial rough classification of ν-CA.

Cylinders, multi-cylinders and the induced action of Aut(Fn)

A cylinder is the set of infinite words with fixed prefix u. A double-cylinder is “the same” for bi-infinite words. We show that for every word u and any automorphism of the free group F the image is a finite union of cylinders. The analogous statement is true for double cylinders. We give (a) an algorithm, and (b) a precise formula which allows one to determine this finite union of cylinders.

A note on the Markoff condition and central words

We define Markoff words as certain factors appearing in bi-infinite words satisfying the Markoff condition. We prove that these words coincide with central words, yielding a new characterization of Christoffel words.

On a zeta function associated with automata and codes

The zeta function of a finite automaton A is exp { ∑ n = 1 ∞ a n z n n } , where a n is the number of bi-infinite paths in A labelled by a bi-infinite word of period n . It reflects the properties of A : aperiodicity, nil-simplicity, existence of a zero. The results are applied to codes.

A tight linear bound on the synchronization delay of bijective automata

Reversible cellular automata (RCA) are models of massively parallel computation that preserve information. We generalize these systems by introducing the class of ω ω bijective finite automata. It consists of those finite automata where for any bi-infinite word there exists a unique path labelled by that word. These systems are strictly included in the class of local automata. Although the synchronization delay of an n -state local automaton is known to be Θ ( n 2 ) in the worst case, we prove that in the case of ω ω bijective finite automata the synchronization delay is at most n − 1 . Based on this we prove that for a one-dimensional n -state RCA where the neighborhood consists of m consecutive cells, the neighbourhood of the inverse automaton consists of at most n m − 1 − ( m − 1 ) cells. Similar bounds are obtained also in [E. Czeizler, J. Kari, A tight linear bound on the neighborhood of inverse cellular automata, in: Proceedings of ICALP 2005, in: LNCS, vol. 3580, 2005, pp. 410–420] but here the result comes as a direct consequence of the more general result. We also construct examples of RCA with large inverse neighbourhoods proving that the upper bounds provided here are the best possible in the case m = 2 .

A characterization of periodicity of bi-infinite words

A finite word is called bordered if it has a proper prefix which is also a suffix of that word. Costa proves in [Theoret. Comput. Sci. 290(3) (2003) 2053–2061] that a bi-infinite word w is of the form fgf ω ω , for some finite words f and g, if, and only if, there is a factorization w = suv , with u ∈ A * such that every factor s ′ uv ′ , with s ′ ≼ s and v ′ ⩽ v , is bordered. We present a shorter proof of that result in this paper.

Periodicity and local complexity

We consider the complexity of bi-infinite words in one and two dimensions. A result of Morse and Hedlund in one dimension states that if the complexity, p ξ ( n ), of a word satisfies p ξ ( n )⩽ n for some n , then the word ξ is periodic. The corresponding question in two dimensions (whether p ξ ( m , n )⩽ mn implies that ξ is periodic) is known as the Nivat conjecture. In this paper, we strengthen the one-dimensional result of Morse and Hedlund and prove a weak form of the Nivat conjecture, namely that if for a bi-infinite two-dimensional word ξ , p ξ ( m , n )⩽ mn /16 then ξ is periodic.

Periodicity and local complexity

We consider the complexity of bi-infinite words in one and two dimensions. A result of Morse and Hedlund in one dimension states that if the complexity, p ξ ( n), of a word satisfies p ξ ( n)⩽ n for some n, then the word ξ is periodic. The corresponding question in two dimensions (whether p ξ ( m, n)⩽ mn implies that ξ is periodic) is known as the Nivat conjecture. In this paper, we strengthen the one-dimensional result of Morse and Hedlund and prove a weak form of the Nivat conjecture, namely that if for a bi-infinite two-dimensional word ξ, p ξ ( m, n)⩽ mn/16 then ξ is periodic.

Asymptotic behaviour of bi-infinite words

We present a description of asymptotic behaviour of languages of bi-infinite words obtained by iterating morphisms defined on free monoids.

Biinfinite words with maximal recurrent unbordered factors

A finite non-empty word z is said to be a border of a finite non-empty word w if w=uz=zv for some non-empty words u and v. A finite non-empty word is said to be bordered if it admits a border, and it is said to be unbordered otherwise. In this paper, we give two characterizations of the biinfinite words of the form ωuvuω, where u and v are finite words, in terms of its unbordered factors.The main result of the paper states that the words of the form ωuvuω are precisely the biinfinite words w=⋯a−2a−1a0a1a2⋯ for which there exists a pair (l0,r0) of integers with l0<r0 such that, for every integers l⩽l0 and r⩾r0, the factor al⋯al0⋯ar0⋯ar is a bordered word.The words of the form ωuvuω are also characterized as being those biinfinite words w that admit a left recurrent unbordered factor (i.e., an unbordered factor of w that has an infinite number of occurrences “to the left” in w) of maximal length that is also a right recurrent unbordered factor of maximal length. This last result is a biinfinite analogue of a result known for infinite words.

A defect theorem for bi-infinite words

We formulate and prove a defect theorem for bi-infinite words. Let X be a finite set of words over a finite alphabet. If a nonperiodic bi-infinite word w has two X-factorizations, then the combinatorial rank of X is at most card(X)−1 , i.e., there exists a set F such that X⊆F + with card(F)< card(X) . Moreover, in the case when the combinatorial rank of X equals card(X) , the number of periodic bi-infinite words which have two different X-factorizations is finite.

Multiple factorizations of words and defect effect

We prove that if X is a finite prefix set and w is a non-periodic bi-infinite word, possessing 3 disjoint X-factorizations, then the combinatorial rank of X is at most card(X)−2 . This is one of the rare cases when a cumulative defect effect is known to hold. Finally, connections to the critical factorization theorem are discussed.

Free profinite locally idempotent and locally commutative semigroups

This paper is concerned with the structure of semigroups of implicit operations on the pseudovariety L Sl of finite locally idempotent and locally commutative semigroups. We depart from a general result of Almeida and Weil to give two descriptions of these semigroups: the first in terms of infinite words, and the second in terms of infinite and bi-infinite words. We then derive some applications.

Defect Effect of Bi-infinite Words in the Two-element Case

Let X be a two-element set of words over a finite alphabet. If a bi-infinite word possesses two X-factorizations which are not shiftequivalent, then the primitive roots of the words in X are conjugates. Note, that this is a strict sharpening of a defect theorem for bi-infinite words stated in \emphKMP. Moreover, we prove that there is at most one bi-infinite word possessing two different X-factorizations and give a necessary and sufficient conditions on X for the existence of such a word. Finally, we prove that the family of sets X for which such a word exists is parameterizable.

On low-complexity bi-infinite words and their factors

Dans cet article on etudie des mots bi-infinis sur deux symboles. On dit qu'un tel mot est de rigidite k si le nombre de facteurs differents de longueur n est egal a n+k pour n grand. Un tel mot est appele k-balance si le nombre d'occurrences du symbole a dans deux facteurs quelconques de meme longueur peuvent differer au plus de k. Dans cet article on donne une description complete de la classe des mots bi-infinis de rigidite k et on montre que le nombre de facteurs de longueur n de cette classe est de l'ordre de n 3 . Dans le cas k = 1 on donne une formule exacte. On considere aussi la classe des mots bi-infinis k-balances. II est bien connu que le nombre de facteurs de longueur n est de l'ordre de n 3 si k = 1. En revanche, on montre que ce nombre est ≥ 2 n/2 si k≥2.

A Note on Palindromicity

Two results on palindromicity of bi-infinite words in a finite alphabet are presented. The first is a simple, but efficient criterion to exclude palindromicity of minimal sequences and applies, in particular, to the Rudin–Shapiro sequence. The second provides a constructive method to build palindromic minimal sequences based upon regular, generic model sets with centro-symmetric window. These give rise to diagonal tight-binding models in one dimension with purely singular continuous spectrum.

On well quasi orders of free monoids

The paper investigates down-sets associated to well quasi orders. Of particular language-theoretic interest is the quasi order u ⩽ s v (resp. u ⩽ ℘v ) of u being a subword (resp. a Parikh subword) of v, as well as their inverses. We establish a number of results about the regularity and effective regularity of the down-sets, in particular, a general condition for the effective regularity of the down-set associated to an inverse well quasi order (Theorem 7.1 ). It is decidable whether or not an arbitrary regular language results as a down-set from an infinite or bi-infinite word, whereas the same problem is undecidable for context-free language. A quasi order being a well quasi order is connected to arbitrary languages being confluent.