Bi-infinite Words Research Articles

Languages obtained from infinite words

We prove that it is decidable whether or not a regular language can be written as the set of all finite factors of an infinite word. The result holds for both right-infinite and bi-infinite words.

Logic, semigroups and automata on words

This is a survey article on the connections between the “sequential calculus” of Buchi, a system which allows to formalize properties of words, and the theory of automata. In the sequential calculus, there is a predicate for each letter and the unique extra non logical predicate is the relation symbol <, which is interpreted as the usual order on the integers. Several famous classes have been classified within this logic. We briefly review the main results concerning second order, which covers classes like PH, NP, P, etc., and then study in more detail the results concerning the monadic second-order and first-order logic. In particular, we survey the results and fascinating open problems dealing with the first-order quantifier hierarchy. We also discuss the first-order logic of one successor and the linear temporal logic. There are in fact three levels of results, since these logics can be interpreted on finite words, infinite words or bi-infinite words. The paper is self-contained. In particular, the necessary background on automata and finite semigroups is presented in a long introductory section, which includes some very recent results on the algebraic theory of infinite words.

Precircular Codes and Periodic Biinfinite Words

The aim of decoding, or factorizing in a single way, biinfinite words with a finitary language leads us to consider two distinct notions of factorization for biinfinite words. For one of these notions, requiring singleness of decoding periodic biinfinite words leads us to study circular codes. The other notion leads us to define the new notion of precircular codes. Precircular codes are defined and analyzed. The family of precircular codes is compared to the family of circular codes and to the family of almost circular codes. The structure of two-element precircular codes is determined. Codes and precircular codes are characterized by their way of acting upon periodic biinfinite words.

FINITE STATE PROCESSES, Z-TEMPORAL LOGIC AND THE MONADIC THEORY OF THE INTEGERS

We introduce four classes of Z-regular grammars for generating bi-infinite words (i.e. Z-words) and prove that they generate exactly Z-regular languages. We extend the second order monadic theory of one successor to the set of the integers (i.e. Z) and give some characterizations of this theory in terms of Z-regular grammars and Z-regular languages. We prove that this theory is decidable and equivalent to the weak theory. We also extend the linear temporal logic to Z-temporal logic and then prove that each Z-temporal formula is equivalent to a first order monadic formula. We prove that the correctness problem for finite state processes is decidable.

Finitely generated biω-languages

In this paper we are interested in the bi-infinite words and mainly in the bi-infinite power of languages of finite words. With the aim of deciding whether, given a rational language R, ω R ω is finitely generated, i.e. whether ω R ω= ω R ω for some finite set F, we study the notions of bilimit and biadherence. The stated results become then efficient tools for solving the initial decision problem.

Cellular automata, ωω-regular sets, and sofic systems

We study and compare several mechanisms for defining sets of biinfinite words (ωω-languages), namely ωω-finite automata, adherences of regular languages and sofic systems. We show that a sofic system is a topologically closed ωω-regular set. We also show that there is a one-to-one correspondence between sofic systems and the adherences of regular languages. We give a complete proof of the closure of the ωω-regular sets under ωω-rational relations and under Boolean operations. Finally, we disprove Hurd's conjecture [15] on bi-extensible subsets of languages, and show that the conjecture would hold if a different definition were used.

Multidimensional unrepetitive configurations

We show that one can associate a letter of a finite alphabet with each lattice-point of the n-dimensional space so that an rth power-free word corresponds to each straight line of the space. This result is obtained as a consequence of the existence of bi-infinite words on finite alphabets which are rth power-free together with a certain class of their subwords. An analogous property is obtained for some avoidable sets of patterns.

Ensembles Reconnaissables de Mots Biinfinis

Introduction. La théorie des automates finis fait partie de ce que l'on appelle aujourd'hui les mathématiques de l'informatique. Comme pour les autres spécialités de ce domaine, elle est née des travaux de logiciens et, pour les automates finis, c'est à Kleene que l'on doit le premier théorème. Cette théorie s'est considérablement développée depuis la période des fondations. La direction principale est l'étude des automates reconnaissant des suites finies ou mots. Elle présente des aspects mathématiques qui la rapprochent de domaines classiques comme par exemple la théorie combinatoire des groupes. Parallèlement, plusieurs auteurs ont étudié le comportement des automates finis sur des objets plus généraux que les mots. C'est le cas notamment pour la théorie des automates reconnaissant des arbres. C'est aussi le cas pour les automates reconnaissant des mots infinis qui sont le sujet de cet article.

An Algorithm for Simple Curves on Surfaces

Let M be a compact orientable surface with non-empty boundary and with X{M) < 0, and let T = nlM. Let C be the free homotopy class of a closed loop on M and let W = W{C) be a word in a fixed set of generators T which represents C. In this paper we give an algorithm to decide, starting with W, whether C has a simple representative, that is a representative without self-intersections. Such a word will be said to be simple. As an application, we begin a study of simple words in F. Our results also apply to infinite geodesies on M, corresponding to biinfinite words in F, where now we ask which finite blocks appear in such a word when the corresponding infinite homotopy class has a simple representative. For finite words there are, of course, other such algorithms, see for example [8, 9, 2, 3, 4]. Our algorithm most resembles that in [2] in that it is purely mechanical and combinatorial. It is simpler than that in [2] but what is more important is that it reveals the underlying mechanism which determines whether self-intersections occur; the combinatorics of that mechanism seem quite interesting and non-trivial. We represent M as U/T where U ^ D is the universal covering space of M, and where D is the unit disc with the Poincare metric and F is a discrete group of hyperbolic isometries. Poincare showed in [7] that C contains a simple representative if and only if the unique smooth geodesic representative C of C is simple, and that C is simple if and only if for each lift y of C to D the curves in the infinite family {fy}fer a r e pairwise disjoint. Now, to see if geodesies yl5 y2 e {fy} are disjoint in D it is enough to know whether the ideal endpoints of yx on 3D separate those of y2. Crucial to our work is a scheme for parametrizing points on 3D by infinite words in F, first developed by Nielsen in [6]. The idea of this paper is to show how information on the order of the points 3yl53y2 on 3D is encoded in Nielsen's 'boundary expansion' (Theorem A) and then to examine consequences. When dM ± 0 the group F is a free group so that each conjugacy class has a unique shortest representative which is obtained by cyclic reduction of any word in the class. However, if dM = 0 the shortest word in the conjugacy class is in general not unique. If dM = 0 and W e T has a shortest representative which does not contain any pieces which are half of the defining relator in F, then the problem of deciding whether W is simple is identical with that on the surface with a disc removed, that is one simply regards F as if it were a free group. On the other hand, the exceptional cases when W contains half a relator involve some subtle points which are not without interest, but are somewhat tangential to the main idea in the paper. For that reason, we shall omit the case in which dM = 0 . Here is an outline of this paper. The tools we need are set up in §§2 and 3 where we prove Theorem A. The algorithm (Theorem B) is given in §4. In §5 we give