Compatibility and Conjugacy on Partial Arrays

Research in combinatorics on words goes back a century. Berstel and Boasson introduced the partial words in the context of gene comparison. Alignment of two genes can be viewed as a construction of two partial words that are said to be compatible. In this paper, we examine to which extent the fundamental properties of partial words such as compatbility and conjugacy remain true for partial arrays. This paper studies a relaxation of the compatibility relation called k-compability. It also studies k-conjugacy of partial arrays.


Introduction
The genetic information in almost all organisms is carried by molecules of DNA. A DNA molecule is a quite long but finite string of nucleotides of 4 possible types: (for adenine), (for cytosine), (for guanine), and (for thymine). The stimulus for recent works on combinatorics is the study of biological sequences such as DNA and protein that play an important role in molecular biology [1][2][3]. Sequence comparison is one of the primitive operations in molecular biology. Alignment of two sequences is to place one sequence above the other [2,4] in order to make clear correspondence between similar letters or substrings of the sequences. Partial words appear in comparing genes. Indeed, alignment of two strings can be viewed as a construction of two partial words that are compatible. The compatibility relation [5] considers two arrays with only few isolated insertions (or deletions). In some cases, it allows insertion of letters which relate to errors or mismatches. A problem appears when the same gene is sequenced by two different labs that want to differentiate the gene expression. Also, when the same long sequence is typed twice into the computer, errors appear in typing.
Partial array of size ( , ) over Σ, a finite alphabet, is partial function : 2 + → Σ, where + is the set of all positive integers. In this paper, we extend the combinatorial properties of partial words to partial arrays. Also, this paper studies a relation called -compatibility where a number of insertions and deletions are allowed as well as -mismatches. The conjugacy result [6] which was proved for partial words is extended to partial arrays. -Conjugacy of partial arrays is discussed.

Preliminaries on Partial Words
In this section, we give a brief overview of partial words [7]. Definition 1. Partial word of length over , a nonempty finite alphabet, is partial map : {1, 2, . . . , } → . If 1 ≤ ≤ , then belongs to the domain of (denoted by Domain( )) in the case where ( ) is defined, and belongs to the set of holes of (denoted by Hole( )), otherwise.
A word [8][9][10] is a partial word over with an empty set of holes.
Let and V be two partial words of length . Partial word is said to be contained in partial word V (denoted by ⊂ V), if Domain( ) ⊂ Domain(V) and ( ) = V( ) for all ∈ Domain( ). Partial words and V are called compatible (denoted by ↑ V), if there exists partial word such that ⊂ and V ⊂ (in which case we define ∨ V by ⊂ ∨ V and V ⊂ ∨V and Domain( ∨V) = Domain( )∪Domain(V)).
As an example, ◊ = ◊◊ and V ◊ = ◊ . The following rules are useful for computing with partial words: (ii) Simplification: If ↑ V and | | = |V|, then ↑ V and ↑ .

Lemma 3.
Let , V, , be partial words such that ↑ V .
Definition 4. Two partial words and V are called conjugate, if there exist partial words and such that ⊂ and V ⊂ .
Definition 5. Two partial words and V are calledconjugate, if there exist nonnegative integers 1 , 2 whose sum is and partial words and such that ⊂

Preliminaries on Partial Arrays
This section is devoted to review the basic concepts on partial arrays [11]. Definition 6. Partial array of size ( , ) over Σ, a nonempty set or an alphabet, is partial function : 2 + → Σ, where + is the set of all positive integers. For 1 ≤ ≤ , 1 ≤ ≤ , and if ( , ) is defined, then we say that ( , ) belongs to the domain of (denoted by ( , ) ∈ ( )). Otherwise, we say that ( , ) belongs to the set of holes of (denoted by ( , ) ∈ ( )).
An array [5] over Σ is a partial array over Σ with an empty set of holes.

Definition 7.
If is a partial array of size ( , ) over Σ, then the companion of (denoted by ◊ ) is total function ◊ : where ◊ ∉ Σ.
The bijectivity of map → ◊ allows defining the catenation of two partial arrays in a trivial way.
By column catenation, we mean By row catenation, we mean ) .

(5)
If and are two partial arrays of equal size, then is contained in denoted by ⊂ if ( ) ⊆ ( ) and Definition 9. Partial arrays and are said to be compatible denoted by ↑ , if there exists partial array such that ⊂ and ⊂ .

Compatibiltiy and -Compatability of Partial Arrays
4.1. Compatibility. The rules mentioned for partial words are also true for partial arrays. Let , , , be partial arrays.
(i) Multiplication: If ↑ and ↑ , then ↑ either by column catenation or by row catenation.

Conjugacy
Definition 19. Two partial arrays and of same order are called conjugate if there exist partial arrays and such that ⊂ and ⊂ using row catenation or column catenation. 0-conjugacy on partial arrays with same order is trivially reflexive and symmetric but not transitive.  Proof. Let , be two partial arrays of same order. Supposing that and are -conjugate, then, by definition, there exist nonnegative integers 1 , 2 whose sum is and partial arrays and such that ⊂ There exist = ( ◊ ) and = ( ◊ ) with ⊂ 3 and ⊂ 2 , = 1 + 2 = 5.

Conclusion
Motivated by compatibility and conjugacy properties of partial words, we define the conjugacy of partial array and derive the compatibility properties of partial arrays. By giving relaxation to the compatibility relation, we discusscompatibility and -conjugacy of partial arrays. We prove that, given partial arrays , and integers , satisfying | | = | | , we find such that ↑ . Also, there exists partial array such that ↑ ≤ .

Disclosure
S. Vijayachitra is a Research Scholar at Department of Science and Humanity Sathyabama University, Chennai, India.