Closure Properties of Intuitionistic Fuzzy Finite

Automata with Unique Membership Transitions on

an Input Symbol

Jency Priya.K, Jeny Jordon.A, Telesphor Lakra and Rajaretnam.T

Department of Mathematics, St.Joseph’s College(Autonomous)

Tiruchirappalli,TamilNadu,India

Email:jencypriya9@gmail.com

Abstract—An intuitionistic fuzzy finite state automaton assignsa membership and nonmembership values in which there is aunique membership transition on an input symbol (IFA-UM) isconsidered. It is proved that the fuzzy behaviors of IFA-UM isclosed under union, product, intersection and reversal.

Index Terms—Intuitionistic fuzzy sets, Intuitionistic fuzzy lan-guages, Intuitionistic fuzzy finite automaton.

I. INTRODUCTION

“Zadeh [21] was the first to propose the theory of fuzzy

sets, as an effective generalization of classical sets, has been

widely used in dealing with problems with imprecision and

uncertainty.” Fuzzy set theory has become more and more

mature in many fields such as fuzzy relation, fuzzy logic, fuzzy

decision-making, fuzzy classification, fuzzy pattern recogni-

tion, fuzzy control and fuzzy optimization. The concept of

fuzzy automaton in the late 1960s was presented by Malik,

Mordeson, Sen, Chowdhry, Wee and Fu as in [12], [17], [19].

“Lee, Zadeh, Thomason and Marinos [9], [18], [22] originated

the research on fuzzy languages accepted by fuzzy finite-

state machines in the early 1970s.” Also, the fuzzy finite

automaton can be applied in many areas such as learning

systems, the model computing with words, pattern recognition,

lattice-valued fuzzy finite automaton and data base theory by

Li, Shi, Pedryez and Ying as in [10], [11], [20].

Finite state automaton, deterministic finite state automaton,

nondeterministic finite state automaton and regular expression

were introduced by Hopcroft and Ullman as in [13]. The

usual fuzzy finite state automaton can have more than one

transition with a membership value on an input symbol. So,

the uniqueness in the membership transition are introduced,

to reduce the number of transitions to at most one transition

where the fuzzy behavior need not be the same, as in [16].

However, it only acts as a deterministic fuzzy recognizer, so

to retain the same fuzzy behavior a condition is incorporated

that the membership function has a unique transition on an

input symbol, as in [14].

Intuitionistic fuzzy sets (IFS) introduced in 1983 are general-

ization of fuzzy sets, in which membership and nonmember-

ship values for every elements are defined by Atanassov, as in

[1]- [5]. “Jun [6]- [8] presented the concept of intuitionistic

fuzzy finite state machines (IFFSM) as a generalization of

fuzzy finite state machines using the notions of IFSs and

fuzzy finite automaton.” The notions of intuitionistic fuzzy

recognizer, complete accessible intuitionistic fuzzy recognizer,

intuitionistic fuzzy finite automaton, deterministic intuitionis-

tic fuzzy finite automaton and intuitionistic fuzzy languages

are introduced by Zhang and Li, as in [23]. “Samuel Eilenburg,

[15] introduced the notion of an automaton and of a set

recognized by an automaton.”

In this paper,the authors consider some of the closure proper-

ties of the fuzzy behaviors of IFA-UMs such as union, product,

intersection and reversal.

II. BASIC DEFINITION

Definition 1: Given a nonempty set Σ Intuitionistic fuzzy

sets (IFS) in Σ is an object having the form A ={(x, μA(x), νA(x)) | x ∈ Σ}, where the μA : Σ → [0, 1]and νA : Σ → [0, 1] denote the degree of membership and the

degree of nonmembership of each element x ∈ Σ to the set A

respectively, and 0 ≤ μA(x)+νA(x) ≤ 1 for each x ∈ Σ. For

the sake of simplicity, use the notation A = (μA, νA) instead

of A = {(x, μA(x), νA(x)) | x ∈ Σ}.

Definition 2: Intuitionistic fuzzy finite automata with

unique transition on an input symbol is an ordered 5-tuple

(IFA-UM) A = (Q,Σ, A, i, f), where

(i) Q is a finite non-empty set of states.

(ii) Σ is a finite nonempty set of input symbols.

(iii) A = (μA, νA) is an intuitionistic fuzzy subset of Q ×Σ×Q.

(a) the fuzzy subset μA : Q × Σ × Q → [0, 1] denotes

the degree of membership and nonmemebership such

that

μA(p, a, q) = μA(p, a, q′) for some q, q′ ∈ Q then

q = q′.

(b) νA : Q × Σ × Q → [0, 1] denotes the degree of

nonmembership is a fuzzy subset of Q.

(iv) i = (iμA, iνA) is an intuitionistic fuzzy subset of Q

i.e. iμA: Q → [0, 1] and iνA : Q → [0, 1] called the

intuitionistic fuzzy initial state.

(v) f = (fμA, fνA) is an intuitionistic fuzzy subset of Q

i.e. fμA: Q → [0, 1] and fνA : Q → [0, 1] called the

intuitionistic fuzzy subset of final states.

Definition 3: Let A = (Q,Σ, A, i, f) be an IFA-UM. Then

the fuzzy behavior of IFA-UM is LA = (LμA, LνA

).

2014 World Congress on Computing and Communication Technologies

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DOI 10.1109/WCCCT.2014.87

142

2014 World Congress on Computing and Communication Technologies

978-1-4799-2876-7/14 $31.00 © 2014 IEEE

DOI 10.1109/WCCCT.2014.87

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2014 World Congress on Computing and Communication Technologies

978-1-4799-2877-4/14 $31.00 © 2014 IEEE

DOI 10.1109/WCCCT.2014.87

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2014 World Congress on Computing and Communication Technologies

978-1-4799-2877-4/14 $31.00 © 2014 IEEE

DOI 10.1109/WCCCT.2014.87

142

Definition 4: Let A = (Q,Σ, A, i, f) be an IFA-UM.

Define an IFSA∗ = (μ∗A, ν

∗A) in Q × Σ∗ × Q as follows:

∀p, q ∈ Q, x ∈ Σ∗, a ∈ Σ.

μ∗A(q, λ, p) =

{1, if p = q

0, if p �= q

ν∗A(q, λ, p) =

{0, if p = q

1, if p �= q

μ∗A(q, xa, p) =

∨{μ∗

A(q, x, r) ∧ μA(r, a, p)|r ∈ Q}

ν∗A(q, xa, p) =∧

{ν∗A(q, x, r) ∨ νA(r, a, p)|r ∈ Q}

Definition 5: Let A = (Q,Σ, A, i, f) be an IFA-UM and

x ∈ Σ∗. Then x is recognized by A if∨{iμA

(p) ∧μ∗A(p, x, q) ∧ fμA

(q) | p, q ∈ Q} > 0 and∧{iνA(p) ∨

ν∗A(p, x, q) ∨ fνA(q) | p, q ∈ Q} < 1.

III. CLOSURE PROPERTIES

Union

Theorem 1: If A and B are two IFA-UM’s with behaviors

LA and LB respectively, then (LμA∪LμB

, LνA∪LνB

) is a

behavior of an IFA-UM.

Proof: The union of A and B (A ∪ B) is the IFA-UM

defined by C = (QC ,Σ, C, iC , fC) (assume that QA ∩QB =φ) where QC = QA ∪QB ,

(i) C = (μC , νC) is an IFS, μC : QC × Σ × QC → [0, 1]and νC : QC × Σ×QC → [0, 1] are defined by

μC(p, a, q) =

⎧⎪⎨⎪⎩μA(p, a, q), if p, q ∈ QA

μB(p, a, q), if p, q ∈ QB

0 , otherwise

and

νC(p, a, q) =

⎧⎪⎨⎪⎩νA(p, a, q), if p, q ∈ QA

νB(p, a, q), if p, q ∈ QB

0 , otherwise

(ii) iC = (iμC, iνC ) is an IFS, iμC

: QC → [0, 1]and iνC : QC → [0, 1] are defined by

iμC(p) =

{iμA

(p), if p ∈ QA

iμB(p), if p ∈ QB

andiνC (p) =

{iνA(p), if p ∈ QA

iνB (p), if p ∈ QB

(iii) fC = (fμC, fνC ) is an IFS, fμC

: QC → [0, 1]and fνC : QC → [0, 1] are defined by

fμC(p) =

{fμA

(p) , if p ∈ QA

fμB(p) , if p ∈ QB

and fνC (p) =

{fνA(p) , if p ∈ QA

fνB (p) , if p ∈ QB

From the definition of C , we have for all x ∈ Σ∗,

μ∗C(p, x, q) =

⎧⎪⎨⎪⎩μ∗A(p, x, q) , if p, q ∈ QA

μ∗B(p, x, q) , if p, q ∈ QB

0 , otherwise

andν∗C(p, x, q) =

⎧⎪⎨⎪⎩ν∗A(p, x, q) , if p, q ∈ QA

ν∗B(p, x, q) , if p, q ∈ QB

0 , otherwise

Let x ∈ Σ∗

LμC(x) =

∨{{iμC

(p) ∧ μ∗C(p, x, q)

∧ fμC(q) | q ∈ QC} | p ∈ QC

}=

(∨{{iμA

(p) ∧ μ∗A(p, x, q)∧

fμA(q) | q ∈ QA} | p ∈ QA

})∨(∨{

{iμB(p) ∧ μ∗

B(p, x, q)∧

fμB(q) | q ∈ QB} | p ∈ QB

})= LμA

(x) ∨ LμB(x)

∴ LμC= LμA

∪ LμB

and

LνC(x) =

∧{{iνC (p) ∨ ν∗C(p, x, q)∨

fνC (q) | q ∈ QC} | p ∈ QC

}=

(∧{{iνA(p) ∨ ν∗A(p, x, q)∨

fνA(q) | q ∈ QA} | p ∈ QA

})∨(∧{

{iνB(p) ∨ ν∗B(p, x, q)∨

fνB (q) | q ∈ QB} | p ∈ QB

})= LνA

(x) ∨ LνB(x)

∴ LνC= LνA

∪ LνB

Thus, the behaviors of IFA-UM is closed under union

Product

Theorem 2: If A and B are two IFA-UM’s with behaviors

LA and LB respectively, then (LμA∧LμB

, LνA∨LνB

) is a

behavior of an IFA-UM.

Proof: The product of A and B(A ×B) is the IFA-UM

defined by C = (QC ,Σ, C, iC , fC) (assume that QA ∩QB =φ) where QC = QA ×QB ,

(i) C = (μC , νC) such that μC = μA × μB and νC =νA × νB ∀p′, q′ ∈ QA, p

′′, q′′ ∈ QB

μC

((p′, p′′), a, (q′, q′′)

)= μA(p

′, a, q′) ∧ μB(p′′, a, q′′)

and νC((p′, p′′), a, (q′, q′′)

)= νA(p

′, a, q′) ∨ νB(p′′, a, q′′)

(ii) iC = (iμC, iνC ) such that

iμC(p′, p′′) =

{iμA

(p′) ∧ iμB(p′′) , if p′ ∈ QA, p′′ ∈ QB

0 , otherwise

143143143143

and

iνC (p′, p′′) =

{iνA(p

′) ∨ iνB (p′′) , if p′ ∈ QA, p′′ ∈ QB

0 , otherwise

(iii) fC = (fμC, fνC ) such that

fμC(p′, p′′) =

{fμA

(q′) ∧ fμB(q′′) , if q′ ∈ QA, q′′ ∈ QB

0 , otherwise

and

fνC (p′, p′′) =

{fνA(q

′) ∨ fνB (q′′) , if q′ ∈ QA, q′′ ∈ QB

0 , otherwise

From the definition of C , we have for all x ∈ Σ∗,

μ∗C

((p′, p′′), x, (q′, q′′)

)=

∨{μ∗

A(p′, x, q′)

∧μ∗B(p

′′, x, q′′)}

and ν∗C((p′, p′′), x, (q′, q′′)

)=

∧{ν∗A(p

′, x, q′)

∨ν∗B(p′′, x, q′′)}

Let ((p′, p′′), x, (q′, q′′)) ∈ (QA × QB) × Σ × (QA × QB),x ∈ Σ∗ then,

LμC(x) =

∨{{iμC

(p′, p′′) ∧ (μ∗C((p

′, p′′), x, (q′, q′′))∧

fμC(q′, q′′) | (q′, q′′) ∈ QA ×QB}

| (p′, p′′) ∈ QA ×QB

}=

∨{{(iμA

(p′) ∧ iμB(p′′))∧{

∨ (μ∗A(p

′, x, q′) ∧ μ∗B(p

′′, x, q′′)}

∧{fμA

(q′) ∧ fμB(q′′)

}| (q′, q′′) ∈ QA ×QB}

}| (p′, p′′) ∈ QA ×QB

}=

[∨{{iμA

(p′) ∧ μ∗A(p

′, x, q′)∧

fμA(q′) | q′ ∈ QA} | p′ ∈ QA}

]∧[∨

{{iμB(p′′) ∧ μ∗

B(p′′, x, q′′)

∧ fμB(q′′) | q′′ ∈ QB} | p′′ ∈ QB}

]LμC

(x) = LμA(x) ∧ LμB

(x)

and

LνC(x) =

∧{{iνC (p

′, p′′) ∨ (ν∗C((p′, p′′), x, (q′, q′′))∨

fνC (q′, q′′) | (q′, q′′) ∈ QA ×QB}

| (p′, p′′) ∈ QA ×QB

}=

∧{{(iνA(p

′) ∨ iνB (p′′))∨{

∧ (ν∗A(p′, x, q′) ∨ ν∗B(p

′′, x, q′′)}

∨{fνA(q

′) ∨ fνB (q′′)}| (q′, q′′) ∈ QA ×QB}

}| (p′, p′′) ∈ QA ×QB

}=

[∧{{iνA(p

′) ∨ ν∗A(p′, x, q′)∨

fνA(q′) | q′ ∈ QA} | p′ ∈ QA}

]∨[∧

{{iνB(p′′) ∨ ν∗B(p

′′, x, q′′) ∨ fνB (q′′)

| q′′ ∈ QB} | p′′ ∈ QB}]

LνC(x) = LνA

(x) ∨ LνB(x)

Hence the theorem

Note: From the above theorem, the behavior of IFA-UM is

closed under product.

Example 1: Consider an IFA-UM A = (QA,Σ, A, iA, fA)where QA = {q1, q2, q3},Σ = {a, b}, A = (μA, νA) such that μA : QA × Σ × QA →[0, 1] and νA : QA ×Σ×QA → [0, 1] are defined as follows:

μA(q1, a, q1) = 0.8μA(q2, a, q3) = 0.7μA(q1, b, q1) = 1.0μA(q3, a, q3) = 0.4μA(q3, b, q3) = 0.3μA(q1, b, q2) = 0.6

νA(q1, a, q1) = 0.2νA(q2, a, q3) = 0.2νA(q1, b, q1) = 0.0νA(q3, a, q3) = 0.5νA(q3, b, q3) = 0.3νA(q1, b, q2) = 0.4

Initial values are defined by iμA(q1) = 1.0, iμA

(q2) = 0.5and iνA(q1) = 0, iνA(q2) = 0.3 respectively . Final values are

defined by fμA(q3) = 1.0 and fνA(q3) = 0 respectively.

The intuitionistic fuzzy behavior of A is LμA: Σ∗ → [0, 1]

and LνA: Σ∗ → [0, 1] such that

LμA(x) =

⎧⎪⎨⎪⎩

0.3 , if x ∈ {a, b}∗ba{a, b}∗

0.3 , if x ∈ a{a, b}∗

0 , otherwise

and LνA(x) =

⎧⎪⎨⎪⎩

0.5 , if x ∈ {a, b}∗ba{a, b}∗

0.5 , if x ∈ a{a, b}∗

0 , otherwise

Consider another IFA-UM B = (Q′B,Σ, B, iB, fB) where

Q′B = {q′1, q

′2, q

′3},Σ = {a, b}, B = (μB, νB) such that

μB : Q′B × Σ×Q′

B → [0, 1] and

νB : Q′B × Σ×Q′

B → [0, 1] are defined as follows:

μB(q′1, a, q

′1) = 0.9

μB(q′2, a, q

′3) = 0.8

μB(q′1, b, q

′1) = 0.6

μB(q′3, a, q

′3) = 0.8

μB(q′1, b, q

′2) = 0.5

μB(q′3, b, q

′3) = 0.9

νB(q′1, a, q

′1) = 0.1

νB(q′2, a, q

′3) = 0.2

νB(q′1, b, q

′1) = 0.3

νB(q′3, a, q

′3) = 0.1

νB(q′1, b, q

′2) = 0.4

νB(q′3, b, q

′3) = 0.1

Initial values are defined by iμB(q′1) = 0.8, iμB

(q′2) = 0.1and iνB (q

′1) = 0.2, iνB(q

′2) = 0.7 respectively. Final values

are defined by fμB(q′3) = 0.3 and fνB (q

′3) = 0.6 respectively.

The intuitionistic fuzzy behavior of B is LμB: Σ∗ → [0, 1]

and LνB: Σ∗ → [0, 1] such that

LμB(x) =

⎧⎪⎨⎪⎩

0.3 , if x ∈ {a, b}∗ba{a, b}∗

0.1 , if x ∈ a{a, b}∗

0 , otherwise

and LνB(x) =

⎧⎪⎨⎪⎩

0.6 , if x ∈ {a, b}∗ba{a, b}∗

0.7 , if x ∈ a{a, b}∗

0 , otherwise

“Fig.1” shows the product of A ×B. The intuitionistic fuzzy

144144144144

(q1,q′

1) (q2,q

′

2) (q3,q

′

3)

b|0.3|0.3

b|0.6|0.3

a|0.8|0.2

b|0.5|0.4 a|0.8|0.2

a|0.4|0.5

Fig. 1. Product of automaton of A and B, i.e.,C = A × B

behavior of C is LμC: Σ∗ → [0, 1] and LνC

: Σ∗ → [0, 1]such that

LμC(x) =

⎧⎪⎨⎪⎩

0.3 , if x ∈ {a, b}∗ba{a, b}∗

0.1 , if x ∈ a{a, b}∗

0 , otherwise

and LνC(x) =

⎧⎪⎨⎪⎩

0.6 , if x ∈ {a, b}∗ba{a, b}∗

0.7 , if x ∈ a{a, b}∗

0 , otherwise

Intersection

Theorem 3: If A and B are two IFA-UM’s with behaviors

LA and LB respectively, then (LμA∩LμB

, LνA∩LνB

) is a

behavior of an IFA-UM.

Proof: The intersection of A and B (A ∩B) is the IFA-

UM defined by C = (QC ,Σ, C, iC , fC) (assume that QA ∩QB = φ) where QC = QA ×QB ,

(i) C = (μC , νC) is an IFS where μC = μA ∧ μB and

νC = νA ∧ νB such that μC : QC × Σ × QC → [0, 1]and νC : QC ×Σ×QC → [0, 1] are defined as follows:

∀p′, q′ ∈ QA, p′′, q′′ ∈ QB.

μC

((p′, p′′), a, (q′, q′′)

)= μA(p

′, a, q′) ∧ μB(p′′, a, q′′)

and

νC((p′, p′′), a, (q′, q′′)

)= νA(p

′, a, q′) ∨ νB(p′′, a, q′′)

(ii) iC = (iμC, iνC ) is an IFS,

iμC: QC → [0, 1] and iνC : QC → [0, 1] are defined by

iμC(p′, p′′) =

{iμA

(p′) ∧ iμB(p′′) , if p′ ∈ QA, p′′ ∈ QB

0 , otherwise

and

iνC (p′, p′′) =

{iνA(p

′) ∨ iνB (p′′) , if p′ ∈ QA, p′′ ∈ QB

0 , otherwise

(iii) fC = (fμC, fνC ) is an IFS,

fμC: QC → [0, 1] and fνC : QC → [0, 1] are defined

by

fμC(p′, p′′) =

{fμA

(q′) ∧ fμB(q′′) , if q′ ∈ QA, q

′′ ∈ QB

0 , otherwise

and

fνC (p′, p′′) =

{fνA(q

′) ∨ fνB (q′′) , if q′ ∈ QA, q

′′ ∈ QB

0 , otherwise

From the definition of C , we have for all x ∈ Σ∗,

μ∗C

((p′, p′′), x, (q′, q′′)

)= μ∗

A(p′, x.q′) ∧ μ∗

B(p′′, x, q′′) and

ν∗C((p′, p′′), x, (q′, q′′)

)= ν∗A(p

′, x.q′) ∨ ν∗B(p′′, x, q′′)

, ∀x ∈ Σ∗, (p′, p′′), (q′, q′′) ∈ QA ×QB .

Using the above result it can be shown that LμC= LμA

∩LμB

and LνC= LνA

∩ LνB

Reversal

Definition 6: A path c in A with |c| = σ1σ2 . . . σk, σi ∈ Σwhere i = 1, . . . , k yields a path c� with |c�| = σkσk−1 . . . σ1

where � : Σ∗ → Σ∗ is the reversal function defined by

�(1) = 1, �(σ) = σ, �(st) = �(t)�(s), where s, t ∈ Σ∗.

Theorem 4: Let A is an IFA-UM and x is recognized by

A , then x� is recognized by some IFA-UM.

Proof:

Let A = (Q,Σ, A, i, f) be an IFA-UM. Define IFA-UM

A � = (Q,Σ, A�, f, i), where

(i) A� = (μA� , νA�) is an IFS of Q× Σ×Q such that

μA�(p, a, q) =

{μA(q, a, p) , if p, q ∈ Q

0 , otherwise

and

νA�(p, a, q) =

{νA(q, a, p) , if p, q ∈ Q

0 , otherwise

(ii) i = (μA� , νA� ) is an IFS of fuzzy subset of Q such that

iμA� (p) =

{fμA

(p) , if p ∈ Q

0 , otherwise

and

iνA� (p) =

{fνA(p) , if p ∈ Q

0 , otherwise

(iii) f = (fμA� , fνA� ) is an IFS of fuzzy subset of Q such

that

fμA� (q) =

{iμA

(q) , if q ∈ Q

0 , otherwise

and

fνA� (q) =

{iνA(q) , if q ∈ Q

0 , otherwise

The intuitionistic fuzzy behavior of A � is defined by

LμA� (x�) = ∨

{{iμA� (q) ∧ μ∗

A�(q, x, p) ∧ fμA� (p)|p ∈ Q}|q ∈ Q}

= ∨{fμA(q) ∧ μA(p, x, q) ∧ iμA

(p)}

= ∨{iμA(p) ∧ μA(p, x, q) ∧ fμA

(q)}

LμA� (x�) = LμA

(x)

and

LνA� (x�) = ∧

{{iνA� (q) ∨ ν∗A�(q, x, p) ∨ fνA� (p)|p ∈ Q}|q ∈ Q

}= ∧{fνA(q) ∨ νA(p, x, q) ∨ iνA(p)}

= ∧{iμA(p) ∨ νA(p, x, q) ∨ fνA(q)}

LνA� (x�) = LνA(x)

Hence the theorem

145145145145

IV. CONCLUSION

In this paper, the authors have made an attempt to study the

closure properties of the intuitionistic fuzzy finite automata

with unique membership transitions on an input symbol IFA-

UM. We have made a humble beginning in this direction,

however, many concepts are yet to be fuzzyfied in the context

of IFA-UM.

ACKNOWLEDGMENT

The author is highly grateful to referees for their valuable

comments and suggestions for improving the paper.

REFERENCES

[1] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst. 20(1), 1986,pp.87-96

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