Sergio Salemi

Star-free trace languages

Abstract Generalizing a classical result of Schutzenberger to free partially commutative monoids, we prove that the family of star-free trace languages coincides with the family of aperiodic trace languages.

Periodicity and the golden ratio

Abstract We prove a periodicity theorem on words that has strong analogies with the Critical Factorization theorem. The Critical Factorization theorem states, roughly speaking, a connection between local and global periods of a word; the local period at any position in the word is there defined as the shortest repetition (a square) “centered” in that position. We here take into account a different notion of local period by considering, for any position in the word, the shortest repetition “immediately to the left” from that position. In this case a repetition which is a square does not suffices and the golden ratio ϑ (more precisely its square ϑ 2 = 2.618 …) surprisingly appears as a threshold for establishing a connection between local and global periods of the word. We further show that the number ϑ 2 is tight for this result. Two applications are then derived. In the firts we give a characterization of ultimately periodic infinite words. The second application concerns the topological perfectness of some families of infinite words.

Binary Patterns in Infinite Binary Words

In this paper we study the set P(w) of binary patterns that can occur in one infinite binary word w, comparing it with the set F(w) of factors of the word. Since the set P(w) can be considered as an extension of the set F(w), we first investigate how large is such extension, by introducing the parameter ?(w) that corresponds to the cardinality of the difference set P(w) \ F(w). Some non trivial results about such parameter are obtained in the case of the Thue-Morse and the Fibonacci words. Since, in most cases, the parameter ?(w) is infinite, we introduce the pattern complexity of w, which corresponds to the complexity of the language P(w). As a main result, we prove that there exist infinite words that have pattern complexity that grows more quickly than their complexity. We finally propose some problems and new research directions.

On the centers of a language

Abstract The center of a language has been defined in [7, 8, 9] as the set of all words which have infinite right completions in the language. In this paper we extend this notion by taking into account also left and two-sided completions. Thus, for any language X , we consider the left center C l ( X ), the right center C r ( X ) and two different bilateral centers C 1 ( X ) and C 2 ( X ). Some properties of these centers are derived. In particular the main results of the paper give some general conditions under which C 1 ( X )= C 2 ( X )and C r ( C l ( X ))= C l ( C r ( X )). These conditions deal with ‘strong’ and ‘weak’ iteration properties and ‘periodicity’ of a language.

Patterns in words and languages

A word p, over the alphabet of variables E, is a pattern of a word w over A if there exists a non-erasing morphism h from E* to A* such that h(p)=w. If we take E=A, given two words u, v ∈ A*, we write u ≤ v if u is a pattern of v. The restriction of ≤ to aA*, where A is the binary alphabet {a, b}, is a partial order relation. We introduce, given a word v, the set P(v) of all words u such that u ≤ v. P(v), with the relation ≤, is a poset and it is called the pattern poset of v. The first part of the paper is devoted to investigate the relationships between the structure of the poset P(v) and the combinatorial properties of the word v. In the last section, for a given language L, we consider the language P(L) of all patterns of words in L. The main result of this section shows that, if L is a regular language, then P(L) is a regular language too.

A generalization of Sardinas and Patterson's algorithm to z -codes

Abstract This paper concerns the framework of z-codes theory. The main contribution consists in an extension of the algorithm of Sardinas and Patterson for deciding whether a finite set of words X is a z-code. To improve the efficiency of this test we have found a tight upper bound on the length of the shortest words that might have a double z-factorization over X. Some remarks on the complexity of the algorithm are also given. Moreover, a slight modification of this algorithm allows us to compute the z-deciphering delay of X.

Some Decision Results on Nonrepetitive Words

The paper addresses some generalizations of the Thue Problem such as: given a word u, does there exist an infinite nonrepetitive overlap free (or square free) word having u as a prefix? A solution to this as well as to related problems is given for the case of overlap free words on a binary alphabet.