Aldo de Luca

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 .

Sturmian words: structure, combinatorics, and their arithmetics

Abstract We prove some new results concerning the structure, the combinatorics and the arithmetics of the set PER of all the words w having two periods p and q, p⩽q, which are coprimes and such that ¦w¦= p + q− 2 . A basic theorem relating PER with the set of finite standard Sturmian words was proved in de Luca and Mignosi (1994). The main result of this paper is the following simple inductive definition of PER: the empty word belongs to PER, If w is an already constructed word of PER, then also (aw)(−) and (bw)(−) belong to PER, where (−) denotes the operator of palindrome left-closure, i.e. it associates to each word u the smallest palindrome word u(−) having u as a suffix. We show that, by this result, one can construct in a simple way all finite and infinite standard Sturmian words. We prove also that, up to the automorphism which interchanges the letter a with the letter b, any element of PER can be codified by the irreducible fraction p q . This allows us to construct for any n⩾0 a natural bijection, that we call Farey correspondence, of the set of the Farey series of order n + 1 and the set of special elements of length n of the set St of all finite Sturmian words. Finally, we introduce the concepts of Farey tree and Farey monoid. This latter is obtained by defining a suitable product operation on the developments in continued fractions of the set of all irreducible fractions p q .

Sturmian words, Lyndon words and trees

Abstract We prove some new combinatorial properties of the set PER of all words w having two periods p and q which are coprimes and such that w = p + q − 2 [4, 3]. We show that aPERb ∪{ a , b } = St ∩ Lynd , where St is the set of the finite factors of all infinite Sturmian words and Lynd is the set of the Lyndon words on the alphabet { a , b }. It is also shown that aPERb ∪{ a , b } = CP , where CP is the set of Christoffel primitive words. Such words can be defined in terms of the ‘slope’ of the words and of their prefixes [1]. From this result one can derive in a different way, by using a theorem of Borel and Laubie, that the elements of the set aPERb are Lyndon words. We prove the following correspondence between the ratio p q of the periods p , q , p ⩽ q of w ϵ PER ∩ a{a,b} ∗ and the slope ρ = (¦w¦ b + 1) (¦w¦ a + 1) of the corresponding Christoffel primitive word awb : If p q has the development in continued fractions [0, h 1 ,…, h n − 1 , h n + 1], then ρ has the development in continued fractions [0, h n ,…, h 2 , h 1 + 1]. This and other related results can be also derived by means of a theorem which relates the developments in continued fractions of the Stern-Brocot and the Raney numbers of a node in a complete binary tree. However, one needs some further results. More precisely we label the binary tree with standard pairs (standard tree), Christoffel pairs (Christoffel tree) and the elements of PER (Farey tree). The main theorem is the following: If the node W is labeled by the standard pair ( u , v ), by the Christoffel pair ( x , y ) and by w ϵ PER , then uv = wab , xy = awb . The Stern-Brocot number SB ( W ) is equal to the slope of the standard word uv and of the Christoffel word xy while the Raney number Ra ( W ) is equal to the ratio of the minimal period of wa and the minimal period of wb . Some further auxiliary results are also derived.

Entropy of L-fuzzy sets*

The notion of “entropy” of a fuzzy set, introduced in a previous paper in the case of generalized characteristic functions whose range is the interval [0, 1] of the real line, is extended to the case of maps whose range is a poset L (or, in particular, a lattice). Some of the reasons giving rise to the non-comparability of the truth values and then the necessity of considering poset structures as range of the maps are discussed. The interpretative problems of the given mathematical definitions regarding the connections with decision theory are briefly analyzed.

On the combinatorics of finite words

In this paper we consider a combinatorial method for the analysis of finite words recently introduced in Colosimo and de Luca (Special factors in biological strings, preprint 97/42, Dipt. Matematica, Univ. di Roma) for the study of biological macromolecules. The method is based on the analysis of (right) special factors of a given word. A factor u of a word w is special if there exist at least two occurrences of the factor u in w followed on the right by two distinct letters. We show that in the combinatorics of finite words two parameters play an essential role. The first, denoted by R, represents the minimal integer such that there do not exist special factors of w of length R. The second, that we denote by K, is the minimal length of a factor of w which cannot be extended on the right in a factor of w. Some new results are proved. In particular, a new characterization in terms of special factors and of R and K is given for the set PER of all words w having two periods p and q which are coprimes and such that ¦w¦ = p + q − 2.

Pseudopalindrome closure operators in free monoids

We consider involutory antimorphisms ϕ of a free monoid A* and their fixed points, called ϕ-palindromes or pseudopalindromes. A ϕ-palindrome reduces to a usual palindrome when ϕ is the reversal operator. For any word w ∈ A* the right (resp. left) ϕ-palindrome closure of w is the shortest ϕ-palindrome having w as a prefix (resp. suffix). We prove some results relating ϕ-palindrome closure operators with periodicity and conjugacy, and derive some interesting closure properties for the languages of finite Sturmian and episturmian words. In particular, a finite word w is Sturmian if and only if both its palindromic closures are so. Moreover, in such a case, both the palindromic closures of w share the same minimal period of w. A new characterization of finite Sturmian words follows, in terms of periodicity and special factors of their palindromic closures. Some weaker results can be extended to the episturmian case. By using the right ϕ-palindrome closure, we extend the construction of standard episturmian words via directive words. In this way one obtains a family of infinite words, called ϕ-standard words, which are morphic images of episturmian words, as well as a wider family of infinite words including the Thue-Morse word on two symbols.

A characterization of strictly locally testable languages and its application to subsemigroups of a free semigroup

A syntactic characterization of strictly locally testable languages is given by means of the concept of constant . If S is a semigroup and X a subset of S , an element c of S is called constant for X if for all p, q, r, s e S 1 *[ pcq, rcs eX ⇒ pcs e X ]. The main result of the paper states that a recognizable subset X of a free semigroup A + is strictly locally testable if and only if all the idempotents of the syntactic semigroup S ( X ) of X are constants for X′ = XΦ , where Φ : A + → S ( X ) is the syntactic morphism. By this result some remarkable consequences are derived for recognizable subsemigroups of A + . In particular we prove that if X is a recognizable free subsemigroup of A + and Y = X/X 2 its base then the following conditions are equivalent : (1) X is strictly locally testable. (2) X is locally testable. (3) X is locally parsable and Y is strictly locally testable. (4) X has a bounded synchronization delay and Y is strictly locally testable (5) A positive integer k exists such that all the elements of A + whose length is greater than or equal to k , are constants for X . (6) For all the idempotents e of the syntactic semigroup S ( X ) of X , eS ( X ) e ⊆ e , 0.

On an involution of Christoffel words and Sturmian morphisms

There is a natural involution on Christoffel words, originally studied by the second author in [A. de Luca, Combinatorics of standard Sturmian words, Lecture Notes in Computer Science 1261 (1997) 249-267]. We show that it has several equivalent definitions: one of them uses the slope of the word, and changes the numerator and the denominator respectively in their inverses modulo the length; another one uses the cyclic graph allowing the construction of the word, by interpreting it in two ways (one as a permutation and its ascents and descents, coded by the two letters of the word, the other in the setting of the Fine and Wilf periodicity theorem); a third one uses central words and generation through iterated palindromic closure, by reversing the directive word. We show further that this involution extends to Sturmian morphisms, in the sense that it preserves conjugacy classes of these morphisms, which are in bijection with Christoffel words. The involution on morphisms is the restriction of some conjugation of the automorphisms of the free group. Finally, we show that, through the geometrical interpretation of substitutions of Arnoux and Ito, our involution is the same thing as duality of endomorphisms (modulo some conjugation).

Words and special factors

In this paper we consider sets of factors of a given finite word over a finite alphabet which permit us to reconstruct the entire word. This analysis is based on the notion of special factor. A factor u of a finite word w is called right (resp. left) special if there exist two distinct letters x and y such that ux, uy (resp. xu, yu) are factors of w. A factor is bispecial if it is right and left special. A proper box of w is any factor of w of the kind asb, with a,b letters and s a bispecial factor of w. The initial (resp. terminal) box of w is the shortest prefix (resp. suffix) of w which is an unrepeated factor. A box is called maximal if it is not a proper factor of another box. The main result of the paper is the following theorem (maximal box theorem): Any finite word w is uniquely determined by the initial box, the terminal box and the set of maximal boxes. A consequence is that a finite word w is uniquely determined by the knowledge of its factors up to the length n=max{Rw,Kw}+1, where Kw is the length of the terminal box and Rw is the minimal natural number for which there is no right special factor of length Rw. Some structural properties of boxes are studied. Another important combinatorial notion is that of superbox. A superbox is any factor of w of the kind asb, with a,b letters and such that s is a repeated factor, whereas as and sb are unrepeated factors. A theorem for superboxes similar to the maximal box theorem is proved. Some algorithms allowing us to construct boxes and superboxes and, conversely, to reconstruct the word are given. In this combinatorial frame we give an upper and a lower bound to the number of states of a minimal deterministic automaton recognizing the set of the factors of w. These bounds are sharper than the known bounds.

Special factors, periodicity, and an application to Sturmian words

Abstract. Let w be a finite word and n the least non-negative integer such that w has no right special factor of length