Compatibility and Conjugacy on Partial Arrays

Abstract

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 -compability. It also studies -conjugacy of partial arrays.

1. 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 [1–3]. 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 , a finite alphabet, is partial function , 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.

2. 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 . If , then belongs to the domain of (denoted by ) in the case where is defined, and belongs to the set of holes of (denoted by ), otherwise.
A word [8–10] is a partial word over with an empty set of holes.

Definition 2. Let be a partial word of length over . The companion of (denoted by ) is map defined byThe symbol is viewed as a “do not know” symbol. Word is the companion of the partial word. The length of the partial word is 6. . .

Let and be two partial words of length . Partial word is said to be contained in partial word (denoted by ), if and for all . Partial words and are called compatible (denoted by ), if there exists partial word such that and (in which case we define by and and ). As an example, and .

The following rules are useful for computing with partial words:(i)Multiplication: If and , then .(ii)Simplification: If and , then and .(iii)Weakening: If and , then .

Lemma 3. Let be partial words such that . (i)If , then there exist partial words such that , , and .(ii)If , then there exist partial words such that , , and .

Definition 4. Two partial words and are called conjugate, if there exist partial words and such that and .

Definition 5. Two partial words and are called -conjugate, if there exist nonnegative integers , whose sum is and partial words and such that and .

3. Preliminaries on Partial Arrays

This section is devoted to review the basic concepts on partial arrays [11].

Definition 6. Partial array of size over , a nonempty set or an alphabet, is partial function , where is the set of all positive integers. For , , 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 is a partial array over with an empty set of holes.

Definition 7. If is a partial array of size over , then the companion of (denoted by ) is total function defined bywhere .

The bijectivity of map allows defining the catenation of two partial arrays in a trivial way.

Example 8. Partial array is the companion of partial array of size (3, 3), where ,
LetBy column catenation, we mean By row catenation, we mean

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 .

4. 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.(ii)Simplification: If either by column catenation or by row catenation with and being of same size, then and .(iii)Weakening: If and , then .

Lemma 3’s version for partial arrays can be stated as follows.

Lemma 10. Let be partial arrays such that , either by column catenation or by row catenation. (i)If order of order of , then there exist partial arrays , such that , , and .(ii)If order of order of , then there exist partial arrays , such that , , and .

4.2. -Compatibility

Definition 11. If and are two partial arrays of same size and is nonnegative integer, then is said to be -contained in denoted by if and there exists subset of of cardinality called the error set such that

Definition 12. If and are two partial arrays of same order and is a nonnegative integer, then and are called -compatible denoted by if there exist partial array and nonnegative integers such that(i) with error set ;(ii) with error set ;(iii);(iv).

Example 13. , , then there exists partial array with , and , ; that is, .
Equivalently, and are -compatible, if there exists subset of of cardinality called the error set such that (i);(ii)If and are arrays, then means . We sometimes use notation , if set has cardinality .

Multiplication. If and , then where , and are partial arrays and are nonnegative integers, using column catenation.

Example 14. , , , .
.

Simplification. If and order of is equal to order of , then and for some , satisfying .

Example 15. , , , .
and with 5 + 3 = 8.

Weakening. If and , then .

Example 16. , , .
with .

Theorem 17. Let and be partial arrays of orders and , respectively. If there exist array of order and integers , and such that with error set and with error set , then there exist integers and such that withMoreover, if , then .

Proof. Let and be partial arrays of and , respectively. Let array of order exist such that, by using column catenation, and for some integers , and . Let be the error set of cardinality such that for all and for all and be the error set of cardinality such that for all and for all . We have with error set of cardinality and with error set of cardinality .
Let and for some letter . There are 4 possibilities.
Case 1. If and , then and . It does not give any error, when we align with .
Case 2. If and , then and for some . It gives an error in the alignment of with only when or when .
Case 3. If and , then and for some . It gives an error in the alignment of with only when or when .
Case 4. If and , then for some and for some . It gives an error in the alignment of with only when .
Therefore, if then with and then .

Example 18. , ,
We have with error set , and   with error set . :(i),. (ii), , . , , (iii), . , . . , , , , , , , . :.

5. Conjugacy and -Conjugacy of Partial Arrays

5.1. 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.

Example 20. , , .
By taking and , we get and showing that and are conjugate similarly and, by taking and , we get and showing that and are conjugate. But and are not conjugate.
That is, conjugate relation is not transitive.

Proposition 21. Let and be nonempty partial arrays of same size. If and are conjugate, then there exists partial array such that , either by column catenation or by row catenation.

Proof. Let and be nonempty partial arrays of same order. Suppose and are conjugate and let be partial arrays such that and either by column catenation or by row catenation; then and . So, for , we have .

5.2. -Conjugacy

Definition 22. Two partial arrays and of same order are -conjugate, if there exist nonnegative integers whose sum is and partial arrays and such that and with row or column catenation.

Theorem 23. Let and be nonempty partial arrays of same order. If and are -conjugate, then there exists partial array such that .

Proof. Let be two partial arrays of same order. Supposing that and are -conjugate, then, by definition, there exist nonnegative integers , whose sum is and partial arrays and such that with error set and with error set using row catenation or column catenation accordingly.
Then, with error set and with error set or according to row or column catenation and so, for , we have .

Example 24. Given , .
There exist and with and , .
There exist such that .

6. 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 discuss -compatibility 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.

Competing Interests

The authors declare that they have no competing interests.