MATHEMATICS AND MODELING OF NEUTROSOPHIC LOGIC SYSTEMS

Information Sciences Manuscript Draft Manuscript Number: INS-D-11-924 Title: MATHEMATICS AND MODELING OF NEUTROSOPHIC LOGIC SYSTEMS Article Type: Full length article Keywords: Neutrosophic reasoning; Neutrosophic modeling; Mamdani Neutrosophic Inference system; Sugeno Neutrosophic Inference system; Tsukamoto Neutrosophic Inference system

Cover Letter

To The Editor, Information Sciences, Elsevier Dear Sir, I am attaching my research paper titled: “MATHEMATICS AND MODELING OF NEUTROSOPHIC LOGIC SYSTEMS” to be considered for inclusion in Information Sciences, Elsevier Journal.

a. This manuscript is the authors' original work and has not been published nor has it been submitted simultaneously elsewhere. b. All authors have checked the manuscript and have agreed to the submission.

Thanks and regards Swati Aggarwal

Manuscript (including abstract) Click here to view linked References

MATHEMATICS AND MODELING OF NEUTROSOPHIC LOGIC SYSTEMS A.Q.Ansari a, Ranjit Biswasb, Swati Aggarwalb,* a

Department of Electrical Engineering, Jamia Millia Islamia, New Delhi-110025, India

b

Department of Computer Science Engineering, ITM University, HUDA Sector 23-A, Gurgaon-122017, Haryana, India

* Corresponding Author. Email: swati1178@gmail.com, Phone: +91-9717995716

Abstract Improbability and indeterminacy permeates in real world executions. It is very common to encounter a vague data, incomplete, contradictory or indeterminate data. Fuzzy logic is a dominant component of the soft computing domain; that takes imprecise inputs and generates a precise output, which is a closer representation of real world executions. This paper attempts to suggest employing Neutrosophic logic for inference mechanism, as neutrosophic logic is an extension to existing fuzzy logic and is much more generalized in it’s working.

Fuzzy logic can only deal with fuzzy, vague information but not the incomplete and inconsistent information. So, this paper explores the possibility of extending the capabilities of the existing fuzzy models by incorporating neutrosophic logic in the inference process.

Suggested neutrosophic models in the paper can work with fuzzy, incomplete and inconsistent information without danger of trivialization

Index Terms — Neutrosophic reasoning, Neutrosophic modeling, Mamdani Neutrosophic Inference system, Sugeno Neutrosophic Inference system, Tsukamoto Neutrosophic Inference system

1

1. INTRODUCTION Fuzzy logic was given by Prof. L.A.Zadeh in his seminal paper during second half of last century1965 [8]. While fuzzy logic witnessed weak acceptance and slow start initially, but gradually it has come forward as one of the extensively researched, experimented and strong pillar of soft computing.

Real world information is full of uncertainties, gaps and inconsistent information. Different types of inference systems based on fuzzy logic were developed to handle vague data [1], [5],[6].

Quite recently, Neutrosophic Logic was proposed by Florentine Smarandache [2] which is based on non-standard analysis that was given by Abraham Robinson in 1960s. Neutrosophic Logic was developed to represent mathematical model of uncertainty, vagueness, ambiguity, imprecision,

undefined,

unknown,

incompleteness,

inconsistency,

redundancy

and

contradiction. All the factors stated are very integral to human thinking, as it is very rare that we tend to conclude/judge in definite environments, imprecision of human systems could be due to the imperfection of knowledge that human receives (observation) from the external world.

This paper proposes to use recently developed Neutrosophic logic for inference process, as it is much more generalised in it’s working and an extension to the already existing fuzzy logic.

Section 2, briefly gives introduction to neutrosophic sets and its operations. Section 3 is dedicated to the understanding of neutrosophic reasoning process. Underlying mathematical principles of different types of neutrosophic inference systems are discussed in section 4.

Short descriptions of the steps to be considered while neutrosophic modelling is undertaken 2

are deliberated in section 5. Section 6 gives the possible partition styles for neutrosophic models. The last section concludes the paper while exploring possible future directions in this domain.

2. NEUTROSOPHIC SETS This section provides a brief introduction to and an outline of basic concepts essential to the study of neutrosophic sets. Exhaustive treatments of specific subjects can be found in the reference list.

Sets are one of the fundamental concepts of mathematics. Classical sets are the most oldest of the different variants of sets prevalent. These conventional sets have crisp boundaries and are used to categorize things, like either an element x is a member of a classical set A or it is not at all member of the classical set. There is no semi-belongingness of the type: ―x belongs 30 % to A‖. Classical set can be represented as A  {x | x / 2  0}

(1)

Here there is clear demarcation between the members and non members of set A. Any number which is even 100% belongs to this set, and odd numbers does not belong to this set.

In contrast to classical sets, fuzzy sets were introduced by Prof. Zadeh in 1965 [8]. Here the transition from ―belongingness to a set‖ to ―not belongingness to a set‖ is gradual and this transition is characterized by the membership functions. These membership functions impart the flexibility in modeling the real world linguistic expressions of the type: ―Rose is red‖.

Crisp definition of this statement is not possible as it is difficult to encode red colour due to various shades of the same colour applicable. This statement has fuzzy uncertainty associated with it, and Zadeh [8] pointed out in his seminal paper that such imprecisely defined sets or

3

classes ―play an important role in human thinking, particularly in the domains of pattern recognition, communication of information, and abstraction‖.

Definition 1: Fuzzy sets and membership functions Let U denote a universe of discourse. Then a fuzzy set A in U is defined as a set of ordered pairs

A  { x,  A ( x) | x U }

(2)

where  A : U  [0,1] is a function of A that delivers the grade of membership of x in A.

Definition 2: Neutrosophic sets In general if X be a universe of discourse, and M a neutrosophic set included in X. An element x from X is noted with respect to the set M as x (T, I, F) and belongs to M in the following way:

it is t% true in the set, i% indeterminate (unknown if it is) in the set, and f% false, where t varies in T, i varies in I, f varies in F, where Then T, I, F are called neutrosophic components.

T, I, F can be standard or non-standard real subsets of ]-0, 1+[, with sup T = t_sup, inf T = t_inf, sup I = i_sup, inf I = i_inf, sup F = f_sup, inf F = f_inf, and n_sup = t_sup+i_sup+f_sup, 4

n_inf = t_inf+i_inf+f_inf.

Neutrosophic logic in comparison to other logics, is not confined to range of [0,1], T,I,F can be assigned over boiling values (>1) or under dried values (<0). , but here for practical purposes and to keep the discussion relatively simpler we are assuming the range of [0,1].

So, if X (universe of discourse: UOD) is a collection of objects denoted generically by x, then a neutrosophic set M in X is characterized by three membership functions: TM truth membership function, IM indeterminacy membership function and FM falsity membership function.

M (TM , I M , FM )

(3)

and M  {( x, TM ( x), I M ( x), FM ( x) | x  X }

(4)



TM (x)   0,1



(5)

I M (x) 

 0,1 

(6)

FM (x) 

 0,1 

(7)

There 





is





no

restriction

on

the

sum

of

TM ( x), I M ( x)andFM ( x) ,

so

0  sup(TM ( x))  sup( I M ( x))  sup(FM ( x))  3  .

Obviously the definition of a neutrosophic set is a simple extension of the fuzzy logic in which there are three characteristic function (truth, indeterminacy and falsity) which are permitted to

5

have continuous values between -0 and 1+; in contrast to only one characteristic function in fuzzy logic.

Also fuzzy logic though ensures multiple belongingness of a particular element to multiple classes with varied degree but capturing of neutralities due to indeterminacy is missing, further data representation using fuzzy logic is limited by the fact that membership and nonmembership value of an element to a particular fuzzy set should sum up to 1. In general, the neutrosophic logic generalizes: the Boolean logic (for n = 1 and i = 0, with t, f either 0 or 1), the multi-valued logic (for 0 ≤ t, i, f ≤ 1), the fuzzy logic (for n = 1 and i = 0, and 0 ≤ t, i, f ≤ 1).

Neutrosophic components T, I, F can take values independent of each other’s value. Moreover the sets T, I, F are not necessarily intervals, but may be any real sub-unitary subsets: discrete or continuous; single-element, finite, or (countably or uncountably) infinite; union or intersection of various subsets; etc. They may also overlap.

For universe of discourse X, let A be any single valued neutrosophic set in X. Here we are discussing single valued neutrosophic set because it can be easily used in real scientific and engineering applications.

Definition 3: Neutrosophic set A with discrete X can be defined as n

A

T

A ( x i ),

I A ( x i ), F A ( x i ) / xi | x i  X

(8)

i 1

Definition 4: Neutrosophic set A with continuous X can be defined as A



X

T A ( x), I A ( x), FA ( x) / x | x  X

(9)

6

If X is universe of discourse, consisting of neutrosophic sets A and B. Belongingness of element x to A and B can be defined as x  x(T A ( x), I A ( x), FA ( x)) and x  x(TB ( x), I B ( x), FB(x) ) respectively.

If, after calculations, in the below calculations one obtains values < 0 or > 1, then one replaces them with –0 or 1+ respectively. The summation and integration signs in equations (8) and (9) stand for union of individual ordered combinations of T A ( x), I A ( x), FA ( x) for all x , also ―/‖ is only a marker and does not imply division.

Example 1 discusses the representation of neutrosophic set with discrete X. Example 1: Assume that a questionnaire is filled by three parents in which they reflect their opinion about degree of ―good combination‖, degree of indeterminacy, and a degree of ―poor combination‖ for their respective experience of having two children with following combinations: x1={1 boy, 1 girl}, x2={ 2 girls}, x3={2 boys}. So here X  [ x1 , x 2 , x3 ] .

A, B, C represents three single valued neutrosophic set in X. A  0.3,0.4,0.2 / x1 , 0.8,0.2,0.1 / x 2 , 0.5,0.6,0.4 / x3

B  0.4,0.3,0.3 / x1 , 0.7,0.3,0.2 / x 2 , 0.3,0.4,0.1 / x3

C  0.4,0.5,0.6 / x1 , 0.9,0.2,0.1 / x 2 , 0.3,0.5,0.4 / x3

Neutrosophic set A of example 1 is shown in figure 1.

7

Figure 1: Neutrosophic set A with discrete characteristic function of example 1 Example 2 discusses the representation of neutrosophic set with continuous X. Example 2: Neutrosophic sets with continuous X Let X be the set of possible ages for human beings. Assume that the task is to judge the approximate age of the person by seeing the photograph. Then the one of the neutrosophic set A=â&#x20AC;&#x2013;about 50 years oldâ&#x20AC;&#x2013; which lies in X may be expressed by three continuous characteristic functions: truth, indeterminacy and falsity as shown in figure 2. Crisp input x1can be mapped to the pre-defined neutrosophic set having three membership functions (truth, indeterminacy and falsity membership functions) as shown in figure 2. This process is called as neutrosophication process.

8

Figure 2: Neutrosophic set A with continuous characteristic functions of example 2 Example 1 and 2 suggests that the construction of a neutrosophic set depends on two things: the identification of a suitable universe of discourse and the specification of an appropriate membership function. Though the notion of specification of the membership function is subjective, that indicates that different people would perceive the same concept (say â&#x20AC;&#x2022;intelligenceâ&#x20AC;&#x2013;) differently. The subjectivity and approximation of neutrosophic sets are two major fundamentals of neutrosophic logic. Also as seen in example 1 and 2, there is no dependency and limitation to the summation of t, i and f values.

Next we give the ordinary set operations defined on neutrosophic sets, which were initially defined by Smarandache [3].

Definition 5: Containment or subset

9

Neutrosophic set A is contained in neutrosophic set B (or equivalently, A is subset of B A  B , or A is smaller than or equal to B if and only if: inf T A ( x)  inf TB ( x), sup T A ( x)  sup TB ( x)

(10)

inf FA ( x)  inf FB ( x), sup FA ( x)  sup FB ( x)

(11)

Definition 6: Union (disjunction) The union of two neutrosophic sets A and B is a neutrosophic set C, written as C  A B

TC ( x)  T A ( x)  TB ( x)  T A ( x)  TB ( x)

(12)

I C ( x)  I A ( x)  I B ( x)  I A ( x)  I B ( x )

(13)

FC ( x)  FA ( x)  FB ( x)  FA ( x)  FB ( x)

(14)

for all x in X. Definition 7: Intersection (conjunction) The intersection of two neutrosophic sets A and B is a neutrosophic set C, written as C  A  B or C  A AND B ,

whose membership functions are related to those of A and B by:

TC ( x)  T A ( x)  TB ( x)

(15)

I C ( x)  I A ( x)  I B ( x)

(16)

FC ( x)  FA ( x)  FB ( x)

(17)

for all x in X. 10

Definition 8: Complement (negation) Complement of neutrosophic set, denoted by A or A or NOT A or c (A) is defined as: TC ( A) ( x)  {1 }  T A ( x)

(18)

I C ( A) ( x)  {1 }  I A ( x)

(19)

FC ( A) ( x)  {1 }  FA ( x)

(20)

for all x in X. Neutrosophic set needs to be specified from a technical point of view. To this effect, set theoretic operators on an instance of neutrosophic set SVNS were given by Wang [7]. If A and B are single valued neutrosophic sets in X, then technically following basic set operations can be defined.

Definition 9: Containment A  B if and only if T A ( x)  TB ( x),

I A ( x)  I B ( x),

FA ( x)  FB ( x),

for all x in X. Definition 10: Union C  A  B TC ( x)  max(T A ( x), TB ( x)),

11

I C ( x)  max( I A ( x), I B ( x)), FC ( x)  min(FA ( x), FB ( x)),

for all x in X. Definition 11: Intersection C  A  B TC ( x)  min(T A ( x), TB ( x)), I C ( x)  min(I A ( x), I B ( x)), FC ( x)  max( FA ( x), FB ( x)),

for all x in X. Definition 12: Complement c(A) Tc( A) ( x)  FA ( x),

I c( A) ( x)  {1 }  I A ( x),

Fc( A) ( x)  T A ( x),

for all x in X. It should be noted that the set operations defined and the membership function introduced in this section is by no means exhaustive. Other membership functions can be designed depending on the nature of problem, provided set of parameters are given to specify the appropriate, meanings of the membership functions.

2.1 Neutrosophic If-Then Rules A linguistic input variable x in a universe of discourse Ui is characterized by X 12

X  {x1 , x 2,.......,xi }

(21)

and A( x)  { Ax , Ax ,......Ax } , 1

2

(22)

i

where X is the variable term set of x; that is the set of names of the linguistic values of x with each value x i being a neutrosophic number mapped to the corresponding neutrosophic set Ax defined on X. i

Neutrosophic set A that corresponds to the ith - x variable can be defined as: Ax = ( T A x , I A x , F A x ), i

i

i

(23)

i

where T A xi is the truth membership function defined, I A xi is the indeterminate membership function defined and F A xi is the falsity membership function defined. So A(x) is a semantic rule for associating each value with it’s meaning.

Axi ( xi )  (T Ax

i

( xi ) , I Axi ( xi ) , FAxi ( xi ) )

 (t xi , i xi f xi )

(24)

Axi ( xi ) corresponds to the triplet value of truth t xi , indeterminacy i xi and falsity f xi , once the

ith-input variable x is mapped to its corresponding Axi neutrosophic set. This process is termed as neutrosophication process that maps a crisp number to neutrosophic set.

For example if x indicates size, then X  {x1 , x 2 , x3 }  {(small-t, small-i, small-f), (medium-t, medium-i, medium-f), (large-t, large-i, large-f)}; where small-t, small-i, small-f corresponds to neutrosophic values truth value, indeterminacy value and falsity value respectively in the neutrosophic small set; similarly for medium and large.

13

Following the above definition, the input vector X which includes the input state linguistic variables xi’s, and the output state vector Y which includes the output state linguistic variables yi’s, can be defined as X  {( xi ,U i,{x1i , xi2 ,........., xiki },{ A1xi , Ax2i ,.......Axki }) |i 1,2,......,n }

(25)

Y  {( y i , U i' { y 1i , y i2 ,.......,y i i }, {B1yi , B y2i ,.......B yi }) | i 1,2,......,m }

(26)

i

l

l

i

th

Ax i  (T Aki , I Aki , FAki ) is the k k

i

xi

xi

neutrosophic set mapped to the ith input variable x, defined on

xi

universe of discourse Ui, comprising of three independent truth, indeterminacy and falsity membership functions respectively.

Similarly B xlii  (TBli , I Bli , FBli ) is lth neutrosophic set mapped to the ith output variables defined xi

xi

xi

'

on universe of discourse U i .

A neutrosophic IF-THEN rule (neutrosophic rule, neutrosophic implication or neutrosophic conditional statement) assumes the form: IF x is Axk then y is B ly ,

(27)

where Axk and B ly are linguistic values defined by neutrosophic sets on universes of discourse X and Y respectively.

Often ―x is Axk ”is called the antecedent or premise while ―y is B ly ”is called the consequence or conclusion.

14

Before we can employ neutrosophic IF-THEN rules to model and analyze a system, we first have to formalize what is meant by the expression: ―IF x is Axk then y is B ly ”, which is sometimes abbreviated as Axk  B ly

(28)

Basically this expression describes a neutrosophic relation between two variables x and y; suggesting that neutrosophic IF-THEN rule be defined as a binary neutrosophic relation R on the product space X  Y .

Generally speaking, there are two ways to interpret the neutrosophic rule as Axk  B ly A coupled with B, then  Axk * B ly /( x, y)

R= Axk  B ly = Axk  B ly = 

X Y

(29)



where * is a neutrosophic AND (or more generally N-norm) operator and Axk  B ly is used to represent the neutrosophic relation R. Definition 13: N-norm (Nn) represents the AND operator in neutrosophic logic and intersection operator in neutrosophic set theory [4]. It is defined as:

Nn: ( 0,1   0,1   0,1 ) 2   0,1   0,1   0,1  N n ( Axk , B ly )  N n ((T Ak , I Ak , F Ak ), ((TB l , I B l , FB l ))  x

x

x

y

y

y

(30)

( N n (T Ak , TB l ), N n ( I Ak , I B l ), N n ( F Ak , FB l )) x

y

x

y

x

y

Where ( N n (TAxk , TBly ), N n ( I Axk , I Bly ), N n ( FAxk , FBly )) are the truth/membership, indeterminacy and falsehood/non-membership components.

15

Nn will have to satisfy, for any x1, x2, x3 in the neutrosophic logic/set M of the universe of discourse U, the following axioms: a. Boundary conditions: Nn(x,0)=0, Nn(x,1)=x

(31)

b. Commutativity: Nn(x1,x2)= Nn(x2,x1)

(32)

c. Monotonicity: If x1≤x2, then Nn(x1,x3) ≤ Nn(x2,x3)

(33)

d. Associativity: Nn (Nn (x1,x2), x3)= Nn (x1,Nn (x2,x3))

(34)

Let J є {T,I,F} be a component.

Most known N-norms, as T-norms in fuzzy logic and set are: a. The algebraic product N-norm: Nn-algebraic J(x1,x2)= x1 . x2

(35)

b. The bounded N-norm Nn-bounded J(x1,x2)= max{0,x1+x2-1}

(36)

c. The default (min) N-norm Nn-min J(x1,x2)= min{x1,x2}

(37)

A general example of N-norm would be: Let N n ( Axk , B ly ) be in the neutrosophic set/logic, then N n ( Axk , B ly )  N n ((T Ak , I Ak , F Ak ), ((T B l , I B l , FB l ))  x

x

x

y

y

y

( N n (T Ak , TB l ), N n ( I Ak , I B l ), N n ( F Ak , FB l ))  x

y

x

y

x

(38)

y

((T Ak  TB l ), ( I Ak  I B l ), ( F Ak  FB l )) x

y

x

y

x

y

Where the  operator acting on two (standard or non-standard) sub-unitary sets, is a N-norm (verifying the above N-norms axioms): while the  operator, also acting on two (standard or non standard) sub-unitary sets, is a N-conorm, as discussed below.

16

The commonly used neutrosophic AND operations are intersection and algebraic product. min(TAk , TB l ) x

(TAk  TB l ) 

Hence

x

y

or TAk .TB l

y

x

(39)

y

Definition 14: N-Conorm -Nc: represents the OR operator in neutrosophic logic and respectively the union operator in the neutrosophic set theory [4]. It is defined as:



  0,1   0,1 )

(  0,1 









2



 0,1   0,1   0,1  











N c ( Axk , B ly )  N c ((T Ak , I Ak , F Ak ), ((TB l , I B l , FB l ))  x

x

x

y

y

y

(40)

( N c (T Ak , TB l ), N c ( I Ak , I B l ), N c ( F Ak , FB l )) x

y

x

y

x

y

where ( N c (TAxk , TBly ), N c ( I Axk , I Bly ), N c ( FAxk , FBly )) are the truth/membership, indeterminacy and falsehood/non-membership components.

Nc will have to satisfy, for any x1, x2, x3 in the neutrosophic logic/set M of the universe of discourse U, the following axioms: a. Boundary conditions: Nc(x,1)=1, Nc(x,0)=x

(41)

b. Commutativity: Nc(x1,x2)= Nc(x2,x1)

(42)

c. Monotonicity: If x1≤x2, then Nc(x1,x3) ≤ Nc(x2,x3)

(43)

d. Associativity: Nc (Nc (x1,x2), x3)= Nc (x1,Nc (x2,x3))

(44)

Let J є {T,I,F} be a component.

Most known N-conorms, as T-conorms in fuzzy logic and set are:

17

a. The algebraic product N-conorm: Nc-algebraic J(x1,x2)= x1 +x2- x1 . x2 (45) b. The bounded N-conorm Nc-bounded J(x1,x2)= min{1,x1+x2} (46) c. The default (min) N-conorm Nc-max J(x1,x2)= max{x1,x2} (47) A general example of N-conorm would be: Let N c ( Axk , B ly ) be in the neutrosophic set/logic, then N c ( Axk , B ly )  N c ((T Ak , I Ak , F Ak ), ((TB l , I B l , FB l ))  x

x

x

y

y

y

( N c (T Ak , TB l ), N c ( I Ak , I B l ), N c ( F Ak , FB l ))  x

y

x

y

x

(50)

y

((T Ak  TB l ), ( I Ak  I B l ), ( F Ak  FB l )) x

y

x

y

x

y

Where the  operator acting on two (standard or non-standard) sub-unitary sets, is a Nnorm (verifying the above N-norms axioms): while the  operator, also acting on two (standard or non standard) sub-unitary sets, is a N-conorm, as discussed.

Though  can be any algebraic /bounded or min form, and  any N-conorm, the easiest way would be to consider the min for crisp components or (inf for subset components) and max for crisp components or (sup for subset components).

The commonly used neutrosophic OR operations are union and bounded sum. Hence: max(T Ak , T B l ) (T Ak  T B l )  x

y

x

y

or min(1, T Ak  T B l ) x

(51)

y

18

Definition 15: Neutrosophic Relations Let X and Y, be two neutrosophic sets. A neutrosophic relation R (X, Y) is a subset of the product space X  Y , and is characterized by Truth-membership function TR( x, y ) , Indeterminacy - membership function I R ( x, y ) , Falsitymembership function FR ( x, y ) where x  X , y  Y and TR( x, y ) , I R ( x, y ) , FR ( x, y )  [0,1] Definition 16: Composition functions The membership functions for the composition of neutrosophic relations R(X,Y) and S(Y,Z) are given by TR S ( x, z )  min(TR ( x, y ) , TS ( y, z ) )

(52)

I R S ( x, z )  max( I R ( x, y ) , I S ( y, z ) )

(53)

FR S ( x, z )  max( FR ( x, y ) , FS ( y, z ) )

(54)

For one to one mapping between x and y, then TR ( x, y )  min(Tx , T y )

(55)

I R ( x, y )  max( I x , I y )

(56)

FR ( x, y )  max( Fx , Fy )

(57)

For all x  X , y  Y .

Using equations (55), (56), (57), equation (29) is interpreted as R= Axk  B ly = Axk  B ly = 

X Y

 Axk * B ly /( x, y)  (min(T Ak , TBl ), max( I Ak , I Bl ), max( FAk , FBl )) (58) x

19

y

x

y

x

y

On the other hand, if Axk  B ly is interpreted as A entails B, it can be written as: Material implication: Axk  B ly  ({1 }  T Ak  T Ak  TBl , {1 }  I Ak  I Ak  I B y , {1 }  FAk  FAk  FBl x

x

y

x

x

l

x

x

y

(59)

3. NEUTROSOPHIC REASONING Neutrosophic reasoning is an inference process used to derive conclusions from a set of neutrosophic IF-THEN rules and one or more conditions.

Before introducing neutrosophic reasoning procedure, we would discuss compositional rule of inference which is fundamental to neutrosophic reasoning.

The compositional rule of inference is the simplification of the usual notion.

Sometimes it is advantageous to use two dimensional neutrosophic membership functions, which have with two inputs, each from different universe of discourse. Contrary to this, one dimensional membership function (ordinary MF) have only one input. So the easiest way to extend ordinary neutrosophic MF to two dimensional MF is through cylindrical extension.

Definition 17: Cylindrical extension of one-dimensional neutrosophic sets.

Assuming that Ax is a neutrosophic set of X and R is a neutrosophic relation on X  Y .Then cylindrical extension of Ax neutrosophic set is given by cyl N ( Ax ) cyl N ( Ax )  Ax  R X Y 

A

X Y

x

/( x, y) 



((min T Ax , TRX Y ), max( I Ax , I RX Y ), max( FAx , FRX Y ))

X Y

(60) The projection is reverse of extension, in the sense that it decreases the dimension of the membership functions.

20

Definition 17: Projection of neutrosophic sets.

Let R be two dimension neutrosophic set on X  Y .Then the projections of R onto X and Y axes are defined as: RX 



RY 

R

X

Y

R x y / x 

x y

/y



X

y

y

 ((max T Y

(61)

(( max TRX Y / x), (min I RX Y / x), (min FRX Y / x)

x

RX Y

y

(62)

/ y), (min I RX Y / y), (min FRX Y / y) x

x

By projecting cyl N ( Ax ) onto y axis, we will have a resultant neutrosophic set, let’-s say B in universe of discourse Y. Definition 18: Neutrosophic compositional formula B y  Yprojection(cyl N ( Ax )) 



X Y

(( max (min T Ax , TRX Y )), (min(max( I Ax , I RX Y )), (min(max( FAx , FRX Y ))) x

x

x

(63) This formula is called as neutrosophic compositional formula and represented as B y  Ax  n R xY

Using the compositional formula, we can formalize an inference procedure, called neutrosophic reasoning, using neutrosophic IF-THEN rules. The basic rule of inference in traditional two-valued logic is modus ponens, according to which we can infer the truth of proposition B from the truth A and the implication A  B . For instance, if A is identified with: ―It is hot‖ and B with ―Lower the AC temperature,‖ then if it is true that ―It is hot‖, it is also true that ―Lower the AC temperature‖. This concept is illustrated below: Premise 1 (fact): x is A 21

Premise 2 (rule): IF x is A then y is B -------------------------------------------------Consequence (conclusion): y is B In reality, much of the human reasoning, modus ponens is employed in an approximate manner. For example, if we have the same implication: â&#x20AC;&#x2022;IF it is hot THEN lower the AC temperatureâ&#x20AC;&#x2013;, and a sensor is designated to record the room temperature, which has its own physical limitations of degradation and reliability.

The advantage of using neutrosophic logic in reasoning system is that it is very uncommon that the data acquired by the system would be 100% complete and determinate.

Incompleteness and indeterminacy in the data can arise from inherent non-linearity, timevarying nature of the process to be controlled, large unpredictable environmental disturbances, degrading sensors or other difficulties in obtaining precise and reliable measurements.

Humans can take intelligent decisions in such situations. Though this

knowledge is also difficult to express in precise terms, an imprecise linguistic description of the manner of control can usually be articulated by the operator with relative ease.

The gap in the processing of fuzzy reasoning system stems from the fact that the important concept of range of neutralities is missing in fuzzy logic fundamentals as fuzzy logic reasoning system is concerned about membership and non membership of a particular element to a particular class; and does not deals with indeterminate nature of data acquired that could happen due to various reasons discussed above; also the concept of fuzzy logic says that that non-membership value=1- membership value. This means that non-membership value and membership values are dependent on each other, which might not always be true for real world executions. The freedom of assigning non-dependent values to truth, falsity 22

and indeterminacy factors is very well represented by neutrosophic logic, as the (t,i,f) components in the Neutrosophic sets.

Definition 19: Neutrosophic Modus Ponens If the sensor senses the room temperature as being hot, also taking into account the indeterminacy and falsity of the data captured, then we may infer that, â&#x20AC;&#x2022;Lower the temperature of ACâ&#x20AC;&#x2013;. Now here the output is determined by taking into account the indeterminacy and falsity in the data acquired, apart from the degree of truthness. Output value also reflects the percentages of truthness, indeterminacy and falsity; giving the realistic estimation of sensed environment.

Premise 1 (fact): x is Ax' Premise 2 (rule): IF x is Ax then y is B y -------------------------------------------------Consequence (conclusion): y is B y' where Ax' is the captured neutrosophic set for the input x by applying neutrosophication process discussed in equation (24), that gives a realistic estimation of input captured by incorporating (t ,i, f) components. Similarly B y' is the output neutrosophic set. Ax and B y are neutrosophic sets of neutrosophic universes, the above inference procedure is called neutrosophic reasoning ; and can be termed as neutrosophic modus ponens: with modus ponens as its special case.

Using the neutrosophic composition rule of inference (equation (63)), we can now formulate the inference procedure of neutrosophic reasoning as discussed in the following definition.

23

Definition 20: Neutrosophic reasoning based on Max-min and Min-max composition.

Let Ax , A ' x and B y be the neutrosophic sets of X, X and Y respectively. Assume that the neutrosophic implication : Ax  B y is expressed as a neutrosophic relation R on X  Y .Then the neutrosophic set B ' y induced by ―x is A ' x ” and the neutrosophic rule IF x is Ax then y is B y , is defined by:

B ' y  (( max (min T A' x , TRX Y )), (min(max( I A' x , I RX Y )), (min(max( F A' x , FRX Y ))) x

B

'

y

x

x

 ( x (T A' x  TRX Y ),  x ( I A' x  I RX Y ),  x ( F A' x  FRX Y ))

(64) Or equivalently, B ' y = A ' x  n R = A ' x  n ( Ax  B y )

(65)

Equation (65) is a general expression for neutrosophic reasoning, while equation (64) is an instance of neutrosophic reasoning where min and max are the operators used for neutrosophic AND and OR respectively.

Now we can use the inference procedure of the neutrosophic modus ponens to derive the conclusions, provided that neutrosophic implication Axk  B ly is defined as appropriate binary neutrosophic relation.

3.1 SINGLE RULE WITH SINGLE ANTECEDENT For a single rule with single antecedent, the formula is available in equation (64). A further simplification of the equation yields (using equation 58):

24

B' y

T '   ( (T '  T ))  T ,  w  T  x Ax By By , B , Ax   1   y     I B ' y ,    ( x ( I A' x  I Ax ))  I B y ,   w2  I B y ,  F '  ( ( F '  F ))  F   w  F  Ax By   3 By   B y   x A x

(66)

3.2 SINGLE RULE WITH TWO ANTECEDENTS A neutrosophic IF-THEN rule with two antecedents is usually written as: IF x is Ax and y is By THEN z is Cz. The corresponding problem for neutrosophic reasoning is expressed as Premise 1 (fact): x is Ax' and y is B y' Premise 2 (rule):IF x is Ax and y is By THEN z is Cz ---------------------------------------------------------------------Consequence(conclusion) : z is C z' The neutrosophic rule in premise 2 above can be simplifies as : ― A  B  C ‖. So this neutrosophic rule can be transformed into a ternary neutrosophic relation R which is specified by the following: R( x, y, z )  (TR( x, y, z ) , I R( x, y, z ) , FR( x, y, z ) )  ((Tx  T y  Tz ), ( I x  I y  I z ), ( Fx  Fy  Fz ))

And the resulting C z' is expressed as C z'  ( Ax'  B y' )  n ( Ax  B x  C z ).

(67)

Thus

25

C z'

T ' ,   (T '  T ' )  (T  T  T ),  ( (T '  T ))  ( (T '  T ))  T ,  w1  w2  TC ,  Ax By Cz y By Cz  z Bx A x By   x A x  C z   x, y A x      I C ' ,    x, y ( I A' x  I B ' ))  ( I Ax  I B y  I C z ),    ( x ( I A'  I Ax ))  ( y ( I B '  I B y ))  I C z ,   w1'  w2'  I C z , z y x y  F '   ( F  F ))  ( F  F  F )  ( ( F  F ))  ( ( F  F ))  F   w ''  w ''  F  ' 2 Cz   C z   x, y A' x Ax By Cz   Ax y By Cz   1 B y' B y'    x Ax 

(68) The generalisation to more than two antecedents is straightforward.

3.3 MULIPLE RULES WITH MULIPLE ANTECEDENT The interpretation of multiple neutrosophic rules is usually taken as the union of the neutrosophic relations corresponding to the neutrosophic rules. For instance, given the following fact and rules: Premise 1 (fact): x is Ax' and y is B y' Premise 2 (rule 1): IF x is A1x and y is B1y THEN z is C1z Premise 3 (rule 2): IF x is A2x and y is B2y THEN z is C2z

---------------------------------------------------------------------Consequence(conclusion) : z is C z' If R1  A1x  B1y  C1z and R2  A2 x  B2 y  C 2 z . As  n is distributive over the  operator, so mathematically this neutrosophic inference can be expressed as: C z'  ( Ax'  B y' )  n ( R1  R2 )  [( Ax'  B y' )  n R1 ]  [( Ax'  B y' )  n R2 ]  C1' z  C 2' z

(69) where C1' z , C 2' z are the inferred neutrosophic sets for rule 1 and rule 2 respectively.

26

Suppose that there are N neutrosophic rules; out of which only M are fired, where M  N , then using equation (68), applying N-conorm definition (14),equation (50); equation (69) can be generalised and re-written as:  M  max T , T ' ,  i 1 Ciz'    C z   M  C z'   I C ' ,   min I C ' ,  z iz i 1 F '   M   C z   min FC '  iz   i 1 

(70)

4. NEUTROSOPHIC INFERENCE SYSTEMS Fuzzy inference systems which are based on the principles of fuzzy set theory, fuzzy if-then rules and fuzzy reasoning; have been very popular computing framework. It has been widely researched and applied to wide range of domains like : automatic control, data classification, decision analysis, expert systems and many more.

Neutrosophic logic is relatively a new concept; this section proposes an extension to fuzzy inference systems in the form of neutrosophic inference systems. Architecture, working and possible models of neutrosophic inference systems are discussed here.

Neutrosophic inference is a computer paradigm based on neutrosophic set theory, neutrosophic IF-THEN rules and neutrosophic reasoning. Similar to fuzzy inference systems, the basic structure of neutrosophic inference system would consist of five conceptual components: a neutrosophication module that would transform the crisp inputs to degrees of match with linguistic values, a neutrosophic rule base- consisting of neutrosophic IF-THEN rules; a database or dictionary –that gives the membership functions of the neutrosophic sets used in neutrosophic rules, a reasoning mechanism-which would perform the inference procedure using the neutrosophic rules and a given condition to derive a reasonable output or

27

conclusion and a de-neutrosophication module which would transform neutrosophic results of the inference into crisp output.

Neutrosophic inference systems can take either neutrosophic inputs or crisp inputs, but the outputs it generates are almost always neutrosophic sets. For using neutrosophic inference systems as controller it is essential that a crisp output is generated, so de-neutrosophication method would be utilized to generate the crisp output that best summarizes a neutrosophic set.

A neutrosophic inference system with a crisp output is schematically shown in figure 3, where the boundary indicates the basic neutrosophic inference system with neutrosophic output and the de-neutrosophication block serves the purpose of transforming a neutrosophic output to a crisp one. The function of de-neutrosophication block would be explained at the later point.

Figure 3: Schematic diagram of neutrosophic inference system 28

With crisp inputs and outputs, a neutrosophic inference system implements a nonlinear mapping from its input space to output space. This mapping is accomplished by a number of neutrosophic IF-THEN rules, each of which describes the local behaviour of the mapping. In particular, the antecedent of each rule defines a neutrosophic region of the input space, and the consequent specifies the corresponding rules.

In what follows, we will first introduce three types of neutrosophic inference systems on the lines of the commonly used fuzzy inference systems that have been widely employed in various applications. The differences between these three neutrosophic inference systems lie in the consequents of their neutrosophic rules and thus their aggregation and deneutrosophication procedures differ accordingly.

4.1 MAMDANI NEUTROSOPHIC MODEL The Mamdani fuzzy model [1] was the first attempt to control the steam engine and boiler combination by a set of linguistic control rules derived from experienced human operators.

On the same lines Mamdani Neutrosophic model is proposed.

To completely specify the operation of a Mamdani neutrosophic model, we need to assign a function for each of the following operators: a. AND operator (usually N-norm, definition 13) for the rule firing strength computation with ANDâ&#x20AC;&#x2122;ed antecedents b. OR operator (usually N-conorm, definition 14) for calculating the firing strength of a rule with ORâ&#x20AC;&#x2122;ed antecedents c. Implication operator (usually N-norm, definition 13 and 16, equation 64) for calculating qualified consequent MFs based on given firing strength

29

d. Aggregate operator (usually N-conorm, definition 14, equation 70) for aggregating qualified consequent MFs to generate an overall output MF e. De-neutrosophication operator for transforming neutrosophic set to equivalent fuzzy set and then applying defuzzification technique to generate final de-neutrosophied value.

Algorithm of working: Step 1: Neutrosophication: The inputs captured are mapped to the repository neutrosophic sets, by using equation (24) and diagrammatically illustrated in figure 2.

Step 2: Neutrosophic Inference: Mamdani Neutrosophic model would comprise of neutrosophic IF-THEN rules of the form discussed in equation (27). Then following combinations are possible: a. Single Rule with single antecedent Using concept of Neutrosophic Modus Ponens (definition (19)) and neutrosophic reasoning (definition (20)), and applying equation (64) resultant neutrosophic set is given by equation (66). b. Single Rule with two antecedents Using concept of Neutrosophic Modus Ponens (definition (19)) and neutrosophic reasoning (definition (20)), and applying equation (67) resultant neutrosophic set is given by equation (68). c. Multiple rules with multiple antecedents

30

Using concept of Neutrosophic Modus Ponens (definition (19)) and neutrosophic reasoning (definition (20)), and applying equation (69) resultant neutrosophic set is given by equation (70).

Step 3: De-neutrosophication Results generated by either of the equations 66/68/70 would result into three output membership functions corresponding to truth, indeterminacy and falsity values respectively.

So de-neutrosophication step would be a two step process. a. Transformation of Neutrosophic set to fuzzy set: This step is used to convert the output neutrosophic set to an equivalent fuzzy set. If Ax is the neutrosophic set, then ~

its corresponding equivalent fuzzy set Ax is obtained by applying transformation function f on Ax as discussed in equation (71) and (72). ~ Ax  T A~  f ( Ax )  f (T Ax , I Ax , FAx ) : [0,1]  [0,1]  [0,1]  [0,1]

(71)

~ Ax  T A~  f ( Ax )  f (T Ax , I Ax , FAx )  a..T Ax  b.I Ax / 2  c.(1  I Ax ) / 2  d .FAx

(72)

x

x

Depending on the application domain and the experts input, weights a, b, c, d can be assigned to the three parameters: truth, indeterminacy and falsity, in such a way that 0  a, b, c, d  1 and a  b  c  d  1 .

These weight values are application specific and can

be fine tuned by using existing methods of neural networks or genetic algorithms.

Truth component gives the direct information for the degree of truthness so it contributes by a.T Ax factor.

Indeterminacy value can oscillate between two extremes of 0 and 1. If indeterminacy value is 0, then either of falsity or truthness component is 100%, so in this case 31

indeterminacy component would not be used in the final evaluation, as it would be taken care by truth or falsity component. But if the indeterminacy value is 1, then both truth and falsity have equal chances of 50% each in the final interpretation. So in this case both cases would be considered, indeterminacy in that case would contribute by b.I Ax / 2  c.(1  I Ax ) / 2 .

Falsity component contributes indirectly in the computation of truthness, so it contributes by d .(1  FAx ) . ' Using equation (71) and (72), resultant neutrosophic set C z in equations (68) and (70)

~

' can be transformed to equivalent fuzzy set C z

~ C z'  TC~ '  f (C z' )  z

T ' , C   z   f  I C ' ,  a.TC '  b.I C ' / 2  c.(1  I C ' ) / 2  d .FC ' z z z z z F '    C  z

(73)

Same equations (71) and (72) can be used to generate the equivalent fuzzy set of the resultant neutrosophic set of equation (66). b. Generation of final de-neutrosophied value: The fuzzy set equivalent of the neutrosophic set obtained in the previous step, can be defuzzified to give final deneutrosophied value. Though there are many methods to convert crisp value from the fuzzy set [ ], here we are giving centroid of area method. So equation (73) becomes:

z COA 

T T

~' z Cz

.z.dz

~' z Cz

(74) .dz

32

Here z is final deneutrosophied resultant output that takes into consideration three important components of truthness, indeterminacy and falsity associated with the input data captured during processing of the input information.

4.2 SUGENO NEUTROSOPHIC MODEL The fuzzy model proposed by Takagi and Sugeno [5] is described by fuzzy IF-THEN rules which represents local input-output relations of a nonlinear system. Based on the existing principles of Sugeno Fuzzy model, Sugeno Neutrosophic model can be designed.

On the same lines the main feature of Sugeno neutrosophic model would be to express the local dynamics of each neutrosophic implication (rule) by a linear system model. The overall neutrosophic model of the system is achieved by neutrosophic "blending" of the linear system models. Sugeno neutrosophic model would be the most generalised model which can easily represent all nonlinear dynamical systems to a high degree of precision.

Sugeno neutrosophic model would be similar to Mamdani neutrosophic model in the first two parts of the neutrosophic inference process, neutrosophying the inputs and applying the neutrosophic operator, are exactly the same. The main difference between Mamdani and Sugeno Neutrosophic model would be that the Sugeno output membership functions are either linear or constant.

In Sugeno neutrosophic model, antecedents of the neutrosophic rules would be same as that of Mamadani Neutrosophic model, but the consequent term would be a function. A typical rule in a Sugeno neutrosophic model has the form

If x is Ax and y is By then z=f(x,y), 33

where Ax and By are neutrosophic sets in the antecedents, while z=f(x,y) is a crisp function in the consequent. Normally f(x,y) is a polynomial in the input variables x and y, but it can be any function as long as it can appropriately describe the output of the system within neutrosophic region specified by the antecedent of the rule.

When f(x,y) is a first order polynomial, the resulting neutrosophic inference system is called first order Sugeno neutrosophic model. When f is constant, we then have zero-order Sugeno neutrosophic model, which can be viewed either as a special case of the Mamdani neutrosophic inference system , in which each rule’s consequent is specified by a neutrosophic singleton ( or a pre-de-neutrosophied consequent), or a special case of the Tsukamoto neutrosophic model in which each rule’s consequent is specified by MF of a step function crossing at the constant.

Algorithm of working: Step 1: Neutrosophication: The inputs captured are mapped to the repository neutrosophic sets, by using equation (24) and diagrammatically illustrated in figure 2.

Step 2: Calculation of the firing strength’s of the neutrosophic rules: Consider the following case: MULIPLE RULES WITH MULIPLE ANTECEDENT A neutrosophic IF-THEN rule with two antecedent in Sugeno Neutrosophic model is usually written as : IF x is Ax and y is By THEN z is p1 x  q1 x  r1 The corresponding problem for neutrosophic reasoning is expressed as 34

Premise 1 (fact): x is Ax' and y is B y' Premise 2 (rule 1): IF x is A1x and y is B1y THEN z1 is p1 x  q1 y  r1 Premise 3 (rule 2): IF x is A2x and y is B2y THEN z2 is p 2 x  q 2 y  r2 ---------------------------------------------------------------------Consequence(conclusion) : z The rules firing strength is computed by using AND operator (usually N-norm, definition 13) with AND’ed antecedents, OR operator (usually N-conorm, definition 14) with OR’ed antecedents. So, if wa is the firing strength of rule 1, for the given fact, then  t w    x, y (T '  T ' )  (T A  TB ),  ( x (T '  T ))  ( y (T '  TB )),  w1  w2 , Ax Bx Ax A x By x y y      a    wa   i wa     x, y ( I A' x  I B ' ))  ( I Ax  I B y ),    ( x ( I A'  I Ax ))  ( y ( I B '  I B y )),   w1'  w2' , y x y  f   ( F  F ))  ( F  F )  ( ( F  F ))  ( ( F  F ))   w ''  w ''  ' 2 Ax By   Ax y By  wa   x, y A' x B y' B y'   1   x Ax

(75) Similarly firing strength of rule 2, wb can also be calculated.

Step 3: Calculation of the resultant neutrosophic set is done by performing operation of weighted average, thus generating a neutrosophic singleton instead of three output neutrosophic sets corresponding to truth, indeterminacy and falsity.  t wa z1  t wb z 2     t wa  t wb  tz   i z  i z    w 1 wb 2  z   iz    a   f   i wa  i wb   z  f z  f z  wa 1 wb 2    f wa  f wb 

(76)

Step 4: De-neutrosophication 35

As the results generated by (76) would result in a neutrosophic singleton having three components corresponding to truth, indeterminacy and falsity values respectively. So in Sugeno neutrosophic model the computationally expensive step of transforming Neutrosophic set to fuzzy set and eventual de-fuzzification is not required. Equation (72) can be directly applied on z obtained in (76) to get the final de-neutrosophied value.

4.3 TSUKAMOTO NEUTROSOPHIC MODEL Similar to Tsukamoto fuzzy models [6], in neutrosophic Tsukamoto model also the consequent of each neutrosophic rule would be represented by a neutrosophic set with a monotonical MF. As a result, the inferred output of each rule is defined as a neutrosophic singleton induced by the ruleâ&#x20AC;&#x2122;s firing strength.

Algorithm of working: Step 1 and step 2 are same as that of Sugeno neutrosophic model.

Step 3: Calculation of individual rule consequents MULIPLE RULES WITH MULIPLE ANTECEDENT A neutrosophic IF-THEN rule with two antecedent in Tsukamoto Neutrosophic model is usually written as : IF x is Ax and y is By THEN z is z1 The corresponding problem for neutrosophic reasoning is expressed as Premise 1 (fact): x is Ax' and y is B y' Premise 2 (rule 1): IF x is A1x and y is B1y THEN z1 is z1i

36

Premise 3 (rule 2): IF x is A2x and y is B2y THEN z2 is z 2i ---------------------------------------------------------------------Consequence(conclusion) : z  t z   t w  Tz  1  1i   a  z1i   i z1i    i wa  I z1  f  f F  z1   z1i   wa

(77)

Similarly z 2i , can also be computed using equation (77).

Here it can be seen that as consequent of each neutrosophic rule is represented by a neutrosophic set with a monotonical MF, so the inferred output of each rule is defined as a neutrosophic singleton induced by the rule’s firing strength.

Step 4: Calculation of the resultant neutrosophic set is done by performing operation of weighted average, thus generating a neutrosophic singleton instead of three output neutrosophic sets corresponding to truth, indeterminacy and falsity.  t wa t z1i  t wa  t  z  i    w iz z   i z    a 1i i wa f    z  f f w z  a 1i f wa 

 t wb t z 2 i    t wb   i wb i z2 i    i wb   f wb f z 2 i    f wb 

(78)

Step 4: De-neutrosophication As the results generated by (78) would result in a neutrosophic singleton having three components corresponding to truth, indeterminacy and falsity values respectively. So in Tsukamoto neutrosophic model the computationally expensive step of transforming

37

Neutrosophic set to fuzzy set and eventual de-fuzzification is not required. Equation (72) can be directly applied on z obtained in (78) to get the final de-neutrosophied value.

5. NEUTROSOPHIC MODELING In the previous section we have covered several types of neutrosophic inference systems (NISâ&#x20AC;&#x2122;s). Essentially the design of a neutrosophic inference system would be based on the past known behavior of a target system. A developed NIS should replicate the behavior of the target system. General applications of NISâ&#x20AC;&#x2122;s could be that of replacing the human operator that regulates & controls a chemical reaction, by employing NIS as a neutrosophic logic controller or more specifically designing a specialized NIS that would act as a neutrosophic expert system for medical diagnosis for simulating medical doctor reasoning.

5.1 Constructing NIS for a specific application For constructing application specific NIS, domain knowledge (in the form of linguistic variables) needs to be build by incorporating human expertise about the target system.

General system identification techniques can be utilized for neutrosophic modeling when input-output data of a target system are available (numerical data).

GENERAL GUIDELINES ABOUT NEUTROSOPHIC MODELING Step 1: First step would concentrate on defining the general behavior of the target system by means of linguistic terms.

General surface structure is identified by selecting relevant input-output variables, choosing a specific type of NIS, determining the number of linguistic terms associated with each input & output variables like for example for a Sugeno neutrosophic model, determining the order of consequent equations.

38

Step 2: This step concentrates on detailed designing of the system in which appropriate family of parameterized MF’s are identified, interviewing human experts familiar with the target systems to determine the parameters of the MF’s used in the rule base; and refining the parameters of the MF’s using regression & optimization techniques 6. PARTITION STYLES FOR NEUTROSOPHIC MODELS Similar to fuzzy models, extended neutrosophic models follow the principle of ―divide and conquer‖: wherein the antecedents of neutrosophic rules partition the input space into number of local neutrosophic regions, while the consequents describe the behaviour within a given region through various constituents. The three neutrosophic models discussed in previous section differ in their consequent terms as Mamdani and Tsukamoto neutrosophic model would have a MF as the consequent, Sugeno neutrosophic model would have either a constant (zero-order Sugeno Model) or linear equation (first-order Sugeno Model) as the consequent; but their antecedents are same. So the partitioning methods which are commonly employed by fuzzy models can be easily extended to the neutrosophic models and can be utilized by the three models discussed in the previous section.

Figure 4: Partition styles for neutrosophic models (a) Grid Partition (b) Tree Partition (c) Scatter Partition

39

6.1 Grid Partition In this type of partition, each region is included in a square area like hypercube which are mutually exclusive. Figure 4 shows the grid partitioning of two dimensional input space. This type of partitioning would be suitable for designing neutrosophic controllers as it requires less number of MFâ&#x20AC;&#x2122;s for each input, but it would be difficult to partition the input using the grid in the case of a large number of inputs because for k inputs & m MFs for each the rule base would be exploded with mk rules.

6.2 Tree Partition Figure 4 b shows tree partition. In this type of partition each region can be uniquely specified along a corresponding decision tree and it relives the problem of an exponential increase in the number of rules. This partitioning would ask for more MFs for each input for defining neutrosophic regions.

6.3 Scatter Partition Figure 4 c shows scatter partition. Each region is determined by covering a subset of the whole input space that characterizes a region of possible occurrence of the input vectors; this type of partitioning would also limit the number of rules to a reasonable amount.

7. CONCLUSIONS AND FUTURE RESEARCH Neutrosophic logic is an extended and general framework to measure the truth, indeterminacy, and falsehood-ness that closely resembles human psychological behavior. This paper proposed and presented the mathematical working of three different neutrosophic models; as an extension to already existing fuzzy models. It is 40

expected that the proposed neutrosophic models would be much more efficient and generalized in their working as compared to fuzzy models as they assign a realistic estimation of truthness, falsity and indeterminacy to every proposition that empowers them to handle fuzzy, incomplete, indeterminate and inconsistent information which fuzzy logic is not.

Neutrosophic logic being relatively a new logic and because of itâ&#x20AC;&#x2122;s inherent generalisation, can find application to wide range of domains varying from industrial process control, databases, e-commerce, stock market prediction to medical diagnosis and invariantly to any domain where information is incomplete, incoherent, inconsistent and has gaps.

Future research would focus on demonstrating the practical applications of the proposed models and giving computer simulation of the same.

REFERENCES

[1] E.H. Mamdami, S. Assilina, "An experiment in linguistic synthesis with a fuzzy logic controller", International Journal of Man-Machine Studies, vol. 7(1), pp. 1-13, 1975.

[2] F. Smarandache (1999), Linguistic Paradoxists and Tautologies, Libertas Mathematica, University of Texas fat Arlington, Vol. XIX, 143-154.

[3] F.Smarandache (1999), A Unifying Field in Logics. Neutrosophy: Neutrosophic Probability, Set and Logic. Rehoboth: American Research Press.

[4] F. Smarandache, â&#x20AC;&#x2022;N-norm and N-conorm in Neutrosophic Logic and Set,and the Neutrosophic Topologiesâ&#x20AC;&#x2013;, A Unifying Field in Logics: Neutrosophic Logic; Neutrosophic 41

Set, Neutrosophic Probability and Statistics (fourth edition), 2005; Review of the Air Force Academy, No. 1 (14), pp. 05-11, 2009.

[5] T. Takagi, M. Sugeno. ―Fuzzy Identification of Systems and Its Applications to Modeling and Control‖, IEEE Transaction Systems, Man and Cybernetics, 15(1), pp.116-132, 1985.

[6] Y. Tsukamoto, An Approach to Fuzzy Reasoning Method, Gupta M.M. et al (Eds.), Advances in Fuzzy Set Theory and Applications, pp. 137–149, 1979.

[7] Haibin Wang, Florentin Smarandache, Yanqing Zhang and Rajshekhar Sunderraman, Single Valued Neutrosophic Sets, Proceedings of the 10th International Conference on Fuzzy Theory and Technology (in conjunction with the 8th Joint Conference on Information Systems), Salt Lake City, Utah, July 21-26, 2005.

[8] L. A. Zadeh, Fuzzy sets, Inf. Control 8 (1965), 338- 353.

A.Q.Ansari is working as Professor in the Department of Electrical Engineering at Jamia Millia Islamia, New Delhi. His area of research and specialization are computer networking and data communication, image processing, networks on chip and soft computing.

42

Ranjit Biswas is working as Professor in computer science department at ITM University, Gurgaon. His area of research and specialization are rough set technology & applications, fuzzy logic & applications, soft sets & systems, AI and graph theory.

Swati Aggarwal is a pursuing Ph.D at Jamia Millia Islamia, New Delhi. She has done B.Tech (Computer Science) in 2001 and M.Tech (IT) in 2005. Currently she is working as Assistant Professor in Computer Science Department at ITM University, Gurgaon.

Her interests include neutrosophic logic, fuzzy sets, artificial

intelligence, neural network and soft computing.

43

Figure

Figure 1: Neutrosophic set A with discrete characteristic function of example 1

Figure 2: Neutrosophic set A with continuous characteristic functions of example 2

Figure 3: Schematic diagram of neutrosophic inference system