General Fuzzy Automata

Transcription

1 Chapter 4 General Fuzzy Automata 4.1 Introduction Doostfatemeh and Kremer [15] have kindly revised the developments in fuzzy automata theory and pointed out certain important issues such as i) membership value assignment for the next state ii) multi-membership resolution and iii) significance of the output mapping in automata theory. To resolve these issues they have proposed an improved definition of fuzzy automaton, they call it as general fuzzy automaton. Further, Doostfatemeh and Kremer [17] have used this definition of general fuzzy automaton in defining a membership value mv) for the strings of a fuzzy grammar/language, which have been important issue since the inception of fuzzy automata and fuzzy languages. In [14] they have discussed representation of general fuzzy automata based on second-order recurrent neural networks and it is shown that they are more efficiently representable in second-order recurrent neural networks. Later on the theoretical developments of general fuzzy automata was discussed mainly by Zahedi and Horry [23, 24, 26] and Abolpour and Zahedi [1]. In this chapter we use this definition of general fuzzy automaton to discuss the concepts of language recognition, state-wise minimization and the equivalence between deterministic and 51

2 non-deterministic general fuzzy automaton. Here, we introduce general fuzzy recognizer and their languages, general fuzzy automata with outputs and their homomorphism, state-wise equivalent and minimal general fuzzy automata with outputs, non-deterministic general fuzzy automata with and without Λ moves). We actually establish following results: 1) The class of general fuzzy automaton language is closed under union, intersection, cartesian product, homomorphic and inverse hmomorphic images. 2) Factor general fuzzy automaton M/θ, with respect to congruence relation θ, is a homomorphic image as well as state-wise equivalent) to a general fuzzy automaton M with outputs. 3) Fundamental theorem for general fuzzy automata with outputs 4) Existence of minimal general fuzzy automation 4) Deterministic general fuzzy automaton, non-deterministic general fuzzy automaton and non-deterministic general fuzzy automaton with Λ moves are equivalent. 4.2 General Fuzzy Recognizer and their Language In this section, we discuss general fuzzy recognizer and their fuzzy languages. Definition [15] A general fuzzy automaton is an eight-tuple M = Q, Σ, R, Z, ω, δ, T 1, T 2 ), where i) Q is a finite set of states, Q = {q 1, q 2,..., q n } ii) Σ is a finite non-empty set of input symbols, Σ = {a 1, a 2,..., a m } iii) R is the set of fuzzy start states, R F Q) iv) Z is a finite set of output symbols, Z = {b 1, b 2,..., b k } v) ω : Q Z is the output function, vi) δ : Q [0, 1]) Σ Q [0, 1] is the augmented transition function, 52

3 vii) T 1 : [0, 1] [0, 1] [0, 1] is the membership assignment function, viii) T 2 : [0, 1] [0, 1] is called the multi-membership resolution function. The function T 1 µ, δ) has two parameters, µ and δ, where µ is the membership value of a predecessor and δ is the weight of a transition. In this definition, the process that takes place upon the transition from state q i to q j on input a k is represented as : µ t+1 q j ) = δ q i, µ t q i )), a k, q j ) = T1 µ t q i ), δq i, a k, q j ) ), whenever µ t q i ) > 0. i.e. That the membership value mv) of the state q j at time t + 1 is computed by function T 1 using both the membership value of q i at time t and the weight of the transition. Sometimes the membership value of q j at time t + 1 is not necessarily unique at time t one may in several q is ). To assign single mv to q j at time t + 1, we use function T 2 i.e. multi-membership resolution function ) in the following manner : µ t+1 q j ) = T n 2 [v i ] = n [ T 2 T1 µ t q i ), δq i, a k, q j ) )], i=1 i=1 where n is the number of simultaneous transitions to the active state q j at time t + 1. In the following definition we use for the set of fuzzy transitions. i.e. = {δ q i, a k, q j ) q i, q j Q and a k Σ} Definition [24] Let Q act t) be the set of all active states at time t 0, we have Q act t) = {q, µ t q)) : q Q act t 1), a Σ, δq, a, q) } and Q act t 0 ) = R. Since Q act t) is a fuzzy set, in order to show that a state q belongs to Q act t) and T is a subset of Q act t), we continue to write : q DomainQ act t)) and T DomainQ act t)) as q Q act t) and T Q act t). 53

4 The definition of max-min general fuzzy automaton is given by Abolpour in [1]. We slightly modify this definition for technical reason. We actually consider rather than Q act = Q act t 0 ) Q act t 1 )... Q act = {Q act t 0 ), Q act t 1 ),... } In this section we do not need set of outputs and output function, therefore we consider general fuzzy automaton without them. Definition [24] Let M = Q, Σ, R, Z, w, δ, T 1, T 2 ) be a general fuzzy automaton. Then the max-min general fuzzy automaton as M = Q, Σ, R, Z, w, δ, T 1, T 2 ) such that δ : Q act Σ Q [0, 1], where Q act = Q act t 0 ) Q act t 1 )... and δ q; µ t q)), Λ, p ) 1, if q = p = 0, otherwise Also, if the input at time t be a where a Σ, then δ q; µ t 1 q)), a, p ) = δ q; µ t 1 q)), a, p ) and δ q; µ t 1 q)), a 1 a 2, p ) = δ q; µ t 1 q)), a 1, q ) δ q ; µ t q )), a 2, p )) q Q actt) and recursively δ q; µ t 0 q)), u 1 u 2... u n, p ) = { δ q; µ t 0 ) q)), u 1, p 1 δ p1 ; µ t 1 ) p 1 )), u 2, p 2 δ p n 1 ; µ t n 1 p n 1 )), u n, p ) p 1 Q act t 1 ), p 2 Q act t 2 ),..., p n 1 Q act t n 1 )} To clarify the concepts we consider an example given below. 54

5 Example Consider the following general fuzzy automaton, where { } 0.2 R = q 0, 0.7 q 1 = Q act t 0 ), i.e. µ t 0 q 0 ) = 0.2, µ t 0 q 1 ) = 0.7. a, 0.5 q 5 b, 0.3 start 0.2 q 0 q 4 a, 0.2 a, 0.3 c, 0.8 start 0.7 q 1 c, 0.8 q 3 c, 0.4 q 2 a, 0.1 b, 0.1 Diagram: General Fuzzy Automaton t 0 t 1 t 2 t 3 q 0 q 0, q 4, q 5 q 0, q 5, q 4, q 4 q 0, q 5, q 4, q 4, q 4, q 5, q 4 q 0 q 0, q 1, q 2 q 0, q 1, q 2, q 0, q 5, q 2, q 1 q 0, q 1, q 2, q 0, q 5, q 2, q 1, q 2, q 0, q 1 Active states at time t 1) when T 1 = maxµ, δ), we have fuzzy set Q act t): 55

6 t 0 t 1 t 2 Λ a b c Λ a b c 0.2 q 0 1 q q q q 2 T 2 1, 1) q 0, T 21, 1) q 1 T 2 1, 0.7) q 5 T 2 1, 0.7) q q q 1 1 q q 0 T 2 0.7, 1) q 0, q 2 q 0 q 2 q 2 1 1, q 4 q 5 { 1 Here Q act t 0 ) = R, Q act t 1 ) for Λ =, 1 }, Q act t 1 ) for a = { q 0 q 1 T2 1, 0.7) Q act t 2 ) for a =, 1 } etc. q 5 q 0 { 0.5, 0.7 }, q 5 q 0 2) when T 1 = minµ, δ), we have fuzzy set Q act t): t 0 t 1 t 2 Λ a b c Λ a b c T 2 0.2, 0.3) T 2 0.2, 0.3) T 2 0.2, 0.3) 0.4 q 0 q 0 q 5 q 4 q 2 q 0 q 5 q 4 q q 1 q 1 q 0 q 1 { 0.2 Here Q act t 0 ) = R, Q act t 1 ) for Λ = { T2 0.2, 0.3) Q act t 2 ) for a =, 0.3, 0.2 } etc. q 5 q 0 q q 2, 0.2 q 4, 0.2 q q 0, 0.2 q q q 2 q 0, 0.7 q 1 }, Q act t 1 ) for a = { 0.2, 0.3 }, q 5 q 0 56

7 Definition A general fuzzy recognizer is a pair M r = M, F ), where i) M = Q, Σ, R, δ, T 1, T 2 ) is a max-min general fuzzy automaton, ii) F is a fuzzy subset of Q, called the fuzzy subset of final states. Throughout this chapter we use this modified definition for general fuzzy recognizer. In [7] the language of general fuzzy recognizers are taken as a crisp language where as we consider the language of general fuzzy recognizer as fuzzy language as follows. Definition The fuzzy language L = {x, µ L x)) x Σ }, where µ L x) = {Rq) δ q, µ t 0 q)), x, p) F p) : q, p Q} is called the fuzzy language generated by general fuzzy recognizer M r LM r ) for this fuzzy language L.. Here, onwards we shall write Example Consider the following general fuzzy recognizer, where { } 0.2 R = q 0, 0.3 q 1 = Q act t 0 ) and δ is defined in the following diagram. a, 0.5 q 5 b, 0.3 start 0.2 q 0 1/q 4 a, 0.2 a, 0.3 a, 0.1 c, 0.8 start 0.3 q 1 c, 0.8 q 3 c, 0.4 q 2 a, 0.1 b, 0.1 General Fuzzy Recognizer We have, LMr )x) = {Rq) δ q, µ t 0 q)), x, p) F p) : q, p Q}. 57

8 Then, LM r ) = { } , ba ab)a For, δ q 0, µ t 0 q 0 )), b, q 4 ) δ q 4, µ t 1 q 4 )), a, q 4 ) = = 0.2 and δ q 1, µ t 0 q 1 )), a, q 0 ) δ q 0, µ t 1 q 0 )), b, q 4 ) δ q 4, µ t 2 q 4 )), a, q 4 ) = = 0.1 Definition A fuzzy language L over Σ is said to be a general fuzzy recognizer language, if there is a general fuzzy recognizer M r = Q, Σ, R, δ, T 1, T 2, F ) such that LM r ) = L. Theorem If L 1 and L 2 are general fuzzy recognizer languages over Σ then L 1 L 2, L 1 L 2, L 1 L 2 are general fuzzy recognizer languages over Σ. Proof. We prove 1) only 1) let Mr i = Q i, Σ, R i, δi, T 1, T 2, F i ) be general fuzzy recognizers such that LMr 1 ) = L 1 and LMr 2 ) = L 2 respectively. Construct a general fuzzy recognizer Mr = Q 1 Q 2, Σ, R 1 R 2, δ1δ 2, T 1, T 2, F 1 F 2 ),by making Q 1 Q 2 = φ possibly by renaming elements of either Q 1 or Q 2 ) as follows : δ1 q, µ t q)), a, p), if q, p Q 1 δ1 δ2) q, µ t q)), a, p) = δ2 q, µ t q)), a, p), if q, p Q 2 0, otherwise, R 1 q), if q Q 1 F 1 p), if p Q 1 R 1 R 2 )q) = and F 1 F 2 )p) = R 2 q), if q Q 2 F 2 p), if p Q 2 Now, LMr )x) { = R1 R 2 )q) δ1 δ2) q, µ t q) ), x, p ) F 1 F 2 )p) } q, p Q 1 Q 2 = { R1 R 2 )q) δ1 δ2) q, µ t q) ), x, p ) F 1 F 2 )p) } q, p Q 1 { R1 R 2 )q) δ1 δ2) q, µ t q) ), x, p ) F 1 F 2 )p) } q, p Q 2 58

9 p Q 1 and q Q 2 { R1 R 2 )q) δ 1 δ 2) q, µ t q) ), x, p ) F 1 F 2 )p) } p Q 2 and q Q 1 { R1 R 2 )q) δ 1 δ 2) q, µ t q) ), x, p ) F 1 F 2 )p) } = q, p Q 1 { R1 R 2 )q) δ 1 δ 2) q, µ t q) ), x, p ) F 1 F 2 )p) } q, p Q 2 { R1 R 2 )q) δ 1 δ 2) q, µ t q) ), x, p ) F 1 F 2 )p) } = LM r 1 )x) LM r 2 )x) = L 1 L 2 )x) Thus, L 1 L 2 is a general fuzzy recognizer language. Theorem Let L be a general fuzzy recognizer language and f : Σ Σ a homomorphism. Then fl) and f 1 L) are general fuzzy recognizer languages. Proof. We prove only 1) 1) Let Mr = Q, Σ, R, δ, T 1, T 2, F ) be a general fuzzy recognizer such that L = LMr ). Now, fmr ) = Q, Σ, R, δ, T 1, T 2, F ) is a general fuzzy recognizer, where δ q, µ t q)), x, p) = {δ q, µ t q)), x, p)}. x:x =fx) Then, flm r ))x ) = {LMr )x) / x : x = fx)}, since f is onto. = { Rq) δ q, µ t 0 q) ), x, p ) F p) )} = = x:x =fx) p, q Q q Q actt0 ), p Q, q Q actt0 ), p Q, Rq) x:x =fx) δ q, µ t 0 q) ), x, p F p) { Rq) δ q, µ t 0 q), x, p ) F p) } 59

10 = LfM r ))x ) Thus, fl) is a general fuzzy recognizer language. 4.3 Properties of General Fuzzy Automaton with Outputs. The aim of this section is to study general fuzzy automaton with outputs. We obtain all its homomorphic images with the help of congruence relations defined on its state set. For a given general fuzzy automaton with outputs, we obtain an equivalent minimal general fuzzy automaton with outputs by the use of the concept of a state equivalence relation. We begin with the definition of general fuzzy automaton with outputs along with extended state transition and output functions Definition An 8-tuple M = Q, Σ, R, Z, ω, δ, T 1, T 2 ) is called a general fuzzy automaton with outputs if q Q act t) and a Σ, we have p Q act t + 1), such that δ q, µ t q)), a, p) > 0 z Z such that ω q, µ t q)), a, z) > 0) The implication precisely states that for any given state q at any time t and input a, whenever there is a transition from q to some state p there is a output z and conversely. The mapping δ and ω can be extended respectively as follows δ q, µ t q)), x, p) = 60

11 0, if x = Λ and q p 1, if x = Λ and q = p δ q, µ t q)), a, p), δ q, µ t q)), a, q ) δ q, µ t q )), x 1, p), q Q actt+1) if x = a Σ if x = ax 1, a Σ, x 1 Σ and ω q, µ t q)), x, y) = 0, either x = Λ and y Λorx Λ and y = Λ 1, if x = y = Λ ω q, µ t q)), a, z), { r Q actt+1) ωq,µ t q)),a,z) δq,µ t q)),a,r) ω r,µ t r)),x 1,y 1 ) One may note that, for u 1, u 2,..., u n Σ if x = a Σ, y = z Z }, if x = ax 1, a Σ, x 1 Σ and y = zy 1, z Z, y 1 Z 1) δ q, µ t q)), u 1 u 2..., u n, p) = { δq, µ t q)), u 1, p 1 ) δ p 1, µ t+1 p 1 )), u 2, p 2 )... δp n 1, µ t+n 1) p n 1 )), u n, p) p 1 Q act t + 1), p 2 Q act t + 2),..., p n 1 Q act t + n 1))} and 2) ω q, µ t q)), u 1 u 2..., u n, z 1 z 2... z m ) = { } ωq,µ t q)),u 1,z 1 ) ω p 1,µ t+1 p 1 )),u 2,z 2 ), if m = n ωp n 1,µ t+n 1) p n 1 )),u n,z n) p i Q actt+i) 0, otherwise 61

12 Remark Note that µ t+1 q) plays role in calculating both δ and ω simultaneously. In each case i.e. in δ and ω ) its values are different. For the case of µ t+1 q) in δ we have µ t+1 q j ) = δq i, µ t q i )), a k, q j ) = T 1 µ t q i ), δq i, a k, q j )), whenever µ t q i ) > 0 and for µ t+1 q) in ω, we have µ t+1 q j ) = ωq i, µ t q i )), a k, v) = T 1 µ t q i ), ωq i, a k, v)), where δq i, a k, q j ) > 0. This problem can be resolved just by denoting them by different symbols namely, µ t+1 δ q) and µ t+1 w q). For use of above notation, we use the same notation µ t+1 q) in both δ and ω. Consider the following general fuzzy automaton with outputs: b, 0.3 / z, 0.7 start 0.6 a, 0.4 q 0 q 1 v, 0.3 c, 0.7 v, 0.6 z, 0.1 b, 0.9 b, 0.1 z, 0.8 v, 0.2 c, 0.7 q 3 v, 0.3 c, 0.8 q 2 For input abc we calculate output as δq 0, µ t 0 q 0 )), a, q 1 ) δq 1, µ t 1 q 1 )), b, q 1 ) δq 1, µ t 2 q 1 )), c, q 2 ) = = 0.3 and ω[q 0, µ t 0 q 0 )), a, v) δq 1, µ t 0 q 1 )), a, q 1 )] [ωq 1, µ t 1 q 1 )), b, z) δq 1, µ t 1 q 1 )), b, q 1 )] [ωq 1, µ t 2 q 1 )), c, v) δq 1, µ t 2 q 1 )), c, q 2 )] = [ ] [ ] [ ] = 0.2 Thus, initially when the input is abc, the next state is q 2 of degree 0.3 and at the same time output is vzv of degree 0.2. Theorem Let M = Q, Σ, R, Z, ω, δ, T 1, T 2 ) be a general fuzzy automaton with outputs. Then for a Σ and z Z, we have 1. δ q, µ t q)), a, p) = δq, µ t+1 q)), a, p) 2. ω q, µ t q)), a, z) = ωq, µ t+1 q)), a, z) 62

13 Proof. 1. Let q Q act t), p Q and a Σ. Then, δ q, µ t q)), a, p) = δ q, µ t q)), Λa, p) = δ q, µ t q)), Λ, r ) δ r, µ t+1 r)), a, p ) r Q actt+1) = δ q, µ t q)), Λ, q ) δ q, µ t+1 q)), a, p ) = δ q, µ t+1 q)), a, p ) 2. Let q Q act t), a Σ and p Q. Then, ω q, µ t q)), a, z) = ω q, µ t q)), Λa, Λz) = ω q, µ t q)), Λ, Λ) δ q, µ t q)), Λ, r ) ω r, µ t+1 r)), a, z )) = r Q actt+1) r Q actt+1) = ω q, µ t+1 q)), a, z ) δ q, µ t q)), Λ, r ) ω r, µ t+1 r)), a, z )) Theorem Let M = Q, Σ, R, Z, ω, δ, T 1, T 2 ) be a general fuzzy automaton with outputs. For all q Q act t), x Σ, y Z, if x = y then ω q, µ t q)), x, y) = 0 Proof. Let q Q act t), x Σ, y Z, x y. Without loss of generality, let x > y. We use induction on y to prove the theorem. If y = 0, then y = Λ and x Λ. By definition of ω, we have ω q, µ t q)), x, y) = 0. Suppose that the theorem is true for all x Σ, y Z such that x > y and y = n 1, n 1. Let y = n. Now, write x = x a, y = y z, where a Σ, z Z, x Σ and 63

14 y Z. Then, ω q, µ t q)), x, y ) = ω q, µ t q)), x a, y z ) = ω q, µ t q)), x, y ) δ q, µ t q)), x, r ) ω r, µ t+1 r)), a, z )) r Q actt+1) = 0 By hypothesis. Let M = Q, Σ, R, Z, ω, δ, T 1, T 2 ) be a general fuzzy automaton with outputs. An equivalence relation θ on M is reflexive, symmetric and transitive binary relation on Q. Clearly, D Q = {q, q) q Q} and U Q = {q, p) q, p Q} are equivalence relations on M. We respectively call them the diagonal and the universal equivalence relations on M. An equivalence relation θ on M is called the congruence relation, if for qθq, we have 1) δ q, µ t q)), a, p ) = δ q, µ t q )), a, p ), pθp pθp 2) ω q, µ t q)), a, z) = ω q, µ t q )), a, z) and µ t q) = µ t q ), a Σ, z Z, q, q Q act t) and p, p Q. The factor general fuzzy automaton of M with respect to a congruence relation θ on M is the general fuzzy automaton with outputs M/θ = Q/θ, Σ, R, Z, wθ, δ θ, T 1, T 2 ). where Q/θ = {[q] θ q Q}, [q] θ = {p Q qθp}, µ t [q] θ ) = µ t q), q Q. δθ [q] θ, µ t [q] θ )), a, [p] θ ) = δ q, µ t q)), a, p ), p θp 64

15 ω θ [q] θ, µ t [q] θ ), a, z) = ω q; µ t q)), a, z), where µ t [q] θ Q/θ act t), q Q act t) and p Q. Theorem Let θ be a congruence relation on a general fuzzy automaton M with outputs. Then for any q, q Q act t), p Q, x Σ, y Z, if qθq, then 1) δ q; µ t q)), x, p ) = δ q ; µ t q )), x, p ) and p θp p θp 2) ω q; µ t q)), x, y) = ω q ; µ t q )), x, y) Proof. 1) Case of x = 1 is trivial. Let x = k + 1. Then write x = ax for a Σ and x Σ. Now, δ q; µ t q)), ax, p ) p θp = p θp = { δ q, µ t q)), a, r ) δ r, µ t+1 r)), x, p )} r Q actt+1) r Q actt+1) { = [r ] θ Q/θ r [r ] θ δ q, µ t q)), a, r ) δ r, µ t+1 r)), x, p )} p θp { δ q, µ t q)), a, r ) δ r, µ t+1 r)), x, p )} p θp Using induction hypothesis, we have = { δ q, µ t q)), a, r ) δ r, µ t+1 r )), x, p )} [r ] θ Q/θ rθr p θp = { δ q, µ t q)), a, r ) δ r, µ t+1 r )), x, p )} [r ] θ Q/θ rθr p θp = { δ q, µ t q )), a, r ) δ r, µ t+1 r )), x, p )} [r ] θ Q/θ rθr p θp = { δ q, µ t q )), a, r ) δ r, µ t+1 r )), x, p )} p θp [r ] θ Q/θ rθr 65

16 = [r ] θ Q/θ r [r ] θ = r Q actt+1) = p θp { r Q actt+1) { δ q, µ t q )), a, r ) δ r, µ t+1 r)), x, p )} p θp δ q, µ t q )), a, r ) δ r, µ t+1 r)), x, p )} p θp { δ q, µ t q )), a, r ) δ r, µ t+1 r)), x, p )} = δ q, µ t q )), ax, p ) p θp 2) We use induction on both x and y simultaneously to prove the result. Proof is trivial for x y. The case of x = y = 1 is obvious. Let x = y = k + 1, for k 1. Let x = ax and y = zy for a Σ, x Σ and z Z, y Z. Then, ω q; µ t q)), ax, zy ) { = ω q; µ t q)), a, z ) δ q; µ t q)), a, r ) r Q actt+1) ω r; µ t+1 r)), x, y )} = ω q; µ t q)), a, z ) { δ q; µ t q)), a, r ) r Q actt+1) ω r; µ t+1 r)), x, y )} = ω q; µ t q)), a, z ) { δ q; µ t q)), a, r ) [r ] θ Q/θ r [r ] θ ω r; µ t+1 r)), x, y )} By induction hypothesis, we have ω q; µ t q)), ax, zy ) = ω q; µ t q)), a, z ) { δ q; µ t q)), a, r ) rθr [r ] θ Q/θ ω r ; µ t+1 r )), x, y )} = ω q; µ t q)), a, z ) { [r ] θ Q/θ ω r ; µ t+1 r )), x, y )} = ω q ; µ t q )), a, z ) { 66 [r ] θ Q/θ rθr δ q; µ t q)), a, r ) rθr δ q ; µ t q )), a, r )

17 ω r ; µ t+1 r )), x, y )} = ω q ; µ t q )), a, z ) [r ] θ Q/θ rθr { δ q ; µ t q )), a, r ) ω r ; µ t+1 r )), x, y )} = ω q ; µ t q )), a, z ) { δ q ; µ t q )), a, r ) [r ] θ Q/θ r [r ] θ ω r; µ t+1 r)), x, y )} = ω q ; µ t q )), a, z ) { δ q ; µ t q )), a, r ) r Q actt+1) ω r; µ t+1 r)), x, y )} { = ω q ; µ t q )), a, z ) δ q ; µ t q )), a, r ) r Q actt+1) ω r; µ t+1 r)), x, y )} = ω q ; µ t q )), ax, zy ) Theorem Let M be a general fuzzy automaton with outputs. If θ is a congruence relation on M, then for any q, p Q act t), x Σ and y Z, we have 1) δθ [q] θ; µ t [q] θ )), x, [p] θ ) = δ q; µ t q)), x, p ) p θp 2) ω θ [q] θ; µ t [q] θ )), x, y) = ω q; µ t q)), x, y). Proof. If x = y = 1, then by definition of δθ and w θ, 1) and 2) hold. Assume that these hold for all words x and y of length k N, k 1. Let x Σ, y Z be such that x = y = k + 1. Let x = ax for a Σ, x Σ and y = zy for z Z, y Z. Then, δ θ [q]θ ; µ t [q] θ )), ax, [p] θ ) = [r] θ Q/θ) actt+1) 67 δθ [q]θ ; µ t [q] θ )), a, [r] θ )

18 )) δθ [r]θ ; µ t+1 [r] θ )), x, [p] θ = δ q, µ t q)), a, r ) = [r] θ Q/θ) actt+1) r θr δ r, µ t+1 r)), x, p )) p θp [r] θ Q/θ) actt+1) r θr δ r, µ t+1 r)), x, p )) p θp δ q, µ t q)), a, r ) By induction hypothesis, we have δ θ ) [q]θ ; µ t [q] θ )), ax, [p] θ = = δ q, µ t q)), a, r ) δ r, µ t+1 r )), x, p )) [r] θ Q/θ) actt+1) r θr p θp = δ q, µ t q)), a, r ) δ r, µ t+1 r )), x, p )) [r] θ Q/θ) actt+1) r θr p θp = δ q, µ t q)), a, r ) δ r, µ t+1 r )), x, p )) p θp [r] θ Q/θ) actt+1) r θr = δ q, µ t q)), a, r ) δ r, µ t+1 r )), x, p )) p θp r Q = δ q, µ t q)), ax, p ) p θp Also, ω θ [q]θ ; µ t [q] θ ), ax, zy ) { = ωθ [q]θ ; µ t [q] θ ), a, z ) ) δθ [q]θ, µ t [q] θ )), a, [p] θ [p] θ Q/θ) actt+1) ω θ [p]θ, µ t+1 [p] θ ), x, y )} = [p] θ Q/θ) actt+1) p θp { δ q; µ t q)), a, p ) ωθ p; µ t+1 p)), x, y )} 68

19 ω q, µ t q)), a, z ) { = δ q; µ t q)), a, p ) ωθ p; µ t+1 p)), x, y )} [p] θ Q/θ) actt+1) p [p] θ ω q, µ t q)), a, z ) By induction hypothesis, we have ω θ [q]θ ; µ t [q] θ ), ax, zy ) = { = δ q; µ t q)), a, p ) ωθ p ; µ t+1 p )), x, y )} [p] θ Q/θ) actt+1) p [p] θ w q, µ t q)), a, z ) { = δ q; µ t q)), a, p ) ωθ p ; µ t+1 p )), x, y )} [p] θ Q/θ) actt+1) p Q ω q, µ t q)), a, z ) = ωθ q; µ t q), ax, zy ) To discuss structural preservation one always need to define homomorphism between the concepts. Here, we prove that the quotient of a general fuzzy automaton with outputs preserve the structure of given general fuzzy automaton with outputs. under the homomorphism between them) Definition Let M i = Q i, Σ, R i, Z, ωi, δi, T 1, T 2 ), i = 1, 2 be general fuzzy automata with outputs. Then a mapping f : Q 1 Q 2 is called a homomorphism, if 1) δ 2 fq 1 ), µ t q 1 )), a, p 2 ) = δ 1 q 1 ; µ t q 1 )), a, p 1 ) and fp 1 )=p 2 2) ω 2 fq 1 ), µ t q 1 )), a, z) = ω 1 q 1 ; µ t q 1 )), a, z), q 1 Q 1act t), p 2 Q 2, a Σ and z Z. A surjective homomorphism is called an epimorphism, and a bijective homomorphism is called an isomorphism. In the case of an isomorphism, we write 69

20 M 1 = M2. If f is injective, then one can easily verify that δ 2 fq), µ t q)), a, fp)) = δ 1 q; µ t q)), a, p), q Q act t), p Q and a Σ. We first establish the canonical homomorphism: Theorem Let θ be a congruence relation on a general fuzzy automaton M with outputs. Then M/θ is homomorphic image of M. Proof. Clearly, the mapping φ that maps q Q into [q] θ Q/θ is surjective. For any q Q act t), [p] θ Q/θ, a Σ and z Z, we have δ θ φq), µ t q) ) ), a, [p] θ = δθ [q]θ, µ t q) ) ), a, [p] θ = δ [q] θ, µ t q) ), a, p ) p θp = [q]θ, µ t q) ), a, p ) φp )=[p] θ δ and ω θ φq), µ t q) ), a, z ) = ω θ [q]θ, µ t q) ), a, z ) = ω q, µ t q) ), a, z ) Hence, M/θ is homomorphic image of M. Definition Let φ : M 1 M 2 be a general fuzzy automaton homomorphism. Then the set kerφ = { q, p) Q 1act t) Q 1act t) φq) = φp) and µ t q) = µ t p) } is called the kernel of φ. Example If φ : M 1 M 2 is homomorphism from general fuzzy automaton M 1 into M 2. Then kerφ is a congruence relation on M 1. Clearly, kerφ is an equivalence relation. Let q, q Q 1act t) be such that q, q ) kerφ. Then for any a Σ, z Z and 70

21 p Q 1. Since, q, q ) kerφ it follows that and p kerφ p δ 1 q, µ t q)), a, p ) = δ 2 φq), µ t q)), a, φp) ) = δ 2 φq ), µ t q )), a, φp) ) = δ 1 q, µ t q )), a, p ) p kerφ p ω 1 q, µ t q)), a, z ) = ω 2 φq), µ t q)), a, z ) Thus, kerφ is a congruence relation on M 1. = ω 2 φq ), µ t q )), a, z ), since q, q ) kerφ. = ω 1 q, µ t q )), a, z ) We now establish the fundamental theorem for general fuzzy automata with outputs. Theorem Let M 2 be homomorphic image of M 1. Then, M 1 /kerφ = M 2 Proof. Define ψ : Q 1 /kerφ Q 2 by ψ [q] kerφ ) = φq). Since [q] kerφ = [q ] kerφ if and only if φq) = φq ), we have ψ is well defined and injective. Since ψ is one one, to prove the homomorphism from M 1 /kerφ to M 2, it is suffices to prove δ 2 ψ[q1 ] kerφ, µ t [q 1 ] kerφ )), a, ψ[q 1] kerφ ))) = δ 1kerφ [q1 ] kerφ, µ t [q 1 ] kerφ ), a, [q 1] kerφ )) for all [q 1 ] kerφ, µ t [q 1 ] kerφ ) Q 1 /kerφ, Q 1 /kerφ) act t)), [q 1] kerφ Q 1 /kerφ, a Σ. Then, δ 2 ψ[q1 ] kerφ, µ t [q 1 ] kerφ )), a, ψ[q 1] kerφ ))) = δ 2 φq1 ), µ t [q 1 ] kerφ ) ), a, φq 1) ) = δ 2 φq1 ), µ t q 1 ) ), a, φq 1) ), sinceµ t [q 1 ] kerφ ) = µ t q 1 ) 71

22 = φq )=φq 1 ) δ 1 q1 ; µ t q 1 )), a, q ) = δ 1kerφ [q1 ] kerφ, µ t [q 1 ] kerφ ), a, [q 1] kerφ )), by definition of FGFA. and ω 2 ψ [q 1 ] kerφ, µ t [q 1 ] kerφ )), a, z )) = ω 2 φq1 ), µ t [q 1 ] kerφ ) ), a, z ) = ω 2 φq1 ), µ t q 1 ) ), a, z ), sinceµ t [q 1 ] kerφ ) = µ t q 1 ) = ω 1 q1, µ t q 1 ) ), a, z ), since φ is a homomorphism = ω 1kerφ [q1 ] kerφ, µ t [q 1 ] kerφ ), a, z )) Therefore, ψ is an isomorphism. Theorem Let θ and τ be congruence relations of a general fuzzy automaton M with outputs such that τ θ. Then the relation θ/τ defined on M/τ) by q/τ, p/τ) θ/τ q, p) θ is a congruence relation and M/τ)θ/τ) = M/θ. Definition Let M i = Q i, Σ, R i, Z, ω i, δ i, T 1, T 2 ) be a general fuzzy automaton with outputs, i = 1, 2. Let q 1 Q 1act t), q 2 Q 2act t). Then 1. q 1 and q 2 are said to be equivalent i.e. q 1 q 2 ) for all x Σ, y Z, ω 1 q 1, µ t q 1 )), x, y) = ω 2 q 2, µ t q 2 )), x, y) 2. For each positive integer k, states q 1 and q 2 are said to be k-equivalent i.e. q 1 k q 2 ) for all x Σ of length less than or equal to k, y Z, ω 1 q 1, µ t q 1 )), x, y) = ω 2 q 2, µ t q 2 )), x, y) 3. M 1 and M 2 are said to be equivalent i.e. M 1 M 2 ) q 1 Q 1act t) q 2 Q 2act t) such that q 1 q 2 and q 2 Q 2act t) q 1 Q 1act t) such that q 2 q 1 72

23 4. For each positive integer k, M 1 and M 2 are said to be k-equivalent i.e. M 1 k M 2 ) q 1 Q 1act t) q 2 Q 2act t) such that q 1 k q 2 and q 2 Q 2act t) q 1 Q 1act t) such that q 2 k q 1 If M 1 = M 2 = M, then both relations and k for any fixed k, are equivalence relations. Theorem Let M = Q, Σ, R, Z, ω, δ, T 1, T 2 ) be a general fuzzy automaton with outputs. Then, i) For all q Q act t), a Σ there exists p Q such that δ q, µ t q)), a, p) > 0 = there exists z Z such that ω q, µ t q)), a, z) > 0. ii) For all q Q act t), a Σ there exists z Z such that ω q, µ t q)), a, z) > 0 = there exists p Q such that δ q, µ t q)), a, p) > 0. iii) For all q Q act t), x Σ there exists p Q such that δ q, µ t q)), x, p) > 0 = there exists y Z such that ω q, µ t q)), x, y) > 0. iv) For all q Q act t), x Σ there exists y Z such that ω q, µ t q)), x, y) > 0 = there exists p Q such that δ q, µ t q)), x, p) > 0. Then i) iii) and ii) iv). Proof. Cases iii) = i) and iv) = ii) are straightforward. Hence, we prove i)= iii) and ii)= iv) only. i) = iii) Suppose for each q Q act t) and x Σ there exists p Q such that δ q, µ t q)), x, p) > 0. We use induction on x to prove the claim. Let x = n. If n = 1, then x = a, a Σ and by assumption claim holds. Suppose that the claim holds for all x Σ such that x = n 1, where n > 1. Let x = x a, where a Σ and x Σ. Then x = n 1. Now, δ q, µ t q)), x, p) > 0 implies that δ q; µ t q)), x, r ) δ r; µ t+1 r)), a, p )) > 0 r Q actt+1) 73

24 Hence, δ q; µ t q)), x, r) δ r; µ t+1 r)), a, p) > 0, for some r Q act t + 1). i.e. δ q; µ t q)), x, r) > 0 and δ r; µ t+1 r)), a, p) > 0, for some r Q act t+1). Then by assumption there exists z Z such that ω r; µ t+1 r)), a, z) > 0 and by induction hypothesis, there exists y Z such that ω q, µ t q)), x, y ) > 0. By taking y = y z one has ω q, µ t q)), x, y) = ω q, µ t q)), x a, y z ) = ω q, µ t q)), x, y ) δ q; µ t q)), x, r ) ω r; µ t+1 r)), a, z )) r Q actt+1) ω q, µ t q)), x, y ) δ q; µ t q)), x, r ) ω r; µ t+1 r)), a, z ) > 0 This proves iii). ii) = iv) Let q Q act t), x Σ and let there exists y Z such that ω q, µ t q)), x, y) > 0. Clearly, x = y. If x = 1, then x = a, y = z, a Σ, z Z and hence by assumption i.e. by ii)), there exists r Q act t + 1) such that δ q; µ t+1 q)), a, r) > 0. Suppose that the theorem holds for all x Σ, y Z such that x = y = n 1, n 1. Let ω q, µ t q)), x, y) > 0 for x = y = n, n 1. Write x = x a, y = y z, a Σ, z Z, x Σ and y Z. Then x = y = n 1. Now, [ ] ω q, µ t q)), x, y ) δ q; µ t q)), x, q ) ω q ; µ t+1 q )), a, z)) > 0 implies that q Q actt+1) ω q, µ t q)), x, y ) > 0 and δ q; µ t q)), x, q ) > 0, ω q ; µ t+1 q )), a, z) > 0, for some q Q act t + 1). By ii), there exists p Q such that δ q ; µ t+1 q )), a, p) > 0. 74

25 Hence, δ q; µ t+1 q)), x, p ) = δ q; µ t+1 q)), x a, p ) = δ q; µ t q)), x, l ) δ l; µ t+1 l)), a, p )) l Q actt+1) > δ q; µ t q)), x, q ) δ q ; µ t+1 q )), a, p ) > 0. Definition Let M = Q, Σ, R, Z, ω, δ, T 1, T 2 ) be a general fuzzy automaton with outputs. Let q, p Q act t). Then i) q is indistinguishable from p, if q p Otherwise q and p are said to be distinguishable ii) M is said to be minimal, if q p = q = p) The following theorem clearly assure the existence of the minimal form for any given general fuzzy automaton with outputs. Theorem Let M = Q, Σ, R, Z, ω, δ, T 1, T 2 ) be a general fuzzy automaton with outputs. Then there exists a minimal general fuzzy automaton with outputs equivalent to M. Proof. Let Q m = {[q] m / q Q}. Set µ q) = µ q ), q q. Define δ m : Q m [0, 1]) Σ Q m [0, 1] by δ m [q], µ t [q])), a, [p]) = {δ s, µ t s)), a, t)} and s q,t p ω m : Q m [0, 1]) Σ Z [o, 1] by ω m [q], µ t [q])), a, z) = {ω s, µ t s)), a, z)}. s q We prove that M m = Q m, Σ, Z, ω m, δ m, T 1, T 2 ) is required minimal general fuzzy automaton with outputs. i) First we prove that M m is a general fuzzy automaton with outputs. Let [q] Q mact t), a Σ. Suppose there exists [p] Q mact t + 1) such that 75

26 δ m [q], µ t [q])), a, [p]) > 0. Then by the definition of M m, there exists s q and t p such that δ s, µ t s)), a, t) > 0. Since M is a general fuzzy automaton with outputs, there exists z Z such that ω s, µ t s)), a, z) > 0. Hence, ω m [q], µ t [q])), a, z) = {ω q, µ t q )), a, z)} > 0. q q On the other hand suppose that [q] Q mact t), a Σ and there exists z Z such that ω m [q], µ t [q])), a, z) > 0. Then, there exists s q such that ω s, µ t s)), a, z) > 0. Since M is a general fuzzy automaton with outputs, there exists t Q mact t + 1) such that δ s, µ t s)), a, t) > 0. Thus, δ m [q], µ t [q])), a, [t]) = {δ q, µ t q )), a, p )} > 0. q q,p t This prove that M m = Q/, Σ, Z, ω m, δ m, T 1, T 2 ) is a general fuzzy automaton with outputs. ii) Secondly, we prove that M m is minimal. Let [q] Q mact t), [p] Q mact t + 1), x Σ and z Z. Now, [q] m [p] ω m [q], µ t [q])), x, y) = ω m [p], µ t [p])), x, y) Then there exists s q and t p such that ω s, µ t s)), x, y) = ω t, µ t t)), x, y) Therefore, s t hence, q p Thus, [q] [p]. This implies that M m is minimal. iii) Finally, we prove that M m M Let q Q act t), a Σ and y Z. Then, ω q, µ t q)), x, y) = [ω s, µ t s)), x, y)] = ωm [q], µ t [q])), x, y) s q Thus, q [q]. Hence, M m M. Theorem Every homomorphic image of a general fuzzy automaton M with outputs is equivalent to M Proof. M = Q, Σ, R, Z, ω, δ, T 1, T 2 ) be a general fuzzy automaton with outputs. Suppose φ : M M be onto homomorphism, where M = Q, Σ, R, Z, ω, δ, T 1, T 2). We prove that for any q Q actt), q Q act t) such that for all x Σ, z Z, we have ω q, µ t q )), x, y) = ω q, µ t q)), x, y) 76

27 and conversely for any q Q act t) q Q actt) such that for all x Σ, y Z, we have ω q, µ t q)), x, y) = ω q, µ t q )), x, y) We prove the half part by induction on x = y. Clearly, if x = y = 1, then x = a, a Σ, y = z, z Z, Let q Q actt), Since φ is onto q Q act t) such that φq) = q. Clearly, φ is homomorphism gives ω φq), µ t q )), a, z) = ω q, µ t q)), a, z) i.e. ω q, µ t q )), a, z) = ω q, µ t q)), a, z) We assume that the claim holds for x = y n 1, n > 1. Let x Σ, y Z such that x = y = n. Let x = ax 1, y = zy 1, a Σ, z Z. Then, ω φq), µ t q) ) ), ax 1, zy 1 = ω φq), µ t q) ), a, z ) δ φq), µ t q) ), a, p ) ω p, µ t+1 p) ) )), x 1, y 1 p Q act t+1) = ω φq), µ t q) ), a, z ) δ φq), µ t q) ), a, φq ) ) ω φq ), µ t q ) ) )), x 1, y 1 q Q act t+1) = ω φq), µ t q) ), a, z ) δ q, µ t q) ), a, q ) ω q, µ t q ) ) )), x 1, y 1 q Q actt+1) = ω q, µ t q) ) ), ax 1, zy 1. By Theorem 4.3.6, ω q, µ t q)), x, y) = ωθ [q] θ, µ t [q] θ )), x, y), for a congruence θ on M. According to homomorphism theorem the function ψ : M 1 /kerφ M 2 defined by ψ[q 1 ] kerφ ) = φq 1 ) is an isomorphism. By the already proved claims for isomorphism and factor general fuzzy automaton with outputs, we get ω φq 1 ), µ t q 1 ) ), x, y ) = ω ψ[q 1 ] kerφ ), µ t [q 1 ] kerφ ) ), x, y ) = ωkerφ [q1 ] kerφ ), µ t [q 1 ] kerφ ) ), x, y ) 77

28 = ω q 1 ), µ t q 1 ) ), x, y ) This clearly implies that M and M are equivalent. 4.4 Non-deterministic General Fuzzy Recognizer The aim of this section is to introduce concepts of deterministic general fuzzy recognizer, non-deterministic general fuzzy recognizer and non-deterministic general fuzzy recognizer with Λ moves. As it is expected, we prove that they all are equivalent, in the sense of their language acceptance. For any family λ i, i I, of elements of [0, 1], we write; i Iλ i or {λ i i I} for the supremum of {λ i i I} and i Iλ i or {λ i i I} for the infimum. In particular, if I is finite, then λ i and i are the greatest element and the least i I i Iλ element of {λ i i I} respectively. For any A F X), i.e. A : X [0, 1], the height of A is defined as heighta) = Ax) and for λ [0, 1] the scalar product, λ A, of λ and A is defined by λ A)x) = λ Ax), for every x X. Note that λ A is a fuzzy subset of X. Recall that M = Q, Σ, R, δ, T 1, T 2 ) is a general fuzzy automaton without outputs, then δ q, µ t q)), u, p) = 1, if q = p and u = Λ 0, if q p and u = Λ x X δ q, µ t q)), u, p), δ q, µ t q)), u 1, q ) δ q, µ t q )), v, p), q Q actt+1) if u Σ if u = u 1 v, u 1 Σ, v 1 Σ and recursively, δ q, µ t q)),u 1 u 2, p) = {δq, µ t0 q)), u 1, p 1 ) δp 1, µ t1 q)), u 2, p 1 ) δp n 1, µ tn 1 p n 1 )), u n, p) p i Q act t i )} 78

29 We now introduce deterministic, non-deterministic and non-deterministic with Λ moves general fuzzy recognizers and prove that they are equivalent in the sense of language accepted by them. Remark In classical automata theory if there are at least two next states for a given state and an input i.e. δq, a) = A; A 2), then the automaton is termed as non-deterministic. Otherwise it is called deterministic. In case of general) fuzzy automata theory it is somewhat confusing to distinguish these two types of fuzzy automata. This is due to the fact that one may treat general) fuzzy automaton as deterministic or non-deterministic according way of treating it as a generalization of classical automaton. To be precise, if one define δ : Q Σ Q from δ : Q Σ Q [0, 1] as δq, a) = q if and only if δq, a, q ) = 1, then the fuzzy automaton is generalization of deterministic fuzzy automaton under the restriction that δq, a, q ) = 1, for exactly one q Q only ), but if δq, a) = q if and only if δq, a, q ) > 0, then the fuzzy automaton is generalization of non-deterministic fuzzy automaton. In the rest of this chapter, we try to distinguish between these two classes of general fuzzy automaton in a different way. To make it possible, we use aliter but equivalent) way of defining general fuzzy automaton and the language accepted by it. Definition A deterministic general fuzzy recognizer is a seven-tuple M = Q, Σ, R, δ, T 1, T 2, F ), where i) Q is a finite set of states, Q = {q 1, q 2,..., q n } ii) Σ is a finite non-empty set of input symbols, Σ = {a 1, a 2,..., a m } iii) R : Q [0, 1] is called the set of initial fuzzy states and iv) δ : Q [0, 1]) Σ F Q) is the fuzzy state transition function, v) T 1 : [0, 1] [0, 1] [0, 1] is the membership assignment function, 79

30 vi) T 2 : [0, 1] [0, 1] is called the multi-membership resolution function. vii) F : Q [0, 1] is called the set of final fuzzy states For any q Q act t), p Q act t+1) and a Σ,we can interpret δq; µ t q)), a)p) as the degree to which the automaton in state q at time t and input a enters into state p at time t + 1 Definition Let M = Q, Σ, R, δ, T 1, T 2, F ) be a deterministic general fuzzy recognizer, where µ t+1 p) = δ q, µ t q)), a) p) = T 1 = µ t q), δq, a)p)). Here T 1 can be defined as min, max etc. of µ t and δ. 1) The extended fuzzy transition function from Q [0, 1]) Σ to F Q), denoted by the notation δ is defined inductively as : δ q, µ t q)), Λ) = {q, µ t q))}, δ q, µ t q)), a) = δq, µ t q)), a) and δ q, µ t q)), ax) = δq, µ t q)), a)p) δ p, µ t+1 p)), x)), q x Σ and a Σ. p Q Q act t), 2) The language LM) accepted by M is a fuzzy subset of Σ with the membership function defined by LM)x) = Rq) height δ q, µ t 0q)), x) F )), x Σ. q Q actt 0 ) The membership LM)x) is the degree to which x is accepted by M. We have, F Q) is the set of all fuzzy sets of Q We now use P F Q)) for the crisp power set of fuzzy set of Q. This above aliter way of defining general fuzzy recognizer enables us to distinguish non-deterministic general fuzzy recognizer as : Definition A non-deterministic general fuzzy recognizer is a seven-tuple M = Q, Σ, R, δ, T 1, T 2, F ), where i) Q is a finite set of states, Q = {q 1, q 2,..., q n } ii) Σ is a finite non-empty set of input symbols, Σ = {a 1, a 2,..., a m } 80

31 iii) R : Q [0, 1] is called the set of initial fuzzy states and iv) δ : Q [0, 1]) Σ P F Q)) is the fuzzy transition function, v) T 1 : [0, 1] [0, 1] [0, 1] is the membership assignment function, vi) T 2 : [0, 1] [0, 1] is called the multi-membership resolution function. vii) F : Q [0, 1] is called the set of final fuzzy states In contrast with definition 4.4.2, the unique difference between a non-deterministic general fuzzy recognizer and deterministic general fuzzy recognizer lies in the fuzzy transition function. For each a Σ, every state q in a deterministic general fuzzy recognizer at time t has exactly one transition, labeled bya where as in non-deterministic general fuzzy recognizer there may be more than one transition. In other words, a deterministic general fuzzy recognizer does not allow non-deterministic choices between transitions involving the same input symbol. Clearly, deterministic general fuzzy recognizer arise as a special case of non-deterministic general fuzzy recognizer by identifying each fuzzy transition function δ : Q [0, 1]) Σ F Q) as δ : Q [0, 1]) Σ P F Q)), where δ q, µ t q)), a) = δq, µ t q)), a). We now explain the role of T 1 and T 2 in case of non-deterministic general fuzzy recognizer: µ t+1 p) = { δ q, µ t q) ), a ) p) } = { T 1 µ t q), δq, a)ap) A δq, µ t q)), a) ) and Ap) > 0 }. Then µ t+1 p) has many degrees for activeness at time t However, the unified activeness degree at time t + 1 will be given by the formula µ t+1 p) = T 2 T1 µ t q), δq, a)ap) A δq, µ t q)), a) ) and Ap) > 0 ) 81

32 Example Consider a non-deterministic general fuzzy recognizer M r = Q, Σ, R, δ, T 1, T 2, F ) as given in the following tables: Table 1 q 1 a b c { } { q 0, 0.9 } φ q 1 q 1 q 3 { } { } φ q 1 { } { q 2, 0.3 q 1 q 1 q 2 q 3 φ φ Table 2 } { 0.5 q 3, 0.4 q 1 q 2 } { 0.8,, 0.7 } q 3 q 2 { } 0.7 q 1 t 0 t 1 t 2 Λ a b c Λ a b c 0.6 q q q q 1 T 2 0.4, 0.1) 0.4 T 2 0.1, 0.3, 0.1) T 2 0.4, 0.1) q 1 q 1 q 1 q q q q q q 3 82

33 t 3 Λ a b c T 2 0.4, 0.1) T 2 0.4, 0.1) 0.6) T 2 0.4, 0.1) 0.1) T 2 q 2 q 1 q 1 [ T2 0.4,0.1) 0.5, T 2 0.4,0.1) 0.8 T 2 0.4, 0.1) 0.3) q 2 T 2 0.4, 0.1) 0.4) q 1 q 3 ] T 2 0.4, 0.1) 0.7) q 2 Definition Let M = Q, Σ, R, δ, T 1, T 2, F ) be a non-deterministic general fuzzy recognizer. 1) The extended fuzzy transition function from Q [0, 1]) Σ to P F Q)), denoted by the notation δ is defined inductively as : δ q, µ t q)), Λ) = q, µ t q)), δ q, µ t q)), a) = δq, µ t q)), a) and δ q, µ t q)), ax) = {Ap) δ p, µ t+1 p)), x) p Q, A δq, µ t q)), a)}, where α B 1, B 2,, B n = α B 1, α B 2,, α B n, q Q act t), x Σ and a Σ. 2) The language LM) accepted by M is a fuzzy subset of Σ with the membership function defined by LM)x) = {Rq) heighta F ) A δ q, µ t 0q)), x)}, x Σ. q Q actt 0 ) The membership LM)x) is the degree to which x is accepted by M. Definition A fuzzy language L over Σ is said to be a non-deterministic general fuzzy recognizer language if there is a non-deterministic general fuzzy recognizer M = Q, Σ, R, δ, T 1, T 2, F ) such that LM) = L. Lemma Let M = Q, Σ, R, δ, T 1, T 2, F ) be a non-deterministic general fuzzy recognizer. Then LM)x) = {Rq) heighta F ) A δ q, µ t 0q)), x)} q Q actt 0 ) 83

34 = q Q actt 0 ) [ ) )] Rq) height A F, x Σ. A δ q,µ t 0 q)),x) Proof. For any given x Σ, assume that δq, µ t q)), x) = p i, µ t ip i )) i I, i 1. Then we need to verify that q Q actt 0 ) Consider q Q actt 0 ) {Rq) heighta F ) A δ q, µ t 0q)), x) } = q Q actt 0 ) Rq) height {Rq) heighta F ) A δ q, µ t 0q)), x) } = q Q actt 0 ) = q Q actt 0 ) = q Q actt 0 ) = q Q actt 0 ) = q Q actt 0 ) A δ q,µ t 0 q)),x) A F Rq) A F ) A δ q,µ t 0 q)),x) Rq) Ap) F p)) A δ q,µ t 0 q)),x),p Q Rq) Ap) F p) p Q A δ q,µ t 0 q)),x) Rq) A p) F p) p Q A δ q,µ t 0 q)),x) Rq) height F A δ q,µ t 0 q)),x) A The aliter way of defining general fuzzy recognizer in Definition not only allows us to distinguish deterministic and non-deterministic general fuzzy recognizer, but also to distinguish non-deterministic general fuzzy recognizer with moves Definition ). 84

35 Theorem For any non-deterministic general fuzzy recognizer M, there exists a deterministic general fuzzy recognizer M such that LM) = LM ). Proof. Let M = Q, Σ, R, δ, T 1, T 2, F ). We construct a deterministic general fuzzy recognizer M = Q, Σ, R, δ, T 1, T 2, F ) as follows : All the components are same as those in M except the fuzzy transition function, which is δ : Q [0, 1]) Σ F Q ) defined as A, if δq, µ t q)), a) φ δ q, µ t q)), a) = x δq,µ t q)),a) φ, otherwise for any q Q actt) and a Σ. This construction gives rise to a deterministic general fuzzy recognizer M. It remains to check that LM) = LM ). By Lemma 4.4.8, [ for any given x Σ, we have ) )] LM)x) = Rq) height F A δ q,µ t 0 q)),x) A On the other hand, it follows from the Definition 4.4.3, that for any x Σ, LM )x) = R q) height δ q, µ t 0q)), x) F )) q Q act t 0) = q Q actt 0 ) Rq) height δ q, µ t 0q)), x) F )) Therefore, to prove LM) = LM ), it is sufficient to prove that δ q, µ t q)), x) = A, for any q Q actt) and a Σ. 4.1) A δq,µ t q)),x) The basis step is for string of length 0, namely x = Λ. In this case, it follows from the definition that for any q Q act t) δq, µ t q)), Λ) = q, µ t q)) and thus δ q, µ t q)), Λ) = A. A δq,µ t q)),λ) 85

36 Therefore, the basis step is true. The induction hypothesis is that the result 4.1) holds for all strings x with length n. We now show that for any a Σ δ q, µ t q)), ax) = A 4.2) A δq,µ t q)),ax) Be definition and induction hypothesis, we have δ q, µ t q)), ax) = p Q = p Q = p Q = p Q = p Q = p Q = δ q, µ t q)), a)p) δ p, µ t+1 p)), x) ) A p) δ p, µ t+1 p)), x) A δq,µ t q)),a) A δq,µ t q)),a) A δq,µ t q)),a) A δq,µ t q)),a) A δq,µ t q)),a) Ap) δ p, µ t+1 p)), x) Ap) δ p, µ t+1 p)), x) [Ap) B] B δ p,µ t+1 p)),x) B δ p,µ t+1 p)),x) A δq,µ t q)),a) p Q B δ p,µ t+1 p)),x) Ap) B) Ap) B) = { Ap) B) A δq, µ t q)), a), p Q, B δ p, µ t+1 p)), x) } = δq, µ t q)), ax) = A A δq,µ t q)),ax) Hence equation 4.2) holds. Definition A non-deterministic general fuzzy recognizer with Λ- moves M = Q, Σ, R, δ, T 1, T 2, F ) is a 7- tuple, where all the components are same as that in definition except the fuzzy transition function δ which is δ : Q [0, 1]) Σ Λ) P F Q)) i.e. allows Λ moves in transition. 86

37 We define Λ closure for a state q at any time t and a subset A Q as follows: Denote δ q, µ t q)), Λ) = δ q, µ t q)), Λ) and δ q, µ t q)), ΛΛ n ) = {Ap) δ p, µ t+1 p)), Λ n ) p Q, A δq, µ t q)), Λ)}, for all q Q act t), p Q act t + 1) and n N. Then, for any q Q, we define Λ closure of q Q at time t as t Λ q) = {{δ q, µ t q)), Λ)}, {δ q, µ t q)), Λ 2 )},, {δ q, µ t q)), Λ n )}} For any A F Q), the Λ closure of A is defined as t Λ A) = {A, {Aq) B q Q, B t Λ q)}} Here, role of T 1 and T 2 is analogous to that of in non-deterministic general fuzzy recognizer. Definition Let M be a non-deterministic general fuzzy recognizer with Λ moves. 1 ) The extended fuzzy transition function δ from Q [0, 1]) Σ to P F Q)) is defined as : δ q, µ t q)), Λ) = t Λ q) and δ q, µ t q)), ax) = { ) t Λ Ap) δ p, µ t+1 p)), x) for all q Q act t), p Q act t + 1), x Σ and a Σ. } A δq, µ t q)), a), p Q, 2 ) The language LM) accepted by M is a fuzzy subset of Σ with the membership function defined by LM)x) = ) Rq) heighta F ) A δq, µ t 0q)), x), q Q, for all x Σ. The membership LM)x) is the degree to which x is accepted by M. Note that, in general, δq, µ t 0q)), a) δq, µ t q)), a), since a = aλ n, n 1. In a same way it is not necessary that δq, µ t 0q)), Λ) δq, µ t q)), Λ). Lemma Let M = Q, Σ, R, δ, T 1, T 2, F ) be a non-deterministic general fuzzy recognizer with Λ moves. Then 87

38 LM)x) = { } Rq) heighta F ) A δq, µ t 0q)), x), q Q = Rq) height F q Q, x Σ Proof. Similar to that of Lemma A δ q,µ t 0 q)),x) A Lemma Let M = Q, Σ, R, δ, T 1, T 2, F ) be a non-deterministic general fuzzy recognizer with Λ moves. Then δ q, µ t q)), ax ) = { t Λ Ap) δ p, µ t+1 p)), x )) } A δq, µ t q)), a), p Q { = Ap) δ p, µ t+1 p)), x ) } A δq, µ t q)), a), p Q 4.3) for all q Q act t), p Q act t + 1), x Σ and a Σ Proof. To prove equation 4.3), it is sufficient to verify that Ap) δ p, µ t+1 p)), x )) Let α p Q A δq,µ t q)),a) = p Q t Λ p Q A δq,µ t q)),a) t Λ A δ q,µ t q)),a) and A 1 δ p, µ t+1 p )), a) such that Ap) δ p, µ t+1 p)), x )) 4.4) ) Ap) δ p, µ t+1 p)), x) Then there exists p Q α t Λ A 1 p ) δ p, µ t+1 p )), x )) {{ = A 1 p ) δ p, µ t+1 p )), x )}, {[ A 1 p ) δ p, µ t+1 p )), x ) ] q ) }} B q Q, B t Λq ) 88

39 Therefore, [ ] α = A 1 p ) δ p, µ t+1 p )), x) or α = A 1 p ) δ p, µ t+1 p )), x) q ) B for some q Q and B δ q µ t q )), Λ k) t Λ q ). For the first case, it follows immediately that α belongs to the right side of equation 4). For the second case, take A = A 1 and p = p. Noting that δ p, µ t+1 p )), x) q ) B δ p, µ t+1 p)), x). we also find that α belongs to the right side of equation 4). Consequently, p Q A δq,µ t q)),a) t Λ Ap) δ p, µ t+1 p)), x )) p Q Ap) δ p, µ t+1 p)), x )) A δ q,µ t q)),a) Conversely, suppose that α belongs to the right side of equation 4). Then there are A 1 δq, µ t q)), a), p Q such that α = A 1 p ) δ p, µ t+1 p )), x). Because δ p, µ t+1 p )), x ) = r Q A δp,µ t+1 p )),x) t Λ Ar) δ r, µ t+2 r)), Λ )), there exists r Q and A δ p, µ t+1 p )), x) such that δ p, µ t+1 p )), x ) t Λ Ar) δ r, µ t+2 r)), Λ )) {{ = Ar) δ r, µ t+2 r)), Λ )}, {[ Ar) δ r, µ t+2 r)), Λ ) ] }} q ) B q Q, B t Λq ) We thus get that either δ p, µ t+1 p )), x) = Ar) [ ] δ r, µ t+2 r)), Λ) or = Ar) δ r, µ t+2 r)), Λ) q ) B For some q Q and B δ q, µ t+3 q )), Λ k) t Λ q ). For both of the cases, take A = A 1 p ) C, p = r and δ p, µ t+1 p)), Λ) = δ r, µ t+1 r)), Λ) 89

40 Since A = A 1 p ) C δ q, µ t+1 q)), Λ), we see that α belongs to the left side of equation 4) in either the case. Hence, equation 4) holds. We now prove that the non-deterministic general fuzzy recognizer with Λ moves and non-deterministic general fuzzy recognizer withoutλ moves accept the same class of fuzzy languages. Theorem For any non-deterministic general fuzzy recognizer M with Λ moves, there exists a non-deterministic general fuzzy recognizer M such that LM) = LM ). Proof. Let M = Q, Σ, R, δ, T 1, T 2, F ). WE construct a non-deterministic general fuzzy recognizer M as follows: Q = Q, Σ = Σ, T 1 = T 1, T 2 = T 2, R = R and δ : Q [0, 1]) Σ P F Q )) and F are defined [ by δ q, ) µ t q)), ] a) = δ q, µ t q)), a), F q) = height F, for any q Q actt), a Σ, where δ is the A t Λ q) A extended fuzzy transition function introduced in definition Then, M is a non-deterministic general fuzzy recognizer without Λ moves. We show that LM) = LM ). For x = Λ, by Lemma , we get LM)Λ) = Rq) height = Rq) height A δ q,µ t 0 q)),λ) A A Delta t Λ q) A F F On the other hand, it follows from Lemma and the construction of M that LM )Λ) = R q) height 90 A δ q,µ t 0 q)),λ) A F

41 = F q 0 ) = Rq) height A Delta t Λ q) A F Whence, LM)Λ) = LM )Λ). For the induction step, suppose that LM)x) = LM )x) for every x Σ with length less than or equal to n. Then it follows from Lemmas and that z Q act A δ q,µ t 0 q)),x) [Rq) Az) F z)] = [Rq) Az) By) F y)] z Q act A δ q,µ t 0 q)),x) y Q 4.5) To consider ax, we first prove the following Claim : For any q Q act t), x Σ with x Λ, δ q, µ t q)), x) = δ q, µ t q)), x). Again, we use induction on the length of x. For x Σ, the claim follows directly from the definition of δ. The induction hypothesis is that δ q, µ t q)), x) = δ q, µ t q)), x) for all strings x with length less than or equal to n. We now, prove the same for strings of the form ax. By definition 4.4.6, we have δ q, µ t q)), ax) = {Ap) δ p, µ t+1 p)), x) p Q, A δ q, µ t q)), a)} and by Lemma ref { we see that } δq, µ t q)), ax) = Ap) δp, µ t+1 p)), x) p Q, A δq, µ t q)), a) Therefore, we obtain by the basis step and the induction hypothesis that δ q, µ t q)), ax) = δ q, µ t q)), ax). This proves the claim. Now, we are ready to verify that LM)ax) = LM )ax). By Lemmas and and by equation 4.5), one has 91

42 LM )ax) = q Q actt) = q Q actt) = q Q actt) = [R q) Az) F z)] A δ q,µ t 0 q)),ax) Rq) Az) By) F z) A δ q,µ t 0 q)),ax) y Q B t Λ z) [Rq) By) F y)] A δ q,µ t 0 q)),ax) y Q B t Λ z) [ Rq) Ap) δ p, µ t+1 p)), x) q Q actt) A δ q,µ t 0 q)),a) p Q y Q B t Λ z) By) F y)] = q Q actt) A δ q,µ t 0 q)),a) p Q = q Q actt) A δ q,µ t 0 q)),a) p Q = q Q actt) = LM)ax) [ Rq) Ap) δ p, µ t+1 p)), x) F z) ] [ ] Rq) Ap) δp, µ t+1 p)), x) F z) A δ q,µ t 0 q)),ax) [Rq) Az) F z)] This completes the proof. Thus, we have Corollary Let L be a fuzzy language. Then the following statements are equivalent i) There exists a deterministic general fuzzy recognizer M such that LM) = L ii) There exists a non-deterministic general fuzzy recognizer M such that LM) = L 92

43 iii) There exists a non-deterministic general fuzzy recognizer M with Λ moves such that LM) = L. 93

44 94