Permutation Groups in Automata Diagrams

Automata act as classical models for recognition devices. From the previous researches, the classical models of automata have been used to scan strings and to determine the types of languages a string belongs to. In the study of automata and group theory, it has been found that the properties of a group can be recognized by the automata using the automata diagrams. There are two types of automata used to study the properties of a group, namely modified finite automata and modified Watson-Crick finite automata. Thus, in this paper, automata diagrams are constructed to recognize permutation groups using the data given by the Cayley table. Thus, the properties of permutation group are analyzed using the automaton diagram that has been constructed. Moreover, some theorems for the properties of permutation group in term of automata are also given in this paper. | Finite automata | Watson-Crick finite automata | Cayley table | Permutation group | Automata diagram | ® 2013 Ibnu Sina Institute. All rights reserved. http://dx.doi.org/10.11113/mjfas.v9n1.79


INTRODUCTION
Since finite automata were first studied by Kleene in 1950 [1], finite automata theory have been widely used in the theories and applications of computer science and mathematics [2,3]. Mainly, finite automata are considered as recognition devices for the strings of formal languages.
A classic model of automaton consists of a finite set of states, an input alphabet (a finite set of symbols), a transition function, the initial state and a finite set of final states. However, the "classic" automata have the limitation of computational power as they can recognize only the regular languages [3]. Therefore, many types of finite automata have been developed to overcome this limitation. A new variant of finite automata with two reading heads and two input tapes that has been developed based on the concept of the classical model of finite automata, called Watson-Crick finite automata, for the recognition of double stranded strings which are the abstractions of DNA molecules [4].
In our previous papers [5,6], we have studied various properties of groups using finite automata and Watson-Crick finite automata. In particular, we have shown that the data given in the Cayley tables of Abelian groups can be recognized by both types of automata.
We have found that some properties of Abelian groups can also be obtained from automata transition diagrams. In this paper, we investigate the recognisability of the permutation groups by finite automata and Watson-Crick finite automata.
Moreover, we analyse some properties of permutation groups using modified deterministic finite automata over permutation groups and modified Watson-Crick finite automata over permutation groups. In addition, we present an example of a finite automaton (the transition diagram) which recognizes the permutation group of the symmetric group of order six, S 3 .
The paper is organized as follows: in Section 2 we recall some basic definitions and notations which are used in sequence. We introduce the modified versions of deterministic finite automata and Watson-Crick finite automata over permutation groups in Section 3. We show that several properties of permutation groups can be analysed using modified deterministic finite automata and modified Watson-Crick finite automata in Section 4. Finally, we summarize our results in Section 5.

PRELIMINARIES
We assume that the readers are familiar with the basic concepts of formal language theory and group theory. For further information, readers can refer to [7][8][9] for formal language theory and [10][11][12] for group theory. − Q is a finite set of internal states, − Σ is a finite set of symbols called an input alphabet, − q 0 ∈ Q is the initial state, − F ⊆ Q is a set of final states, − δ: Q × Σ →Q is a transition function.

Definition 3 ([7])
A transition diagram is a diagram with a finite number of vertices and arrows for the edges between two vertices, such that vertices represent the states, the arrows represent the transition functions and the labels on the edges are current values of the input symbol.

Definition 4 ([7])
A string is said to be accepted by an automaton if the automaton is in one of its final states when the end of the string is reached. For the string which is not accepted by automaton then it is said to be rejected.
Next, we cite the definitions of modified deterministic finite automata and Watson-Crick finite automata over Abelian groups, and explain the recognition of permutation groups by these types of automata in examples.

Definition 5 ([6])
Let G be an Abelian group. A modified deterministic finite automaton over G are defined as where, − Q ⊆ G is a finite set of internal states, − Σ ⊆ G is a finite set of symbols called an input alphabet, − q 0 ∈ Q is the initial state, − F ⊆ Q is a set of final states, − δ: Q × Σ →Q is a transition function such that with the Cartesian product of Q and Σ, a binary operation " * " of the group is associated.

Definition 6 ([6])
Let G be an Abelian group. A modified Watson-Crick finite automaton over G are defined as →Q is a transition function such that with the Cartesian product of Q and (Σ, Σ), a binary operation " * " of the group is associated.
For the Cayley table, all data of the group are written in the table. So, a group is said to be accepted by an automaton if the Cayley table of the group is recognized by the automaton.
In the following examples, we construct the automata diagrams of a modified finite automaton and modified Watson-Crick finite automaton over Abelian groups, which recognize the Abelian group of ℤ 2 and the direct product of Abelian group ℤ 1 × ℤ 4 respectively.

Example 1
For the group ℤ 2 = {0, 1}, its Cayley table is illustrated in Table 1. Then, the group ℤ 2 can be recognized by an automaton whose transition diagram is shown in Fig. 1.
Using the Cayley table, we can construct an automaton recognizing the group. The transition diagram is shown in Fig. 2.

AUTOMATA DIAGRAMS FOR PERMUTATION GROUPS
Before we relate automata to permutation groups, we give the definition of a permutation group.

Definition 7 ([10])
Given a non-empty set X; a permutation of X is a bijection α: X → X. The set S x of all permutation of X under the composition of mappings is called permutation group.
Next, we define modified versions of finite automata and Watson-Crick finite automata which recognize permutation groups.

Definition 8
Let P be permutation group. A modified deterministic finite automaton over P is defined as where, − Q ⊆ P is a finite set of internal states, − Σ ⊆ P is a finite set of symbols called an input alphabet, − q 0 ∈ Q is the initial state, − F ⊆ Q is a set of final states, − δ: Q × Σ →Q is a transition function such that with the Cartesian product of Q and Σ, a binary operation "∘" (the compositions of maps) of the group is associated.
It immediately follows that for any permutation group, one can easily construct a modified deterministic finite automaton which accepts the permutation group. As an example, we construct the transition diagram of an automaton accepting the permutation group S 3 .

Example 3
Given a finite set A = {1, 2, 3}. The group of all permutations of A is denoted by S 3 . Thus, For short, we denote the permutation group S 3 by The Cayley table for S 3 is shown in Table 3. Thus, we can construct the transition diagram of an automaton for the permutation group of S 3 based on the Cayley table as illustrated in Fig. 3.
For the direct product of permutation groups we can also construct a modified variant of Watson-Crick finite automata which recognizes the direct product of permutation groups.

Definition 9
Let P be a permutation group. A modified Watson-Crick finite automaton over P is defined as M = (Σ, , Q, P, s 0 , F, δ), where, − Σ ⊆ P is an input alphabet, − ⊆ Σ × Σ is the complementarity relation, − Q ⊆ is a finite set of internal states, − s 0 ∈ Q is the initial state, − F ⊆ Q is a set of final states, − δ: Q × (Σ, Σ) →Q is a transition function such that with the Cartesian product of Q and (Σ, Σ), a binary operation "∘" of the group is associated.
The recognition of modified Watson-Crick automaton is illustrated in Fig. 4 for S 2 × S 2 .

Example 4
Given the direct product of permutation groups S 2 × S 2 , such that The Cayley table for S 2 × S 2 is shown in Table 4. The transition diagram of an automaton for S 2 × S 2 is shown in Fig. 4.

Fig. 4. The transition diagram of an automaton for permutation group S2×S2
Again, it is easy to show that for each direct product of two permutation groups, one can construct a modified Watson-Crick automaton over a permutation group for the direct product of permutation groups. Moreover, we can analyse some properties of permutation groups using the transition diagrams of the corresponding automata.

SOME PROPERTIES OF PERMUTATION GROUPS IN TERMS OF AUTOMATA
In this section, we show that several properties of permutation groups can be expressed in terms of properties of the transition diagrams, for the corresponding modified finite automata.

Lemma 1
For a modified deterministic finite automaton over a permutation group P, the element assigned to a transition is the identity element of the permutation group if and only if the transition is a self-loop transition of a state, i.e., a transition e ij = (a i , a j ) is labelled by the identity element if and only if a i = a j for some a i , a j ∈ P.
Proof. Let b be an element of the group P associated with the transition e ij from node a i to node a j such that b∈{b k : δ(a i ,b k ) = a j , for all a i , a j ∈ Q and b k ∈ Σ}.
By construction, if δ(a i ,b k ) = a j then a j = a i ∘ b k . Thus, b k = a i -1 ∘ a j . Suppose that a i =a j , then b k = a j -1 ∘ a j and we can easily conclude that a j -1 ∘ a j is the identity element of the permutation group P. Therefore, b is the identity element associated with e ij .
Conversely, suppose that the label of the transition from node a i to node a j is the identity element b of the permutation group. By definition, δ(a i , b) = a j . On the other hand, a i ∘ b = a j . Since b is the identity element, a i ∘ b = a i , it follows that a j = a i , correspondingly, δ (a i , b) = a i .

Lemma 2
For a modified deterministic finite automaton M over a permutation group P, if the transition is not self-loop, then the element associated with the transition from node a i ∈Q to a j ∈Q is the inverse of the element associated with transition from node a j ∈Q to a i ∈Q, i.e., if e ij = (a i , a j ) is labelled by b∈P then e ji = (a j , a i ) is labelled by b -1 . Proof. Suppose that a transition e ij = (a i , a j ) is not self-loop (i ≠ j), then there is an element b 1 such that δ(a i , b 1 ) = a j . By definition, On the other hand, for the transition e ji = (a j , a i ), there is an element b 2 such that δ(a j , b 2 ) = a i . By definition, Substitute (2) into (1), By the group associativity law, By cancellation law, Thus, b 2 ∘ b 1 is the identity element, i.e., (b 2 ) -1 = b 1 . The identity of b 1 ∘ b 2 can also be shown by using similar arguments.

Theorem 1
If P is a permutation group of n elements, then P is recognized by a modified deterministic finite automaton M with n! states and (n!) 2 transitions.
Proof. Let P be a permutation group of n elements. Then |P| = n!. Let M be a modified deterministic finite automaton accepting P. By Definition 8, M also has n! states. For each state a i of M, there is one self-loop transition by Lemma 1, one outgoing transition to each state a j with i ≠ j and one incoming transition from each state a j with i ≠ j by Lemma 2. Then the total number of transitions is |Q| × |Q| = |P |× |P| = (n!) × (n!) = (n!) 2 .
The similar relationships between the direct products of permutation groups and modified Watson-Crick finite automata over permutation groups can be obtained by using the same arguments of the proofs of Lemmas 1, 2 and Theorem 1.

Lemma 3
For a modified Watson-Crick finite automaton over a permutation group P 1 ×P 2 , the element assigned to a transition is the identity element of the direct product P 1 ×P 2 if and only if the transition is a self-loop transition of a state, i.e., a transition e ij = (a i , a j ) is labelled by the identity element if and only if a i = a j for some a i , a j ∈ P 1 ×P 2 .

Lemma 4
For a modified Watson-Crick finite automaton M over a permutation group P 1 ×P 2 , if the transition is not self-loop, then the element associated with the transition from node a i ∈Q to a j ∈Q is the inverse of the element associated with transition from a j ∈Q to node a i ∈Q, i.e., if e ij = (a i , a j ) is labelled by b∈P then e ji = (a j , a i ) is labelled by b -1 .

CONCLUSIONS
In this research, the relation between the Cayley tables of permutation groups and finite automata is studied. A modified version of deterministic finite automata and modified Watson-Crick finite automata are defined and the relations are explained in examples. It is also shown that permutation groups and the direct products of permutation groups can be simulated by deterministic finite automata and Watson-Crick finite automata if the set of states and the alphabet of an automaton are defined as subsets of a permutation group. Since this paper is one of the preliminary works in this direction there are many open questions and a lot of interesting topics for future research: • From Theorems 1 and 2 one can notice that the descriptional complexities of automata are not optimal. Thus, it is interesting to check the possibility of minimizing the number of states and transitions of automata constructed in Theorems 1 and 2. • Since this work shows that permutation groups are simulated by modified automata, it is of interest to investigate the properties of permutation groups using automata. • It is also interesting to generalize the ideas considered in this paper and papers [4,5] for any type of finite groups, and to show the possibility of the construction of finite automata accepting finite groups. • The introduction of "group related automata" is also interesting in formal language theory: we can define the families of languages accepted by group which are related to automata and investigate their properties.