Commutative FSM Having Cycles over the Binary Alphabet

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1 Commutative FSM Having Cycles over the Binary Alphabet Dr.S. Jeya Bharathi 1, Department of Mathematics, Thiagarajar College of Engineering, Madurai, India A.Jeyanthi 2,* Department of Mathematics Anna University Regional Center, Madurai, India Abstract In this paper we deal with commutative FSM over the binary alphabet X. In particular, we introduce and study normal commutative FSM and commutative FSM having cycles over the binary alphabet X. Some properties on endomorphism monoids of these automata are given. Also, the representations of normal commutative FSM and commutative FSM having cycles over the binary alphabet X, are provided by S-automata and regular S-automata, respectively. Keywords: Endomorphism monid, commutative FSM, Representation S-automata, 1. Introduction Automata considered in this paper will be always automata without outputs. Clearly, an automaton A = (Q, X, δ) consists of the following data: (i) Q is a finite nonempty set of states (ii) X is a finite nonempty set of binary alphabet (iii) δ is a function, called a state transition function from Q X into Q. Let X * (X + ) denote the free monoid (free semi group) generated by the binary alphabet X. An element of X * is called a word over the binary alphabet X and ε is called the empty word. The state transition function [9] can be extended to the function from Q X * to Q by (i) δ(a, ε) = a for any a Q ; (ii) δ(a, xu) = δ(δ(a, x), u) for any a Q, x=(0,1) X and any u X * 2 Preliminaries Let A = (Q, X, δ) and B = (R, X, γ ) be automaton and let f be a mapping from Q into R. If f(δ(a, x)) = γ (f(a), x) holds for any a Q and any x =(0,1) X, then f is called a homomorphism from A into B. If a homomorphism f is bijective, then f is called an isomorphism. If there exists an isomorphism from A onto B, then A and B are said to be isomorphic to each other and denoted by A B. Moreover, a homomorphism (an isomorphism) from A into itself is called an endomorphism (an automorphism) [20] of A. It is clear that E(A) (G(A)) of all endomorphisms (automorphisms) of A forms a monoid (group) on the usual composition, called the endomorphism monoid (automorphism group) of A. Fig. 1 Commutative Automaton An automaton A = ( Q, X, δ) is said to be connected if for any a, b Q there exist a sequence a = a 0, a 1,...,a n = b of states and a sequence x 0, x 1,..., x n of letters such that δ(a i 1, x i ) = a i or δ(a i, x i ) = a i 1 for i = 1, 2,..., n. More-over, if for any a, b Q there exists a word u X* such that δ(a, u) = b, then A is said to be strongly connected. Fleck proved that if A = (Q, X, δ) is a strongly connected automaton, 8

2 then E(A) = G(A) and G(A) divides Q. 3 Commutative FSM over the binary alphabet X An automaton A = ( Q, X, δ) is said to be commutative if δ(a, uv) = δ(a, vu) for any a Q and for u, v X* a commutative FSM. The following results are basis of this chapter. Proposition 1 over the binary alphabet X and let a, b be a pair of states in Q. If δ (a, u) = b and δ (b, v) = a for some u, v X*, then a = b. over the binary alphabet X and let a, b Q. Suppose that δ (a, u) = b and δ(b, v) = a for some u, v X *. Then δ(a,uv) = δ(b, v) = a. Consider, δ (a, v) = δ (δ (b, v), v) = δ (b, v) = a. Therefore, a = δ (a, uv) = δ(a, vu) = δ(δ(a, v), u) = δ(a, u) = b. over the alphabet X. For any mapping λ u from Q into itself as follows: Fig. 2.State transition diagram of Automaton A=(Q,X, ) u X*, define a Fig.3.Hasse diagram of A ( a Q) λ u (a) = δ(a, u). The set {λ u u X*} is denoted by V(A). Theorem 1. Then V(A) is a commutative idempotent sub monoid of E(A).i.e., V(A) E(A) over the binary alphabet X. Clearly, λ ε E (A). For any nonempty word u X* and any x = (0,1) X, then λ u δ(a, 0) = δ( δ(a, 0), u ) (by the definition of λ u ) = δ(a, 0u) = δ(a, u0) (since A is commutative) = δ (λ u (a), 0). This shows that λ u E(A) and hence V(A) E(A). It is also a routine matter to verify that λ u 2 = λ u and λ u λ v = λ v λ u = λ uv V (A) for any u,v X*. Thus V(A) is a commutative idempotent sub monoid of E(A). 4 Normal commutative FSM over the binary alphabet X In this section normal commutative FSM over the binary alphabet X and S-automata are introduced. Also, the representations of normal commutative FSM over the binary alphabet X are providedby S-automata. over the binary alphabet X. Define a binary relation A on Q as follows: ( a, b Q) a A b δ (b, w) = a for some w X*. Since δ(a, ε) = a holds for any a Q, A is reflexive. It follows immediately from Proposition 1 that A is Antisymmetric. Also, it is a routine matter to verify that A is transitive. Thus, A is a partial order on Q. If the partially ordered set ( Q, A ) is a meet semi lattice and a A b denotes the greatest lower bound of a and b in Q, then A is a binary operation on Q and ( Q, A ) is a commutative idempotent semi group. Recall the translations of a semi group, the 9

3 translation ρ a associated with each element a in Q is given, i.e. ( b Q) ρ a (b) = a A b. The set of all ρ a is denoted by χ(a). Definition 1 A commutative FSM A = ( Q, X, δ) is said to be normal if ( Q, A ) is a meet semi lattice and χ(a) E(A).The class of all normal commutative FSM over the binary alphabet X is denoted by NC - FSM. Example 1 Fig. 2 shows the state transition diagram of a commutative automaton A = (A, X, δ). It is easy to see that the Hasse diagram in Fig. 3 of the partially ordered set (A, A ) is a meet semilattice. In the following there is another example which shows that ρ a may not be an endomorphism of a commutative FSM over the binary alphabet X, B = (R,X,δ), even if (B, B ) is not meet semilattice, where a R. Example 2 Fig. 4 shows the state transition diagram of a FSM over the binary alphabet X, B = (B, X, δ).it is easy to see that the Hasse diagram in Fig. 5 of the partially ordered set (B, B ) is not semi lattice. Also, Fig. 4 State transition Fig. 5 Hasse diagram of B diagram of automaton have ρ a δ(b, 0) = a A δ(b, 0) = a A d = e; δ ( ρ a (b,0) = (a A b,0) = (c,0) = f ρ a δ(b, 1) = a A δ(b, 1) = a A e = f ; (ρ a (b, 1) ) = (a A b,1) = (g,1) = c This implies that ρ a (δ(b, 0)) δ(ρ a (b), 0) and ρ a δ(b, 1) ( ρ a (b, 1) so ρ a is not an endomorphism of automaton B. Theorem 2 Let A = (Q, X, δ) be an automaton in NC- FSM over the binary alphabet X and E(A) the endomorphism monoid of A. Then the following statements are true: (i) χ(a) is a commutative idempotent sub semi group of E(A) (ii) χ(a) V(A) is a commutative idempotent sub monoid of E(A) (iii) χ(a) is an ideal of χ(a) V(A). Let A = (Q, X, δ) NC- FSM. Then χ(a) is a semi group under usual composition of mappings and isomorphic to (A, A ). Thus, χ(a) is a commutative idempotent sub semi group of E(A), since χ(a) E(A). To show that χ(a) V(A) is a commutative idempotent sub monoid of E(A) and χ(a) is an ideal of χ(a) V(A), to prove that λ u ρ a = ρ a λ u χ (A) holds for any u X* and any a Q, since both V(A) and χ(a) are commutative idempotent sub semi groups of E(A). It is true that λ ε ρ a = ρ a λ ε = ρ a for any a Q. Suppose that u X +. Then for any a, b Q, we have λ u ρ a (b) = λ u (a A b) (since ρ a is a translation) = δ(a A b, u) (by the definition of λ u ) = δ( ρ a (b), u ) = ρ a δ(b, u) (since ρ a is an endomorphism) = ρ a λ u (b) and ρ a λ u (b) = ρ a δ(b, u)(by the definition of λ u ) = δρ a (b), u(since ρ a is an endomorphism) = δ(a A b, u) (since ρ a is a translation) 10

4 = δρ b (a), u(since ρ b is a translation) = ρ b δ(a, u)(since ρ b is an endomorphism) = b A δ(a, u)(since ρ b is a translation) = ρδ(a,u) (b) (since ρ δ(a,u) is a translation). This shows that λ u ρ a = ρ a λ u = ρ δ(a,u) χ(a) holds for any u X* and any a Q, as required. In order to provide representations of automaton in NC-FSM over the binary alphabet X, introduce S-automata. 5 Representations of automaton in NC-FSM S- Automaton Definition 2 Let (S, ) be a finite commutative idempotent semi group and let T be an ideal of S. An automaton T = (T, X, δ ϕ ) is called an S-automaton, if state transition function δ ϕ is defined by δ ϕ (t, x) = t ϕ(x) for any t T and any x =(0,1) X, where ϕ is a mapping from X into S. Since X + is the free semi group on X, the mapping ϕ in the above definition can be extended to a semi group homomorphism from X + follows: into S as ( x = (0, 1) X, u X + ), ϕ(xu) = ϕ(x) ϕ(u). It is easy to verify that an S-automaton is a commutative FSM over the binary alphabet X. Theorem 3 Let (S, ) be a finite commutative idempotent semi group and T an ideal of S. If T = (T, X, δ ϕ ) is an S-automaton, then T can be embedded into the endomorphism monoid E(T) of automaton T. Let T = (T, X, δ ϕ ) be an S-automaton. Define a mapping Q s for any s S as follows: Q s : T T, Q s (t) = s t. Then for any x = (0, 1) X, we have Q s (δ ϕ (t, x)= Qs(t ϕ(x)= s t ϕ(x) = δ ϕ (s t, x) = δϕ(q s (t), x). This shows that Q s E(T). Now, to prove that the mapping φ: s Q s is a semi group homomorphism from S into E(T). For this purpose, take u Q u, v Q v. Then for any t T, Q u v (t) = u v t =Q u (Q v (t) = Q u Q v (t). This implies that u v Q holds. Hence, the mapping φ is a homomorphism. Next, to prove that φ T is injective. Suppose now that t, t T and that Q t = Q t. Then t =t t = Q t (t) (by the definition of Q t ) = Q t (t) (since Q t =Q t ) =t t =t t = Q t (t ) =Q t (t ) (since Q t = Q t ) =t t =t. This shows that φ T is injective. Hence, T can be embedded into the endomorphism monoid E(T) of automaton T. Let A = (Q, X, δ) be an automaton in NC- FSM. From Theorem 1, V(A) E(A). Define a mapping ϕ from X* into V (A) ( u X*) ϕ (u) = λ u. We can easily verify that ϕ is a homomorphism from the free monoid X* into E(A). Furthermore, take S = χ (A) V (A) in Definition 2. S-automaton χ (A) = ( χ(a), X, δ ϕ ), since it follows from Theorem 2(iii) that χ(a) is an ideal of χ(a) V(A). Define a mapping θ from Q into χ (A): ( a Q) θ(a ) = ρ a. It is easy to verify that θ is bijective. Now, to prove that θ is a homomorphism from automaton A into automaton χ (A), i.e., θ(δ(a, x)) = δ ϕ (θ(a), x) holds for any a Q and any x=(0,1) X. In fact, for any b Q, then (θ(δ(a, x))(b) = ρ δ(a,x) (b) 11

5 = b A δ(a, x)(since ρ δ(a,x) is a translation) = ρ b (δ(a, x)) (since ρ b is a translation) = δ (ρ b (a), x (since ρ b is an endomorphism) = δ (a A b, x) where x = (0,1) X and (δ ϕ (θ(a), x))(b) = δ ϕ (ρ a, x) (b) since θ(a) = ρ a ) = (ρ a ϕ(x)) (b) (by Definition 2) = (ρ a λ x ) (b) (by the definition of ϕ) = ρ a (δ(b, x) (by the definition of λ x ) = δ(ρ a (b), x (since ρ a is an endomorphism) = δ (a A b, x) (since ρ a is a translation), where x= (0,1) X This implies that θ (δ(a, x)) = δ ϕ (θ(a), x) holds for any a Q and any x X. Therefore, θ is an isomorphism from the automaton A onto the automaton χ(a). 6 Isomorphism with Commutative Idempotent Semigroup Theorem 4 Let A = (Q, X, δ) be an automaton in NC- FSM and let S be the commutative idempotent semi group χ (A) V (A). Then A is isomorphic to the S- automaton (χ (A), X, δ ϕ ). The following lemma is obvious and its proof is omitted. Lemma 1 Let S and S be two finite commutative idempotent semi groups. If S is isomorphic to S, then for any S-automaton T = (T, X, δ ϕ ), there exists an S -automaton T = (T, X, δ ϕ ) such that T T. From the above Lemma 1 and Theorem 4, then Corollary 1 Let A be an automaton in NC-FSM and let S be a finite commutative idempotent semi group such that S χ(a) V(A).Then A is isomorphic to some S-automaton. 7 Commutative FSM having cycles over the binary alphabet X In this section introduce and study commutative FSM having cycles over the binary alphabet X and regular S-automata. It is shown that every commutative FSM having cycles over the binary alphabet X is normal. Also, the representations of commutative FSM having cycles are provided by regular S-automata. Let A = (Q, X, δ) be an automaton. Recall that a state g in Q is called a generator of A if for any a Q there exists u X* such that δ(g, u) = a. The set of all generators of A is denoted by Gen (A). An automaton A is said to be cyclic if Gen (A). The class of all commutative automaton having cycle is denoted by CAC. Let A = (Q, X, δ) be an automaton in CAC. It is true that A has a unique generator. In fact, if g,h are generators of A, then there exist u, v X* such that δ(g, u) =h and δ(h, v) = g. Thus, it follows from Proposition 1 that g =h. Lemma 2 If A is an automaton in CAC-FSM having cycle over the binary alphabet X, then A has a unique generator. Some properties of an endomorphism of automaton A in CAC-FSM having cycle over the binary alphabet X are given as follows. Theorem 5 Let A = (Q, X, δ) be an automaton in CAC and g the unique generator of A. If ρ is an endomorphism of A, then the following statements are true. (i) If ρ o (g) = ζ(g) for some ζ E(A), then ρ o = ζ ; (ii) ρ o is an order-preserving mapping from partially ordered set (Q, A) into itself, i.e., for any a, b Q if a A b, then ρ o (a) A ρ o (b); 12

6 (iii) ρ o (a) A a for any a Q Let A=(Q, X, δ) CAC and g be the unique generator of A. Given an endomorphism ρ o of A, Then for any a Q, there exists u X* such that a = δ (g, u). (i) Suppose that ζ E(A) and ρ o (g) = ζ(g). Then we have ρ o (a) = ρ o (δ(g, u)) = δ(ρ o (g), u) (since ρ is an endomorphism) = δ(ζ(g), u) since ρ o (g) = ζ(g)) = ζ(δ(g, u)) (since ζ is an endomorphism) = ζ(a). This implies that ρ o = ζ, as required. (ii) Suppose that a, b Q and a A b. Then there exists u X* such thatδ (b, u) =a. Since ρ o E(A), we have δ(ρ o (b), u) = ρ o (δ(b, u)) = ρ o (a). This is, ρ o (a) A ρ o (b). Thus, ρ o is order-preserving, as required. (iii) Since g is the generator of A, it is true that for any a Q, there must exist some u, v X* such that δ(g, u) =a and δ(g, v) = ρ o (a). Thus, we have δ(a, u) = δ(δ(g, u), u) = δ(g, u) =a; δ(ρ o (a), u) = ρ o (δ(a, u)) = ρ o (a). Then δ(a, v) = δ(δ(g, u), v) = δ(g,uv) = δ(g, vu) = δ(δ(g, v), u) = δ(ρ o (a), u) = ρ o (a). This implies that ρ o (a) A a, as required. Let A = (Q, X, δ) be an automaton. For any x= (0,1) X*, denote the set {y X* ( a A) δ(a, x) = δ(a, y)} by, x and denote the set { x x X*} by C(A). Then C(A) is a monoid under the operation defined by x y = xy. It is called the characteristic monoid of automaton A. Lemma 3 having cycles over the binary alphabet X. Then (i) E(A) C(A) (ii) E(A) = A. Also, it is easy to prove that C(A) V(A). Thus by Lemma 3, show that E(A) V(A). In fact, from Theorem 1 immediately have lemma, Lemma 4 If A is an automaton in Commutative FSM having cycles over the binary alphabet X, then E(A) = V(A) and E(A) is a commutative idempotent monoid. Let A be an automaton in Commutative FSM having cycles over the binary alphabet X, CAC. Then from the above Lemma 3 hat E(A) is a meet semi lattice under the natural partial order,, defined as follows: ( ρ o,ζ E(A) ρ o ζ ρ o ζ = ζ ρ o = ρ o Theorem 6 If A = (Q, X, δ) is an automaton in then the partially ordered sets (Q, A ) and (E(A), ) are order isomorphic, i.e., there exists a bijection θ from Q onto E(A) such that both θ and θ 1 are orderpreserving. Let A = (Q, X, δ) CAC and g be the unique generator of A. By Lemma 4, E(A) = V(A). Define a mapping θ from Q into E(A) as follows: ( a A), θ (a) = λ u, where λ u (g) =a, u X* First, to show that θ is well defined. Noticing that g is a generator, that for any a Q, there exists u X* such that λ u (g) = δ(g, u) = a. Moreover, if there exist λ u,λ v V(A) such that λ u (g) 13

7 = λ v (g), then by Theorem 3.5(i), then λ u = λ v. Thus, θ is well defined. Now, to prove that θ is a bijection. Given a, b Q. Suppose that θ(a) = λ u and θ(b) = λ v for some u, v X*. If λ u = λ v, then a = λ u (g) = λ v (g) = b. Hence, θ is injective. Since Q is finite, we have by Theorem 3 that θ is also surjective. Thus, θ is a bijection. Next, to prove that θ is order-preserving. Suppose that a A b. That is to say, δ (b, w) = a for some w X*. Moreover, assume that θ(a) = λ u and θ(b) = λ v. It is clear from Theorem 1 that λ u λ v = λ v λ u = λ uv. Also, we have λ v (b) = δ(b, v) = δ(δ(g, v), v) = δ(g, v) =b. Thus, λ uv (g) = δ(g,uv) = δ(δ(g, u), v) = δ (a, v) = δ(δ(b, w), v) since δ(b, w) =a) = δ (δ(b, v), w) (since A is commutative) = δ (b, w) since δ(b, v) = b =a = λ u (g). Then by Theorem 1, then λ u λ v = λ v λ u = λ uv = λ u. This implies that λ u λ v and hence θ is order-preserving. Finally, to prove that θ 1 is also orderpreserving. Given λ u, λ v V(A) such that λ u λ v. That is to say, λ u λ v = λ v λ u = λ u. Assume that θ 1 (λ u ) =a and θ 1 (λ v ) =b. Then we have a = λ u (g) (since θ 1 (λ u ) =a) = λ u λ v (g) = λ u (b) (since θ -1 (λ v ) = b) = δ(b, u). This shows that a A b. Hence, θ 1 is orderpreserving. Therefore, (Q, A ) and (E(A), ) are order isomorphic to each other. Lemma 5 Let A = (Q, X, δ) be an automaton in Commutative FSM having cycles over the binary alphabet X, Commutative FSM having cycles over the binary alphabet X, and g the unique generator of A. Then ρ o (g) is the maximum fixed point of ρ o in (Q,, A ) for any ρ o E(A), i.e., ρ o (g) = max{a Q ρ o (a) =a}. Let A = (Q, X, δ) be an automaton in Commutative FSM having cycles over the binary alphabet X., and g the unique generator of A. Suppose that ρ o E(A). Then, from Lemma 4 that ρ o 2 = ρ o. In particular, ρ o 2 (g) = ρ o ρ o (g). That is to say, ρ o (g) {a A ρ o (a) =a}. For any b Q, there exists u X* such that δ(g, u) =b, since g is a generator. If b is a fixed point of ρ o, then δ(ρ o (g), u) = ρ o δ(g, u)) = ρ o (b) =b. That is to say, b A ρ o (g) and hence ρ o (g) = max{a Q ρ o (a) =a}. 8 Representations of automaton in CAC-FSM Regular S- Automaton Theorem 7 If A is an automaton in Commutative FSM having cycles over the binary alphabet X., then E(A) = χ(a) holds. Let A = (Q, X, δ) Commutative FSM having cycles over the binary alphabet X., and g be the unique generator of A. Since (E(A), ) is a meet semilattice, from Theorem 6 that (Q, A ) is also a meet semilattice. Thus, (Q, A ) is a commutative idempotent monoid and is isomorphic to E(A). (Q, A ) is also isomorphic to χ(a). Then we have E(A) χ(a) and so E(A) = χ (A). In the following to prove that E(A) = χ(a) recall the bijection from A onto E(A) defined in Theorem 6. ( a Q) θ(a) = λ u, where λ u (g) =a, u X* 14

8 Given a Q and suppose that θ(a) = λ u. Then λ 2 u = λ u and λ u (a) = λ 2 u (g) = λ u (g) = a. To show that λ u = ρ a. First, to prove that λ u (b) = a A b holds when a A b. If a A b, then from Theorem 5(ii) that a = λ u (a) A λ u (b) A λ u (g) =a. Thus λ u (b)=a=a A b. Next, to prove that λ u (b) =a A b holds when b A a. If b A a, then δ(a, w) =b for some w X*. Hence, λ u (b) = λ u (δ(a, w)) = δ(λ u (a), w) (since λ u is an endomorphism) = δ(a, w) since λ u (a) =a = b = a A b. Suppose now that a and b are incomparable in (Q, A ). Furthermore, assume that a A b =d and λ u (b) =d d. Then, from Theorem 5(iii) shows that, d A b. Since λ u (d ) = λ 2 u ( (b) = λ u (b) = d, d is a fixed point of λ u. Then by Lemma 5, d A λ u (g) = a. This implies that d is a lower bond of a and b. thus have d < A d. On the other hand, from d A a, hence λ u (d)=a A d = d. Then λ u (b) = d < A d = λ u (d), which is a contradiction with the fact that λ u is orderpreserving (Theorem 5(ii)). That is to say, λ u (b) = a A b holds when b and a are incomparable in (Q, A ). Therefore, λ u = ρ a and hence E(A) χ(a). Since E(A) = χ(a) = A and A is finite, χ(a) = E(A), as required. From Theorem 6 and Theorem 7 the next proposition, Proposition 2 Commutative FSM having cycles over the binary alphabet X., is a subclass of NC-FSM. In order to provide representations of automaton in Commutative FSM having cycles over the binary alphabet X, and introduce regular S-automata. 9 Isomorphism with regular S-automaton Definition 3 Let (S, ) be a finite commutative idempotent monoid. An S-automaton T = (T, X, δϕ) is said to be regular if T = S and S E(T). Let A = (Q, X, δ) be an automaton in CAC. From Theorem 3.4, there exists an S-automaton χ(a) = (χ(a), X, δ ϕ ) isomorphic to the automaton A, where S is the commutative idempotent monoid χ(a) V(A) and ϕ(u) = λ u for any u X*. Then it is clear that the endomorphism monoid E(χ(A)) of the automaton χ(a) is isomorphic to E(A). By Lemma 4 and Theorem 7, then E(A)=χ(A)=V(A). That is to say, S = χ (A) = E(A) and hence S E(χ(A)). Thus, the automaton χ(a) is a regular S-automaton. Then from Lemma 1 we have proposition, Proposition 3 Let A be an automaton in CAC over binary alphabet X and let S be a semi lattice such that S E(A). Then A is isomorphic to some regular S- automaton. References [1] Bavel.Z, Structure and transition-preserving functions of finite automata, J. ACM 15 (1968) [2] Fleck.A.C, Isomorphism groups of automata, J. ACM 9 (1962) [3] Fleck.A.C., On the automorphism group of an automaton, J. ACM 12 (1965) [4] Ito.M., Algebraic Theory of Automata and Languages, World Scientific Publishing Co. Pte. Ltd., [5] Weeg.G.P., The structure of an automaton and its operation-preserving transformation group, J. ACM 9 (1962)