Filippo Mignosi

Some combinatorial properties of Sturmian words

Abstract In this paper we give a characterization of finite Sturmian words, by palindrome words, which generalizes a property of the Fibonacci words. We prove that the set St of finite Sturmian words coincides with the set of the factors of all the words w such that w = AB = Cxy with A, B, C palindromes, x , y ϵ{ a,b } and x ≠ y . Moreover, using this result we prove that St is equal to the set of the factors of all words w having two periods p and q which are coprimes and such that | w | ⩾ p + q − 2. Several other combinatorial properties concerning special and bispecial elements of St are shown. As a consequence we give a new, and purely combinatorial, proof of the enumeration formula of St .

Automata and forbidden words

Abstract Let L ( M ) be the (factorial) language avoiding a given anti-factorial language M . We design an automaton accepting L ( M ) and built from the language M . The construction is effective if M is finite. If M is the set of minimal forbidden words of a single word ν, the automaton turns out to be the factor automaton of ν (the minimal automaton accepting the set of factors of ν). We also give an algorithm that builds the trie of M from the factor automaton of a single word. It yields a nontrivial upper bound on the number of minimal forbidden words of a word.

Infinite words with linear subword complexity

Abstract In Section 1 we study the relations among some combinatorial properties of infinite words, especially in the case of infinite words with linear subword complexity; the main result of Section 1 concerns a permutation property of infinite words with linear complexity. In Section 2 we investigate the special case of the sturmian infinite words; the main result is that a sturmian infinite word associated to a real number α contains no k th powers iff α has a continued fraction expansion with bounded partial quotients. In Section 3 it is shown that some of the results of Sections 1 and 2 are optimal.

Fine and Wilf's theorem for three periods and a generalization of Sturmian words

We extend the theorem of Fine and Wilf to words having three periods. We then define the set 3-PER of words of maximal length for which such result does not apply. We prove that the set 3-PER and the sequences of complexity 2n + 1, introduced by Arnoux and Rauzy to generalize Sturmian words, have the same set of factors.

The expressibility of languages and relations by word equations

Classically, several properties and relations of words, such as “being a power of the same word” can be expressed by using word equations. This paper is devoted to a general study of the expressive power of word equations. As main results we prove theorems which allow us to show that certain properties of words are not expressible as components of solutions of word equations. In particular, “the primitiveness” and “the equal length” are such properties, as well as being “any word over a proper subalphabet”.

Minimal Forbidden Words and Symbolic Dynamics

We introduce a new complexity measure of a factorial formal language L: the growth rate of the set of minimal forbidden words. We prove some combinatorial properties of minimal forbidden words. As main result we prove that the growth rate of the set of minimal forbidden words for L is a topological invariant of the dynamical system defined by L.

If a D0L Language is k-Power Free then it is Circular

We prove that if a DOL language is k-power free then it is circular. By using this result we are able to give an algorithm which decides whether, fixed an integer k≥1, a DOL language is k-power free; we are also able to give a new simpler proof of a result, previously obtained by Ehrenfeucht and Rozenberg, that states that it is decidable whether a DOL language is k-power free for some integer k≥1.

Words and forbidden factors

Given a finite or infinite word v, we consider the set M(v) of minimal forbidden factors of v. We show that the set M(v) is of fundamental importance in determining the structure of the word v. In the case of a finite word w we consider two parameters that are related to the size of M(w): the first counts the minimal forbidden factors of w and the second gives the length of the longest minimal forbidden factor of w. We derive sharp upper and lower bounds for both parameters. We prove also that the second parameter is related to the minimal period of the word w. We are further interested to the algorithmic point of view. Indeed, we design linear time algorithm for the following two problems: (i) given w, construct the set M(w) and, conversely, (ii) given M(w), reconstruct the word w. In the case of an infinite word x, we consider the following two functions: gx that counts, for each n, the allowed factors of x of length n and fx that counts, for each n, the minimal forbidden factors of x of length n. We address the following general problem: what information about the structure of x can be derived from the pair (gx,fx)? We prove that these two functions characterize, up to the automorphism exchanging the two letters, the language of factors of each single infinite Sturmian word.

Simple real-time constant-space string matching

String matching is the classical problem of finding all occurrences of a pattern in a text. A real-time string matching algorithm takes worst-case constant-time to check if a pattern occurrence ends at each text location. We derive a real-time variation of the elegant Crochemore-Perrin constant-space string matching algorithm that has a simple and efficient control structure. We use observations about the locations of critical factorizations to deploy two tightly-coupled simplified real-time instances of the Crochemore-Perrin algorithm that search for complementary parts of the pattern whose simultaneous occurrence indicates an occurrence of the complete pattern.

On a conjecture on bidimensional words

We prove that, given a double sequence w over the alphabet A (i.e. a mapping from Z2 to A), if there exists a pair (n0, m0) ∈ Z2 such that pw(n0, m0) < 1/100n0m0, then w has a periodicity vector, where pw is the complexity function in rectangles of w.