NAFIPS 2005 - 2005 Annual Meeting of the North American Fuzzy Information Processing Society

On Using Type-I Fuzzy Set Mathematics to DeriveInterval Type-2 Fuzzy Logic Systems

Jerry M. Mendel*, Robert I. John** and Feilong Liu*

* Signal and Image Processing Institute, Department of Electrical Engineering, University ofSouthern Califomia, 3740 McClintock Ave., Los Angeles, CA 90089-2564**Centre for Computational Intelligence, Department of Computer Science, Faculty of ComputingScience and Engineering, DeMontfort University, Leicester LEI 9BH, U. K.

Abstract-In this paper, we demonstrate that it isunnecessary to take the route from general type-2 fuzzy set tointerval type-2 fuzzy set, and that all of the results that areneeded to implement an interval type-2 fuzzy logic system can beobtained using type-i fuzzy set mathematics. As such, this papermakes an interval type-2 fuzzy logic system much moreaccessible to the fuzzy logic community, and we can nowdevelop an interval type-2 fuzzy logic system in a much morestraightforward way.

I. INTRODUCTIONType-2 fuzzy sets (T2 FSs), originally introduced by Zadeh[13], provide additional design degrees of freedom inMamdani and TSK fuzzy logic systems (FLS), which can bevery useful when such systems are used in situations wherelots of uncertainties are present [8]. The resulting type-2fuzzy logic systems (T2 FLS) have the potential to providebetter performance than a type-I (T1) FLS (e.g., [2], [4]-[6],[7; also, see refs. therein], [10]-[12]). To-date, because of thecomputational complexity of using a general T2 FS, mostpeople only use interval T2 FSs in a T2 FLS, the result beingan interval T2 FLS (IT2 FLS) [4]. The computationsassociated with interval T2 FSs are very manageable, whichmakes an IT2 FLS quite practical.

Unfortunately, there is a heavy educational burden even tousing an IT2 FLS, namely, one must first become proficientabout a TI FLS (this does not change as a result of thispaper), then one must become proficient about general T2FSs, operations performed upon them (T2 FS mathematics-join, meet, negation), T2 fuzzy relations (extended sup-starcomposition), and T2 FLSs, after which one can then focuson interval T2 FSs, their associated operations and relations,and IT2 FLSs, all as examples of the more general results. Toobtain such a level of proficiency, one has to make a verysignificant investment of time, something that manypracticing engineers do not have.

In retrospect, we believe that requiring a person to use T2FS mathematics represents a barrier to the use of IT2 FSsand FLSs. Here we demonstrate that it is unnecessary to takethe above route, from general T2 FS to IT2 FS, and that aUlofthe results that are needed to implement an IT2 FLS canbe obtained using Ti FS mathematics. As such, we hopethat this paper makes IT2 FLSs much more accessible to allreaders of this paper.

II. INTERVAL TYPE-2 FUZZY SETSIn this section we defme an IT2 FS and some importantassociated concepts, so as to provide a simple collection ofmathematically well-defined terms that will let us effectivelycommunicate about such sets. Our motivation is that thismaterial is used extensively in the rest of the paper. To beginwe locate an IT2 FS in the taxonomy of a general T2 FS

Def. 1: A type-2 fuzzy set, denoted i:, is characterized by atype-2 MFM,(x,u) , where x C X and u [ Jx C [0,1], i.e.,

A4 {((x,u),MyX(x,u))IF xD X,2 u o Jix L [0,1]} (1)

in which 0 LI yu,4x, u) L 1 . i can also be expressed as

' ax QD 25E7i1yX(U)/(X,U) Jx: [0,1] (2)

where [D denotes union over all admissible x and u. Fordiscrete universes of discourse D is replaced by 01 . [

Def. 2: When all yur(x, u) = 1 then A is an interval T2 FS(IT2 FS). ElAlthough the third dimension of the general T2 FS is no

longer needed because it conveys no new information aboutthe IT2 FS, the IT2 FS can still be expressed as a special caseof the general T2 FS in (2), as

43X ETJX 1 / (X, U) Jx ° [0, 1] (3)In the rest of this paper we will only be interested in IT2 FSs.Note, however, that in order to introduce the remainingwidely used terminology of a T2 FS we temporarily continueto retain the third dimension for an IT2 FS.

Def. 3: At each value of x, say x = x1l, the 2D plane whoseaxes are u and xu,xL u) is called a vertical slice ofM,ux, u) .A secondary MF is a vertical slice ofuy(x, u) . It is

yu4(x=xLu) for x02 X and I u0EJx]L [0,1],i.e.,

Y,fx=4=Xu),x)OSXC= 1J/u Jx3°0 [0,1] (4)Because 0 x00 X, we drop the prime notation on Pu,x[), andrefer to M,(x) as a secondary MF; it is a TI FS, an intervalFS, which we also refer to as a secondary set. HBased on the concept of secondary sets, we can reinterpret

an IT2 FS as the union of all secondary sets, i.e., using (4),we can re-express i in a vertical-slice manner, as:

0-7803-9187-X/05/$20.00 ©2005 IEEE. 528

A7 {(X,YX)) I D X D X} (5)or, alternatively as

,1 Lj X1(x)/x =[7 [[ 1/ul]/x Jx 5 10,1] (6)Def. 4: The domain of a secondary MF is called the

primary membership of x. In (6), Jx is the primarymembership ofx, where Jx D [0,1] for D x D X. 0

Def. 5: The amplitude of a secondary MF is called asecondary grade. The secondary grades of an IT2 FS are allequal to 1. 0

If X and JX are both discrete (either by problemformulation or by discretization of continuous universes ofdiscourse), then the right-most part of (6) can be expressed as

F] F[10 1/ uE/X = gVUlk Dxi +F +[Ee 1UNkDXXDX[1oJh D/ D[]k1 nk=1 5/

(7)Def. 6: Uncertainty in the primary memberships of an IT2

FS, i, consists of a bounded region that we call the footprintof uncertainty (FOU). It is the union- of all primarymemberships, i.e.,

FOU(XP= [D/xnx

An example of an embedded IT2 FS is depicted in Fig. 1;it is the wavy curve for which its secondary grades (notshown) are all equal to 1. Other examples ofAe are 1/ ji (x)and 1 /ljux), S x9X, where in this notation the secondary

grade equals 1 at all values of Xt)(x) and 1j) .

Def. 9: For discrete universes of discourse X and U, anembedded Tl FS A, has N elements, one eachfrom Jx, .JX2_., and JxN X namely u,,u2, ... and uN, i.e.,

N

A[=0 ui/xi ui Jx U=[0,1]i=1

(12)

Set Ae is the union of all the primary memberships of set A,N

in (11), and, there are 0] Mi A,. Note that A, acts as thes='

domain for A,. 0An example of an embedded Ti FS is depicted in Fig. 1; it

is the wavy curve. Other examples of A, are Tt)(x)andy)u(x), xDxX.

I-(8)

This is a vertical-slice representation of the FOU, becauseeach of the primary memberships is a vertical slice. 0

The shaded region on the x - u plane in Fig. I is a FOU.Because the secondary grades of an IT2 FS convey no newinformation, theFOU is a complete description ofan IT2 FS.The uniformly shaded FOU of an IT2 FS denotes that there isa uniform distribution that sits on top of it.

Def. 7: The upper membership function (UMF) and lowermembershipfunction (LMF) ofA are two Ti MFs that boundthe FOU (e.g., see Fig. 1). The UMIF is associated with theupper bound of FOU(Ag and is denoted d,(x), D xt7X, and

the LMF is associated with the lower bound of FOU(AP andis denoted yJ(x), D xDX, i. e.

Ax) D FOU(P- x 0 X (9)lj,-(x) FQU(AW OXD0 X (10)

Def. 8: For discrete universes of discourse X and U, anembedded IT2 FS A has N elements, where F containsexactly one element from J,J,X2.. and JXN , namelyu, U2,..., and UN each with a secondary grade equal to 1:

N

=0 [1lui]1xi uiDJ,D U =[0,1] (I11)i=1

N

Set ,e is embedded in A, and, there are l[J M, Xe. 0i=1

UMF(A)

- Embedded FS

LMF(A)-.4 X

Fig. 1. FOU (shaded), LMF (dashed), UMF (solid) and an embedded FS(wavy line) for 1T2 FS F.

Comparing (11) and (12), we see that the embedded IT2FS )P can be represented in terms of the embedded Ti

FSA.,as41= A, (13)

with the understanding that this means putting a secondarygrade of 1 at all points ofA.. We will make heavy use of thisnew way to represent -, in the sequel.So far we have emphasized the vertical-slice representation

(decomposition) of an IT2 FS as given in (6). Next, weprovide a different representation for such a fuzzy set that isin terms of so-called wavy slices. This representation, whichmakes very heavy use of embedded IT2 FSs (Def. 8), wasfirst presented in [9] for an arbitrary T2 FS, and is thebedrock for the rest of this paper.Theorem 1 (Representation Theorem): For an IT2 FS,

for which X and U are discrete, A is the union of all of itsembedded IT2 FSs, i.e.

(14)I For a continuous IT2 FS, although there are an uncountable-infinitenumber of embedded IT2 FSs, the concept of an embedded IT2 FS (as wellas of an embedded Ti FS (Def. 9)) is still a theoretically useful one.

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j=l

where( j=1,..., nA)N

]=L [1Iuj]/xi uJOJ,D[ U=[O,1] (15)i=l

andN

nA-LI Mi (16)i=1

in which Mi denotes the discretization levels of secondaryvariable u' at each ofthe N xi. [Comments: (1) This theorem expresses K as a union of

simpler T2 FSs, the P. They are simpler because theirsecondary MFs are singletons. Whereas (6) is a vertical slicerepresentation ofK, (14) is a wavy slice representation ofK.(2) A detailed proof of this theorem appears in [9]. Althoughit is important to have such a proof, we maintain that theresults in (14) are obvious using the following simplegeometric argument: The ME of an IT2 FS is three-dimensional (3D). Each of its embedded IT2 FSs is a 3Dwavy slice (a foil). Create all of the possible wavy slices andtake their union to reconstruct the original 3D MW. Samepoints, which occur in different wavy slices, only appearonce in the set-theoretic union.With reference to Fig. 1, (14) means collecting all of the

embedded IT2 FSs into a bundle of such T2 fuzzy sets.Equivalently,-because of (13), we can collect all of theembedded TI FSs into a bundle of such TI FSs.

Corollary 1: Because all of the secondary grades of anIT2 FS equal 1, we can also express (14) and (15) as

JP-1 / FOU(P (17)where:

nA {/X)) ... A(X)} DXL XdFOU(G=[I Aj (18)

j=and seeX)(12)X)] Oxand [see (12)/

N

Aj=] u 'ixU1Jx D U=I[O,1]i=1

(19)

The top line of (18) is for a discrete universe of discourse,Xd, and contains n, elements (functions), where nA isgiven by (16), and the bottom line is for a continuousuniverse of discourse and is an interval set of functions,meaning that it contains an uncountable number of functionsthat completely fills the space between ji4,x) LIugx)forLIxE[ X.

Proof: From (13), each AP in (14) can be expressedasl/A' ; hence,

K- /(IAi) ij A w I/FOU(AJ (20)J=1 j=l

which is (17). Note that, as already mentioned, juq,x) andt,,.x) are two legitimate elements of the nA elements of A,,'.

In fact, they are the lower and upper bounding functions,respectively for these nA functions. For discrete universes ofdiscourse, we can therefore express FOU(Ai as in the topline of (18), whereas for continuous universes of discoursewe can express FOU(A as in the bottom line of (18). IEquation (18) is a new wavy-slice representation of

FOU(AP, because all of its elements are functions, i.e. theyare wavy-slices. We will see in the sequel that we do notneed to know the explicit natures of any of the wavy slices inFOU(Xp other than ,u,x) and )i(,x).

III. INTERvAL TYPE-2 FLSA. IntroductionA general T2 FLS [7] is depicted in Fig. 2. The OutputProcessor block consists of type-reduction followed bydefuzzification. Type-reduction maps a T2 FS into a TI FS,and then defuzzification, as usual, maps that Tl FS into acrisp number. Here we assume that all the antecedent andconsequent fuzzy sets in rules are T2; however, this need notnecessarily be the case in practice. All results remain valid aslong as just one FS is T2. This means that a FLS is T2 aslong as any one of its antecedent or consequent (or input)FSs is T2.

Typ.-2 FLSOutput Processing

Rules | Defuzzifier i t* P y

a x :j i " uSet (Type-1)

Fuzzy Fuzzy_Ifrence

input sets output sets

Fig. 2. Type-2 FLS.

The structure of rules remains exactly the same in the T2case as in the TI case, but now some or all of the FSsinvolved are T2. The T2 FLS has p inputs xl ] XI,...,xp 0 Xp , and one output y L Y, and, is characterized by Mrules, where the rule has the form (I = 1,...,M )

R': xl is and F- and xp is P- THEN y is (21)When all of the antecedent and consequent T2 fuzzy sets areIT2 FSs, then we call the resulting T2 FLS an interval T2FLS (IT2 FLS). These are the FLSs that we focus on in therest of this paper.Because of space limitations, we shall focus on a single

rule (i.e., I = 1 ) that has either one antecedent or multipleantecedents and that is activated by a crisp number (i.e.,singleton fuzzification (SF)).

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----------------------------------------

B. Singleton Fuzzification (SF) and One AntecedentIn the rule "IF xl is (,THEN y is &:," let K be an IT2 FSin the discrete universe of discourse Xld for the antecedent,and : be an IT2 FS in the discrete universe of discourse Ydfor the consequent. Decompose F into nr embedded IT2FSs e (Ji = 1,..., n, ), whose domains are the embedded TI

FSs F," , and decompose ( into nG embedded IT2 FSs ((j=1,...,nG), whose domains are the embedded TI FSs GI.According to Representation Theorem 1 and Corollary 1, wesee that t and 6: can be expressed as:

nrip] PI = 1I FOU( ;}

j,=l

a set of nF, O nG functions2:, where

&4Y) = 'nf(IB(jl,j)(Y)) O] y O Yd

ft5.y) = SUP(YB(j,j)(Y)) ] y D YdDlj,,j

(29)

(30)

denote the lower bounding and upper bounding functionsofB(y), respectively.

x1

I y

G.e * IB01n)(Y)

.()) B(y)G, - p- PB(n, j) (Y) By

nG 00 PB(n; ,ng ) (Y)

Ifl

(22)

nF, nfi N.,FOU(&;=, F Fljl = R R ujl /xli '(23)

j,=1 j1=1 i=1

(ul D Jx 3 U = [0, 1] ), andnG

i1 F] de = 1I/FOU(&5 ' (24)j=1

nG nG NyFOU(dS= D Gj =]] Wj IYk (25)

j=1 j=1 k=1

(wk 0 D U =[0, 1]). Consequently, we have n, ] nGpossible combinations of embedded TI antecedent andconsequent FSs so that the totality of fired output sets for allpossible combinations of these embedded TI antecedent andconsequent FSs will be a bundle of functions B(y) asdepicted in Fig. 3, where

nF1 nG

B(y) E[ ] sB(j,,j) (Y) O y 01 Yd (26)j1=1 j=1

in which the summations denote union. The relationshipbetween the bundle offunctions B(y) in (26) and the FOU ofthe T2 fired output FS is summarized by the following:Theorem 2: The bundle of functions B(y) in (26),

computed using Ti FS mathematics, is the same as the FOUof the T2 fired output FS, which is computed using T2 FSmathematics.

Proof: From Fig. 3, we see that the fired output of thecombination of the jlth embedded TI antecedent FS and the1h embedded TI consequent FS can be computed for SFusing the well-known Mamdani implication for SF (e.g., [7]),i.e.

PB(jJ,j)(Y) = F (XPX /"G' (y) OyO Yd (27)

Since for any j, andj, uB(j,j (y) in (27) is bounded in [0, I],B(y) in (26) must also be a bounded function in [0, 1], whichmeans that (26) can be expressed as

B(y) D jS{l'Y), .**iY)} Dy [1Yd (28)

Fig. 3. Fired output FSs for all possible n1 = nF, O nG combinations of theembedded Ti antecedent and consequent FSs for a single antecedent rule.

Let j4(x,) and ,u(x1) denote the upper and lower MFs

fort, and ftdy) and _u(y) denote the upper and lower

MFs for (. Additionally, let jF (xl) and pF (X1) denote the

embedded Ti FSs associated with f,(x1,) and _u,(xlrespectively, and 1G (y) and MG (y) denote the

corresponding embedded Tl FSs of f4,,y) andy',437),respectively. From (27), we see that to compute the infimumof PB(j,1)(y) we need to choose the smallest embedded TIFS of both the antecedent and consequent, namely pF (XPand MG (y), respectively. By doing this, we obtain thefollowing equation for _,u(y):

/.F,Y) = PF, (XXo 'G (Y)' ° Y O Yd (31 )Similarly, to compute the supremum of uB(JQJ)(y), we need

to choose the largest embedded Ti FS of both the antecedentand consequent, namely,fF (xp and, G(y), respectively. Bydoing this, we obtain the following equation for,4fgy):

Mf.Y)MFj(XPOIG(Y), E y l Yd (32)Obviously, when the sample rate becomes infinite, the

sampled universes of discourse Xld and Yd can beconsidered as the continuous universes of discourse X1 andY, respectively. In this case, B(y) contains an infinite anduncountable number of elements, which will still be bounded

2We choose to call the lower and upper bounding functions in B(y) pu(y)and f(y) rather than pB (y) and PB (y) because doing so wil let us moreeasily connect our T1 FS derivation with the already known IT2 FS results.

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below and above by ,up(y) and ft,uy), respectively, wherethese functions are still given by (31) and (32) (withYd El Y ), such that (28) can be expressed as

B(y) = Bs(y), fA.ly)H, [ y O Y (33)Comparing (33) and the second line of (18), we see that

B(y) = FOU(by (34)FOU(fi= I§Sy), 4ly)7) E1yR Y (35)

and by (17) we conclude thatXI 1/FOU(#i] (36)

The combined results of (35), (36), (31) and (32) are exactlythe same as those in [4]; hence, we have been able to obtainthe FOU of the T2 fired output FS using Ti FS mathematics.U

B. SF and Multiple AntecedentsIn the rule (21) (in which I = 1, and we have omitted thisindex for notational simplicity), let . be IT2 FSsin discrete universe of discourses Xld, X2d, ..., Xpd.respectively, and ( be an IT2 FS in universe ofdiscourseYd Decompose each t into its nF (i=l,...,p)embedded IT2 FSs fe, i.e.

(37)nF,

faE-F] Pf=1/FOU(19 i= 1,...,Ipii=l

The domain of each 9 is the embedded TI FS F,,'i . As inthe preceding section, we decompose C- into nG embeddedIT2 FSs , whose domains are the embedded Ti FSs G,respectively, so (24) and (25) remain unchanged for thiscase.The Cartesian product of f...P, D9] 11L]El,

P

has ] nF, combinations of the embedded Ti FSs, Fii. Leti=l

Fe' denote the nt combination of these embedded TI FSs,i.e.

p

JY=FIJ=F' E[LFj lElnEl nandlEl EnF 38e e pe 1 Fi (38)i=1

This equation requires a combinatorial mappingfrom (ji, i2,..., jp) E n; however, in the sequel we will notneed to perform the specific mapping. All we need tounderstand is that it is theoretically possible to create such amapping. To represent this mapping explicitly, weshow(j1,ij2... I ji )E (ji (n), j2(n),..., jp (n)), so that (38) can

pbeexpressedas (1E nE] nF)

i=1

Fen = le(n) E [ Fpe(n) 1E ji(n)EnFl (39)

in which case ( x = col(x,..., xp)

MF,(X)Tr=llFJ()(Xm) = T jm (n)ElnFAdditionally, let

p

nFEl [ nFmm=l

(40)

(41)

With nG embedded Ti FSs for the consequent, we obtainnF nG1 combinations of antecedent and consequentembedded TI FSs, which generate the bundle of nF D nGfired output consequent TI FS functions, i.e. (E y E Yd1)

nF nGB(y) = F] [] YB(n,j)(Y) (42)

n=1 j=1

Theorem 2 is valid for this case, but in its proof thefollowing changes must be made:

1) In (27), instead of computing MB(j,j) (Y), we mustnow compute PB(n,j)(Y) as (El y E Yd )

MB(.,j)(Y) = Tr=lI(FJm (Xt) / i, (Y) (43)2) (28) is unchanged.3) In (29) and (30), replace index j, by the index n

( n =l,..., nF )'4) Let ft,(Xm) and ky,(Xm) denote upper and lower

MFs for. Additionally, let fF (xm) and pF (Xm)

denote embedded TI FSs associated with jii,,(xm)andyM(Xm), respectively; ftF (xm) and /F (X,)

are two of the nF embedded TI FSs associatedwithP, and will be the ones that are used in thenext step.

5) (31) and (32) are changed to (Ely E Yd)

P() = nf (yB(f,J)(y)) = Tm=i/Ir(XP; p)JG(Y) (44)

JP.Y) = Sup (UB(,jl)(y)) HTm=lfFP (x~)HJ l,G(y) (45)6 n,j

6) Equations (33)-(36) remain unchanged.

C. Other SituationsThe journal version of this paper will also describe thesituations of TI non-singleton fuzzification (NSF) and T2NSF for multiple antecedents, as well as multiple rules. In T2NSF, the number of functions in the bundle of functions in(42) increases because each of the T2 input FOUs has to bedecomposed into its own bundle of functions, i.e. the doublesummation in (42) becomes a triple summation.

D. OutputProcessingWith reference to the T2 FLS depicted in Fig. 2, we nowexplain how to perform output processing. Type-reduction,the first step of output processing computes the centroid ofan IT2 FS, where the specific IT2 FS that it does this for is

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one that is associated with the IT2 fired-rule output FSswhose formulas have just been obtained in Section B. UsingRepresentation Theorem 1 we defme the centroid, Cif, of an

IT2 FS i as the collection of the centroids of all of itsembedded IT2 FSs. From (17) and (18), we see that thismeans we need to compute the centroids of all of the nBembedded Ti FSs contained within FOU(fiq. The results ofdoing this will be a collection of nB numbers, and thesenumbers will have both a smallest and largest element,c1 (AED c, and cr (OD cr, respectively. That such numbersexist is because the centroid of each of the embedded TI FSsis a bounded number. Associated with each of these numberswill be a membership grade of 1, because the secondarygrades of an IT2 FS are all equal to 1. This means

CP= lC1/{c,,* Cr} (46)N /

c1 = min jJ)'i1 / OL (47)

Cr = max] 1 YiLi j] gi (48)

In general, there are no closed-form formulas for cl and Cr;however, Karnik and Mendel [3] have developed two verysimple and easy to implement iterative algorithms forcomputing these end-points exactly, and, they can be run inparallel. Space does not permit us to provide the details ofthese algorithms here. There are as many type-reductionmethods as there are Ti defuzzification methods becauseeach of the former is associated with one of the latter. Karnikand Mendel [3] (see, also, [7]) have developed centroid,center-of-sums, height, modified-height and center-of-setstype-reducers.

Regardless of which type-reduction method one uses,defuzzification-which follows type-reduction-is based onusing the average of cl and Cr , i.e.,

y(x) = [ cl (x) + Cr (X)] / 2 (49)Note that the explicit dependencies on x occur through thecalculations in (31) and (32). We have now completed all ofthe computations that characterize an IT2 FLS.

E. CommentIt is worth re-iterating that although we used the concept ofan embedded TI FS to derive the fired-rule output FS ofvarious kinds of IT2 FLSs, we never actually had to computethe nF I nG IB(.,J)(y), a number of computations that couldbe astronomical. Instead, in all cases, we showed that weonly need to compute two functions, ,g,(y) and f,t(y) . It isthis tremendous reduction in computations that distinguishesan IT2 FLS from a general T2 FLS. For the latter, one mustnot only compute the FOU of each fired rule but also thesecondary grade at each value of y. At present, suchcalculations are not practical, but in the future, research on

efficient ways to perform them may make them practical(e.g., [1]). Such research must not only address thecalculations of the fired-rule T2 FSs but also type-reductionfor general T2 FSs.

IV. CONCLUSIONSWe have shown that all of the results that are needed toimplement an IT2 FLS can be obtained using Ti FSmathematics. The key to doing this is the Mendel-JohnRepresentation Theorem for a T2 FS. We can now developan IT2 FLS in a much more straightforward way. Since anIT2 FLS models higher levels of uncertainty than does a TiFLS, this opens up an efficient way of developing improvedcontrol systems and for modeling human decision making.We believe that the results in this paper will make IT2

FLSs much more accessible to practitioners of FL since thetime and effort now required to learn about IT2 FLSs is verysmall. We also believe that the approach taken in this papercan be used to extend many existing Ti FS results to IT2FSs. Whether or not comparable results can be obtained forgeneral T2 FLSs is an open question.

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