Compatibility and Binary Correlations of Fibonacci Partial Words

The sequences of Fibonacci words plays an important role in formal language theory and combinatorics on words. Fibonacci partial words, arrays and their combinatorial properties such as palindromes and primitivity was established. In this paper, we extend some fundamental results about Fibonacci words to Fibonacci partial words such as compatibility, periodicity and also binary correlations which turns out to be an effective technique of portraying sets of periods of Fibonacci partial words briefly with comprehensive scope.


Introduction
The study of sequence of Fibonacci word is a great interest in some aspects of formal language theory. In theory of combinatorics, the sequence of Fibonacci words which is uniformly recurrent play a vital role due to their remarkable properties, some of which have been studied by Knuth [13] in relation with string matching problems and by Duval [10] in the study of "periodicity"of words. Aldo De Luca [9] discussed primitivity and palindromic properties of Fibonacci words and later on S.S.Yu and Yu-Kuang Zhao [16] discussed d-primitive and palindrome properties of Fibonacci words and have also shown that Fibonacci languages are regular free. The urge for the introduction of partial words came from the molecular biology of nucleic acids. There, among other things, the properties of the DNA sequences experienced in the genome of organisms are determined. These are considered as strings over the alphabet {A, C, G, T } of four bases respectively. In DNA sequencing, some part of information may be absent or unseen. This can be revealed by positions denoting missing symbols in a word. Thus, instead of complete words, partial words are considered. Fischer and Paterson [11] introduced partial words as strings containing don't care letters. Berstel and Boasson [3] initiated the study of combinatorics on partial words and was later pursued by Blanchet-Sadri.et.al [4,5,6,7]. The critical factorization theorem and Fine and Wilf theorem [6,8] are the two basic results on periodicity which is one of the most used and known results on words, has extensions to partial words. In [14] the notion of Fibonacci partial words, arrays and their combinatorial properties such as palindromes and primitivity are established. Guibas et.al [12] represented their results using binary vectors called correlations which 2 turned out to be a convenient way of describing periods sets briefly with comprehensive scope. The motive for recent studies on combinatorics of words is the study of molecules such as DNA that play a major part in molecular biology. Therefore several basic results on words have been extended to partial words [1,2,4,7,9,12,14,15]. In this paper, we extend some basic results about Fibonacci words to Fibonacci partial words. An outline of this paper follows. Some basic definitions are recollected in section 2. In section 3, correlations and compatibility of Fibonacci partial words are discussed. Finally section 4 concludes this paper.

Preliminaries
Here we recall fundamental notions of partial words, correlations of partial words and Fibonacci words. Let Σ be a non-empty finite set of letters. These letters are also termed as symbols and the set is termed as an alphabet. Any string over Σ is called a word or a total word. If a word u is of the form xy, where x is empty then u R = yx is called a rotation of u of degree |u|. The correlation of a word u over a word v is defined to be binary vector of the same length as u, composed as follows. The ith bit (from the left) of the correlation is determined by placing v under u so that the leftmost character of v is under the ith character of u (from the left). Then, if all pairs of characters that are directly over each other match, the ith bit of the correlation is 1, else it is 0. The same process is applicable for bianary correlation of a word over itself. The sequence or word that contains a number of "do not know "symbols or "holes"denoted as ♦ is termed as partial word. The symbol ♦ does not belong to the alphabet Σ. A partial word over Σ ♦ with length n is a partial function r ♦ : where D(r) and H(r) are the domain set and hole set of r respectively. A partial word r over Σ ♦ is p − periodic if a non-negative integer p exists such that i ≡ jmodp whenever r(i) = r(j) for every i, j ∈ D(r). If two partial words say x and y are of equal length and if all the elements in domain of x are also in domain of y with x(i) = y(i) for every i ∈ D(x), then x is contained in y and is denoted by x ⊂ y. Two partial words x and y are compatible, denoted by x ↑ y if x(i) = y(i) for all i ∈ D(x) ∩ D(y). The least upper bound of x and y is denoted by x ∨ y where D(x ∨ y) = D(x) ∪ D(y) Consider an alphabet Σ with |Σ| ≥ 2. The F ibonacci sequence of words {f n }, n ≥ 1 over the alphabet Σ is defined inductively as f n = f n−1 f n−2 , f 1 = a, f 2 = b where a = b.Length of the Fibonacci sequence of words |f n | is denoted by F (n). Also |f 1 | = |f 2 | = 1 and |f n | = |f n−1 | + |f n−2 | for all n ≥ 2.

Compatibility and Binary correlations of Fibonacci partial words
Here we define sequence of Fibonacci partial words and discuss their compatibility and binary correlations. Binary correlations of Fibonacci partial words are the binary vectors specifying the periods of Fibonacci partial words.

Definition 3.1 [14] Consider an alphabet
Here p is a positive integer termed as a strong period.

Definition 3.4 Two Fibonacci partial words f
. Property 3.1 Let f ♦ m ,f ♦ n and f ♦ p be Fibonacci partial words of equal length in the sequences f ♦ m , f ♦ n and f ♦ p over Σ ♦ . The following three properties proves that compatibility on Fibonacci partial words is an equivalence relation. Reflexive property: Trivial Symmetric property: Proof. Let f ♦ n and g ♦ n be two sequence of Fibonacci partial words over Σ ♦ with |Σ ♦ | ≥ 2. The above statement is provable if the initial partial words of both the sequences are of equal length say Let us prove by contradiction. Let f ♦ r ↑ g ♦ r ∀ r ≤ n. and let f ♦ r ↑ g ♦ r ∀1 ≤ r ≤ 2, From the notion of sequence of Fibonacci partial words, But this contradicts the fact that ∀1 ≤ r ≤ 2, Theorem 3.2 Consider the Fibonacci partial words f ♦ x ,f ♦ y ,f ♦ m and f ♦ n of equal length in the sequences Then the domain of f ♦ m f ♦ x and f ♦ n f ♦ y is exactly equal to the union of domains of f ♦ m ,f ♦ x ,f ♦ n and f ♦ y .
Similarly we can show that f ♦ x ↑ f ♦ y which completes the proof.
Proof. Since f ♦ m and f ♦ n are compatible, f ♦ m and f ♦ n are contained in For instance let f ♦