Recognition of Simple Splicing Systems using SH-Automaton

Splicing language is the language which results from a splicing system. Splicing system was first introduced by Tom Head in 1987 as the mathematical model of systems of restriction enzymes acting on initial DNA molecules. Splicing languages are closely related to automata theory. Simple splicing systems can be recognized by SH-automata diagrams due to the regularity of splicing languages. SH-automaton defines exactly one language which is the language generated by the simple splicing system. In this paper, the concept of firm and maximal firm subwords are introduced. Some examples are then given to illustrate the maximal firm subwords of a word in a simple splicing system. Taking the SH-automata concept, which is a short compact way of encoding normal non-deterministic automata in the special case of SH systems, the maximal firm subwords of the initial words of an SH systems serve as the labels for the associated SH-automaton. Some examples which will show the maximal firm subwords of the words in the initial set I, the regular expression for the language generated by the given splicing system and the simplest nondeterministic automaton that recognizes the corresponding splicing system are also given | Splicing system | Splicing language | SH-automaton | Maximal firm subwords | Regular expression |


Introduction
Splicing system was developed as a mathematical model of systems of restriction enzymes acting on initial DNA molecules [1].DNA molecules are made up of nucleotides, and these nucleotides differ from each other by their bases, namely Adenine (A), Cytosine (C), Guanine (G) and Thymine (T) respectively.DNA molecules can be cut by restriction enzymes at particular sequence of the nucleotides.After cutting, the resulting fragments can be pasted together by a ligase, thus forming a molecule of recombinant DNA.The set of double-stranded DNA molecules that may arise from a splicing system is called the splicing language.The definitions of a splicing system and a splicing language are stated in the following.Definition 1. [1] (Splicing System and Splicing Language) A splicing system S = (A, I, B, C) consists of a finite set of alphabet A, a finite set of initial strings I in A*, and finite sets B and C of triples (c, x, d) with c, x and d in A*.Each such triple in B or C is called a pattern.For each such triple the string cxd is called a site and the string x is called a crossing.Patterns in B are called left patterns and patterns in C are called right patterns.L(S) is the language generated by a splicing system S which consists of the strings in I and all strings that can be obtained by adjoining the words ucxfq and pexdv to L whenever ucxdv and pexfq are in L, and (c, x, d) and (e, x, f) are patterns of the same hand.A language L is a splicing language if there exists a splicing system S for which L = L(S).
A particular type of splicing system discussed in this paper is the simple splicing (SH) system.The formal definitions of a simple splicing system and a simple splicing language are given in the following.Definition 2. [2] (Simple Splicing System and Simple Splicing Language) Let S = (A, I, R) be a splicing system in which all rules in R have the form (a, 1; a, 1), where a is in A. Then S is called a simple splicing system.A splicing language L is said to be a simple splicing language if L can be generated by a simple splicing system.Splicing languages are regular, but not all regular languages are splicing languages [3].The definition of a regular language is given in the following.

Definition 3. [4] (Regular)
A language L is called regular if and only if there exist a deterministic finite accepter M such that L = L(M).
For the definition of regular language, deterministic finite accepter is used instead of deterministic finite automaton since deterministic finite automaton [5] is also known as deterministic finite accepter in [4].Therefore, since simple splicing languages are regular, they can be recognized by automata diagrams.Labels for an SH-automaton are actually the maximal firm subwords of a word.In the next section, the concepts of firm and maximal firm subwords of a word are discussed.

Maximal Firm Subwords
For the simple splicing language L we say that a word w in L is firm if it contains no occurrence of a letter in the rule set R. Recall that by a factor of a word w in A* meant any word y, there are words x and z for which w = xyz, where x, z in A*.So a maximal firm subword of a word w should be a factor y of w in which no letter in R occurs in y but if w = uayz with a in A, then a is in R and if w = xybv with b in A, then b is in R.
If w = uayz and a is not in R, then ay would be a firm factor of w that is one letter longer than y and so y would not be a maximal firm subword of w.If w = xybv and b is not in R, then yb would be a firm factor of w that is one letter longer than y and so y would not be a maximal firm subword of w.A word y is a maximal firm subword of w as long as y can get, and stay, inside w.In other words, a maximal firm subword of w is the longest possible factor of w that contains no letter in R.
A word w in A* that contains no letter in R cannot be cut at all, thus it is considered to be firm.If w = xry with neither x nor y null then w can be cut into two pieces and should therefore be not firm.For simple splicing languages, a word w is firm if and only if it contains no element of R. Some examples to illustrate the maximal firm subwords of a word in a simple splicing system are given next.They differ in terms of the number of rules and initial strings involved.Example 1 involves a splicing system with only one rule with the initial set I left unspecified.

Example 1.
Let A = {a, b}, B = {b}, I will be left unspecified.There is only one rule, namely r = (b, λ, b, λ).A word w in A* is firm with respect to r if (and only if) w is in a*.The maximal firm subwords of the word aabaaabaaba are indicated via underscores: aabaaabaaba.
In the next example, maximal firm subwords of a word in a splicing system having two rules with initial set I unspecified are given.

Example 2.
Let A = {a, b, c, d, e}, B = {b, d}, I will be unspecified.There are only two rules, namely r = (b, λ, b, λ ) and r' = ( d, λ, d, λ).These two rules can be abbreviated as b and d respectively.A word w in A* is firm with respect to r if w in {a, c, d, e}*.A word w in A* is firm with respect to r' if w in {a, b, c, e}*.A word w in A* is firm with respect to R = {r, r'} if and only if w is in {a, c, e}*.The maximal firm subwords of the word aabacadcccbceeab are indicated via underscores: aabacadcccbceeab.
Example 3 and Example 4 show the maximal firm subwords of words in a splicing language L generated by a splicing system involving a specified set I, one and two rules respectively.The language generated is listed in recursive form and the maximal firm subwords of words in that language are indicated via underscores.

Example 3.
Let A = {a, b}, B = {b} and I = {aaabaa, aba}.The language generated by (A, B, I) is L = {aaabaa, aba, aaaba, abaa}.The maximal firm subwords of words in L are indicated via underscores: aaabaa, aba, aaaba, abaa.In order to visualize and represent finite automata, transition diagrams which involve vertices and edges are used [4].An automaton recognizes a language L if each string accepted by the automaton is in L and each string in L is accepted by the automaton [6].The simple splicing automata (SH-automata) concept is explained in the following.

SH-automata Concept
Taking the SH-automata concept, which is a short compact way of encoding normal non-deterministic automata in the special case of SH systems, the maximal firm subwords of the initial words of an SH systems serve as the labels for the associated SH-automaton.
One trip through the SH-automaton is an arbitrary walk through the graph entering via an entrance following any path until exit.The language is an infinite set.We can generate a lot of different words from an SH-automaton, but SH-automaton defines exactly one language which is exactly the language generated by the simple splicing system.Some examples are given in the following which will show the maximal firm subwords of the words in the initial set I, the regular expression for the language generated by the given splicing system and the simplest nondeterministic automaton that recognizes the corresponding splicing system.The examples differ in the number of initial strings and rules involved.
Example 6 involves a splicing system with only one string in I and one rule in R.

Example 6.
Suppose A = {a, b}, I = {aabaaaaabaaabaa} and R = {b}.The maximal firm subwords of the only word in I are aa, aaaaa, aaa.The regular expression for this language is aab(aaab+aaaaab)*aa.The simplest nondeterministic automaton that recognizes L (A, I, R) is given in Figure 1.Example 7 involves a splicing system with two strings in I and one rule in R.

Example 7.
Suppose A = {a, b}, I = {abaaabaaabaa, aabaaaaabaaaaabaaaaa} and R = {b}.The maximal firm subwords of I are a, aaa, aa, aaaaa.The regular expression for this language is (a+aa)b(aaab+aaaaab)*(aa+aaaaa).Notice that we will get the same language generated if I were {abaaabaa, aabaaaaabaaaaa} or if I were {abaaaaa, aabaaaaabaaabaa}.The simplest non-deterministic automaton that recognizes L (A, I, R) is given in Figure 2.   Therefore, simple splicing systems can be recognized by SH-automaton using transition diagrams as illustrated in this section.

Conclusion
In this paper, the concept of maximal firm subwords is introduced.The maximal firm subwords of the initial words of a simple splicing system are shown to serve as the labels for the associated SH-automaton.Also,