Arturo Carpi

On Dejean's conjecture over large alphabets

The (maximal) exponent of a non-empty finite word is the ratio of its length to its period. Dejean (1972) conjectured that for any n≥5 there exists an infinite word over n letters with no factor of its exponent larger than n/(n-1). We prove that this conjecture is true for n≥33.

Overlap-free words and finite automata

Abstract A method to represent certain words on a binary alphabet by shorter words on a larger alphabet is introduced. We prove that overlap-free words are represented by the words of a rational language. Several consequences are derived concerning the density function of the set of overlap-free words on a binary alphabet and the prolongability of overlap-free words. In particular, efficient algorithms are obtained computing the density function of the set of overlap-free words on a binary alphabet, testing whether a word on a binary alphabet is overlap-free and computing its depth.

On synchronized sequences and their separators

We introduce the notion of a k -synchronized sequence, where k is an integer larger than 1. Roughly speaking, a sequence of natural numbers is said to be k -synchronized if its graph is represented, in base k , by a right synchronized rational relation. This is an intermediate notion between k -automatic and k -regular sequences. Indeed, we show that the class of k -automatic sequences is equal to the class of bounded k -synchronized sequences and that the class of k -synchronized sequences is strictly contained in that of k -regular sequences. Moreover, we show that equality of factors in a k -synchronized sequence is represented, in base k , by a right synchronized rational relation. This result allows us to prove that the separator sequence of a k -synchronized sequence is a k -synchronized sequence, too. This generalizes a previous result of Garel, concerning k -regularity of the separator sequences of sequences generated by iterating a uniform circular morphism.

The Synchronization Problem for Strongly Transitive Automata

The synchronization problem is investigated for a new class of deterministic automata called strongly transitive. An extension to unambiguous automata is also considered.

Words, univalent factors, and boxes

Abstract. A factor u of a word w is (right) univalent if there exists a unique letter a such that ua is still a factor of w. A univalent factor is minimal if none of its proper suffixes is univalent. The starting block of w is the shortest prefix

On factors of synchronized sequences

\overline{h}_w

The Synchronization Problem for Locally Strongly Transitive Automata

of w such that all proper prefixes of w of length

Strongly transitive automata and the Černý conjecture

\geq |\overline{h}_w|

Multidimensional unrepetitive configurations (note)

are univalent. We study univalent factors of a word and their relationship with the well known notions of boxes, superboxes, and minimal forbidden factors. Moreover, we prove some new uniqueness conditions for words based on univalent factors. In particular, we show that a word is uniquely determined by its starting block, the set of the extensions of its minimal univalent factors, and its length or its terminal box. Finally, we show how the results and techniques presented can be used to solve the problem of sequence assembly for DNA molecules, under reasonable assumptions on the repetitive structure of the considered molecule and on the set of known fragments.

Harmonic and gold Sturmian words

Let k>=2 be an integer. A sequence of natural numbers is k-synchronized if its graph is represented, in base k, by a right-synchronized rational relation. We show that the factor complexity and the palindromic complexity of a k-synchronized sequence are k-regular sequences. We derive that the palindromic complexity of a k-automatic sequence is k-automatic.