Type2representationandreasoningforCWW - Semantic Scholar · Type2representationandreasoningforCWW...

Fuzzy Sets and Systems 127 (2002) 17–36www.elsevier.com/locate/fss

Type 2 representation and reasoning for CWWI. Burhan T(urk*sen ∗

Information=Intelligent Systems Laboratory, Department of Mechanical and Industrial Engineering,University of Toronto, Toronto, Ont., Canada M5S 3G8

Abstract

Computing with words (CWW) is enriched by Type 2 fuzziness. Type 2 fuzziness exists and provides a richer knowledgerepresentation and approximate reasoning for computing with words. First, it has been shown that membership functions,whether (1) they are obtained by subjective measurement experiments, such as direct or reverse rating procedures whichcaptures varying degrees of membership and hence varying meanings of words or else (2) they are obtained with theapplication of modi/ed fuzzy clustering methods, where they all reveal a scatter plot, which captures varying degrees ofmeaning for words in a fuzzy cluster. Secondly, it has been shown that the combination of linguistic values with linguisticoperators, “AND”, “OR”, “IMP”, etc., as opposed to crisp connectives that are known as t-norms and t-conorms and standardnegation, lead to the generation of Fuzzy Disjunctive and Conjunctive Canonical Forms, FDCF and FCCF, respectively. Inthis paper, we /rst discuss how one captures Type 2 representation. Then we concentrate on Type 2 reasoning that rests onType 1 representation. Next, we show how one computes Type 2 reasoning starting with Type 2 representation. It is to beforecasted that in the new millennium more and more researchers will attempt to capture Type 2 representation and developreasoning with Type 2 formulas that reveal the rich information content available in information granules, as well as exposethe risk associated with the graded representation of words and computing with words. This will entail more realistic systemmodel developments, which will help explore computing with perceptions, and computing with words by exposing graded:exibility as well as uncertainty embedded in meaning representation. Crown Copyright c© 2002 Published by ElsevierScience B.V. All rights reserved.

1. Introduction

In most current investigations, fuzzy set represen-tations and their logical combinations are based onType 1 schema for both the knowledge representationand approximate reasoning. First, Type 1 representa-tion is a “reductionist” approach for it discards thespread of membership values by averaging or curve/tting techniques and hence, camou:ages the “uncer-

∗ Tel.: +1-416-978-1298; fax: +1-416-978-3453.E-mail address: [email protected] (I.B. T(urk*sen).

tainty” embedded in the spread of membership val-ues. Therefore, Type 1 representation does not pro-vide a good approximation to meaning representationof words and does not allow computing with words aricher platform. Secondly, Type 1 approximate reason-ing relies just on the fuzzi/ed version of the “shortest”forms of the classical Boolean Normal Form formulas,with the “assumption” that the linguistic “AND” cor-responds to a t-norm and linguistic “OR” correspondsto a t-conorm in a one-to-one mapping. In Type 1 ap-proximate reasoning, “AND”ness is simply mappedto disjunctive normal form (DNF), and “OR”ness is

0165-0114/02/$ - see front matter Crown Copyright c© 2002 Published by Elsevier Science B.V. All rights reserved.PII: S 0165 -0114(01)00150 -6

18 I.B. T(urks*en / Fuzzy Sets and Systems 127 (2002) 17–36

simply mapped to conjunctive normal form (CNF),etc. This is again a reductionist, as well as, “myopic”approach to approximate reasoning. It is an old habitfrom two-valued to fuzzy-valued theory. At least twointuitive expectations come to mind in the light ofthe limitations of Type 1 representation and reasoningoutlined above.First, intuitively, it is natural to expect that a com-

bination of two imprecise Type 1 memberships shouldproduce new membership functions that capture thecompounded increase of imprecision, i.e., produce aType 2 membership function. Thus, one needs to de-velop formulas that represent Type 2 uncertainty thatarises out of the combination of Type 1 membershipswith linguistic connectives, “AND”, “OR”, “IMP”,etc.Second, again intuitively, we would expect that

the combination of two Type 2 memberships shouldproduce new membership functions that capture thecompounded increase of imprecision realizable withadditional granulation of information.

2. Sources of Type 2 fuzziness

In a historical perspective, it is to be noted that,Zadeh [49] outlined the importance of interval-valued(a special case of Type 2) fuzzy sets in decision pro-cesses. Furthermore, Zadeh [50] discussed Type 2fuzzy set representation and its potential in approxi-mate reasoning. A number of researchers began to in-vestigate Type 2 fuzzy sets and their properties afterZadeh’s introduction of this topic. Some of these are asfollows: Mizumoto and Tanaka [23] (1981); Niemien[24]; Yager [47]; Hisdal [12]; Wagenkneckt and Hart-mann [46]; Roy and Biswas [30]; Burillo and Bustince[5,6]; Bustince and Burillo [7]; John [13]; John etal. [14]; Karniak [15]; Karniak and Mendel [16,17];Liang and Mendel [19,20].There are two aspects of Type 2 fuzziness in these

investigations: First, it is not explicitly stated how theType 2 fuzzy set memberships are acquired with theexception of (i) John et al. [14], which indicates thatType 2 fuzzy sets are obtained via neuro-fuzzy clus-tering of radiographic images and (ii) Mendel et al.,which indicate that they are computed by mean �, and� of the scatter points obtained from questionnaires.Secondly, most of these works discuss axiomatic pro-

Fig. 1. Direct rating for “tall” man (subject #3).

perties and related approximate reasoning results, anduse the myopic formulas of Type-1 reasoning in de-veloping inference schemas, e.g., Mendel et al., i.e.,they do not consider FDCF and FCCF formulas inreasoning schemas.In contrast, in our investigations, we have dealt with

interval-valued Type 2 representation and reasoningstarting in the early 80s. First, in experimental stud-ies of measurement of membership functions, it wasshown by Norwich and T(urk*sen [25–27] that directmeasurement experiments, conducted in order to ex-tract membership functions for “tall” men, “pleasing”houses, reveal a Type 2 membership representationthat can be identi/ed with the mean and spread of scat-ter points, with standard deviations (Figs. 1 and 2).It should be observed that in these graphs, the spreadof scatter points come from the same subject. It is nat-ural to conjecture that the spread would be larger ifthe scatter points come from a number of subjects in ameasurement experiment. It should be noted that thesame individual when asked in an experiment givesdiQerent membership values to the same height of anindividual. The replicated experiments are designed sothat there is no memory. Type 2 fuzziness is a naturaloutcome of the measurement theory. Since individualsgive diQerent membership values, the spread of mem-bership values re:ects on the one hand the uncertainty

I.B. T(urks*en / Fuzzy Sets and Systems 127 (2002) 17–36 19

Fig. 2. Direct rating for “pleasing” house (subject #1).

associated in the meaning of a word, say, “tall”, androbustness association with meaning representation oflinguistic variables.Secondly, in modi/ed FCM experiments which we

have began to carry out recently, we have observedthat the dilemma concerning the choice of (m; c)pair, can be resolved in the following way. Instead

Fig. 3. Type 2 fuzzy sets: (a) a set of membership functions of a cluster; (b) interval-valued membership functions of a cluster.

of searching the optimal (m; c) pair (m∗; c∗), we canidentify a set of m’s, {m} that minimize a c, sayc∗. Thus having identi/ed c∗, we can then choose alower value, m∗

L =minm{m; c∗} and an upper valuem∗U =maxm{m; c∗}.Thus, for diQerent levels of fuzziness, i.e., order of

fuzzy overlaps,m’s, we get a scatter plot that gives us aground for Type 2 representation. This means that wecan acquire Type 2membership functions in inductive,unsupervized, learning experiments with training data(Fig. 3).Before we continue, it should be noted that there are

essentially two types of Type 2 fuzziness: (i) interval-valued Type 2 and (ii) full Type 2.

(i) Interval-valued Type 2 fuzziness is a specialType 2 fuzziness where the upper and lower boundsof membership are identi/ed and the spread of mem-bership, either fuzzy or probabilistic and where thespread of membership distribution is ignored with theassumption that membership values between upperand lower values are uniformly distributed or scatteredwith a membership value of “1” on the �(�(·)) axisas shown in Fig. 4. Thus, the upper and lower boundsof interval-valued Type 2 fuzziness specify the rangeof uncertainty about the membership values.(ii) Full Type 2 fuzziness identi/es upper and

lower membership values as well as the spread of


Fig. 4. Interval-valued Type 2.

Fig. 5. Full Type 2.

membership values between these bounds either prob-abilistically or fuzzily. That is there is a probabilisticor possibilistic distribution of membership values thatare between upper and lower bound of membershipvalues in the �(�(·)) axis as shown in Fig. 5. Thus,

we obtain a graded distribution of uncertainty betweenthe bounds.

3. Interval-valued fuzzy sets

In conjunction with Type 2 fuzziness found in mea-surement experiments, interval-valued fuzzy sets werediscovered in combination of concepts and hence ap-proximate reasoning by T(urk*sen [32–35] where it wasshown that disjunctive and conjunctive normal forms(DNF) and (CNF) of Boolean logic are no longerequivalent due to the fact that the law of excludedmiddle (LEM), and its dual the law of contradiction(LC) do not hold in fuzzy set and logic theory whichwere relaxed by Zadeh [48] in his seminal paper.At the same time, it was pointed out that linguistic

operators, “AND”, “OR”, etc., are fuzzy operators andthey do not directly correspond to one of the t-normsand t-conorms, which are crisp operators. Linguisticoperators “AND”, “OR”, etc., being inherently fuzzy,need to be interpreted in a graded manner, whereast-norms, T’s, and t-conorms, �’s, are crisp opera-tors that combine singleton fuzzy degrees of informa-tion granules and give a singleton fuzzy degree, i.e.,�1T�2 = �3 or �1��2 = �4.In fact, Zimmerman and Zysno [53] had shown

that human use of “AND”, “OR” do not directlycorrespond to a t-norm, such as min, or algebraicproduct and to a t-conorm, such as max or algebraicsum, respectively. They had proposed “Compensatory‘AND’”, which either linearly or exponentially com-bines a t-norm and a t-conorm, such as min–max,or algebraic product and sum with a compensationoperator, �; i.e.,

(i) Convex linear compensation:

F1(�A; �B; �1) = �1(�A ∧ �B)

+(1− �1)(�A ∨ �B)

or(ii) Exponential compensation:

F2(�A; �B; �2) = (�A ∧ �B)�2 (�A ∨ �B)(1−�2);

where 06�1; �261.Later, T(urk*sen [37] showed that in fact � opera-

tors of Zimmermann and Zysno [53] were compen-sation weights that combined DNF and CNF values


of “AND”ness, “OR”ness that T(urk*sen discovered in1986. That is, the degrees of “AND”ness should becaptured, for the case of complex linear compensationas

FLAND(�A; �B;�LAND)

= �LAND�DNF(AAND B) + (1− �LAND)�CNF(AAND B);

where “LAND” stands for linear ‘AND’ compensa-tion; the degrees of “OR”ness are captured as

FLOR(�A; �B; �LOR)

= �LOR�DNF(AOR B) + (1− �LOR)�CNF(AOR B);

where “LOR” stands for linear ‘OR’ compensation;and the degrees of “AND=OR”ness are captured as

FL(AND=OR)(�A; �B; �L(AND=OR))

= �L(AND=OR)�CNF(AAND B)

+(1− �L(AND=OR))�DNF(AOR B);

where “L(AND=OR)” stands for linear ‘AND=OR’compensation.In a similar manner, the degrees of “AND”ness for

the exponential compensation are captured as

FEAND(�A; �B; �EAND)

= (�DNF(AAND B))�EAND(�CNF(AAND B))(1−�EAND);

where ‘EAND’ stands for exponential ‘AND’ com-pensation; the degrees of “OR”ness are captured as

FEOR(�A; �B; �EOR)

(�DNF(AOR B))�EOR (�CNF(AOR B))(1−�EOR);

where ‘EOR’ stands for exponential ‘OR’ compensa-tion; and the degrees of “AND=OR”ness are capturedas

FE(AND=OR)(�A; �B; �E(AND=OR))

=(�CNF(AAND B))�E(AND=OR) (�DNF(AOR B))(1−�E(AND=OR));

where ‘E(AND=OR)’ stands for exponential ‘AND=OR’ compensation.These investigations were conducted within the

spirit of the “reductionist” approach and therefore

reduced Type-2 uncertainty generated by DNF andCNF formulae to Type-1 membership functions withthe assignment of � operators. In Fig. 6, “exponentialcompensatory ‘AND’” is shown for �a=0:8, and�b=0:6.But there remained a fundamental question that

needed to be answered from a theoretical point ofview. It was known that Boolean formulae, DNF(·)≡ (·) CNF (·) for all 16 possible combinationsof any two two-valued sets. But, it was shown thatBoolean DNF and CNF formulae are no longer equiv-alent, when used with fuzzy membership values [35].That is, they give results such that DNF (·) �=

CNF (·), where (·) stands for any of the sixteencombinations of any two linguistic values of any twolinguistic variables (see Table 1). In fact, it is alsoshown that DNF⊆CNF for the well-known casesof t-norm and t-conorm based De Morgan Triplesformed with standard negation such as (∧;∨;−),(⊗;⊕;−), (L⊗; L⊕;−), (D⊗; D⊕;−), where (∧;∨)stand for min–max, (⊗;⊕) stand for algebraic productand sum, (L⊗; L⊕) stand for Lukasiewicz intersectionand union, and (D⊗; D⊕) stand for drastic intersectionand union operators [40].The fundamental question at this juncture was

whether or not fuzzy disjunctive and conjunctivecanonical forms can be derived from fuzzy truth ta-bles. In 1970, it was argued that fuzzy truth tablescould not be formed [21] for it would require a tablewith in/nite number of rows. In a series of papers[38–42], it was progressively shown that fuzzy truthtables can be constructed and fuzzy disjunctive andconjunctive canonical forms (FDCF) and (FCCF)can be derived from the fuzzy truth tables. Morerecently, Resconi and T(urk*sen [29] have shown thatthere are deeper reasons why FDCF and FCCF arerealized.Along the way, other researchers began to investi-

gate such topics “AND=OR” intervals of possible de-grees of belief [54], and Fuzzy normal forms [8,9].

4. FDCF and FCCF expressions

The application of the normal form derivation al-gorithm [see Appendix A] gives us the derivationof FDCF and FCCF expressions for “A AND B”,shown in row 6 of Table 1, from the fuzzy truth table,


Fig. 6. IVFS “A AND B”, “A OR B”, expressed in DNF and CNF and “exponential compensatory AND” using algebraic sum-product DeMorgan Triple, (⊕;⊗;−) for realization of t-conorm and t-norm for (�A =0:8; �B =0:6 and �∈ [0; 1].

Table 5, as

FDCF (A AND B)

= (A ∩ B); A ⊇ B XOR A ⊂ B is T;

FCCF (A AND B)

= (A ∪ B) ∩ (B ∪ c(A) ∩ (c(B) ∪ A);

A ⊇ B XOR A ⊂ B is T

for all t-norm, t-conorm and standard complement-based De Morgan Triples that correspond to symbolicset operation “∩”, “∪”, “c” in the numerical domain,respectively.

As well, with an appropriate modi/cation of the lastcolumn of Table 5, we can drive FDCF and FCCFexpressions shown in Table 2 for all the remaining 15meta-linguistic combinations shown in Table 1. Forexample for “A OR B”, we get

FDCF (A OR B)

= (A ∩ B) ∪ (A ∩ c(B)) ∪ (c(A) ∩ B);

A ⊇ B; XOR A ⊂ B is T;

FCCF (A OR B)

= (A ∪ B); A ⊇ B XOR A ⊂ B is T:


Table 1Meta-linguistic expressions of combined concepts for any A and B

No. Meta-linguistic expressions

1 UNIVERSE2 EMPTY SET3 A OR B4 NOT A AND NOT B5 NOT A OR NOT B6 A AND B7 A IMPLIES B8 A AND NOT B9 A OR NOT B10 NOT A AND B11 A IF AND ONLY IF B12 A EXCLUSIVE OR B13 A14 NOT A15 B16 NOT B

For another example, for “A IMPLIES B”, (A → B),we get

FDCF (A → B)

= (A ∩ B) ∪ (c(A) ∩ B) ∪ (c(A) ∩ c(B));

A ⊆ B; XOR A ⊂ B is T;

FCCF (A → B)

= (c(A) ∪ B); A ⊇ B; XOR A ⊂ B is T:

Again these formulae hold, true for all t-norm, t-conorm and standard complementation operations inthe numerical domain that correspond to symbolic setoperations “∩”, “∪”, and “c”, respectively.It should be noted that for pseudo t-norm, t-conorm

and standard complementation, we need to abide bythe order of A ⊇ B XOR A ⊂ B and adjust all theformulae accordingly, since they are not commutativeand associative.

5. Impact of FDCF and FCCF

The discovery that we ought to use FDCF and FCCFin fuzzy combination of linguistic concepts and henceapproximate reasoning, opens up at least three avenuesfor the formation of the combination of linguistic con-

Table 2Classical disjunctive normal and fuzzy disjunctive canonical forms(DNF) and (FDCNF) and classic conjunctive normal and fuzzyconjunctive canonical forms (CNF) and (FCCF), where ∩ is aconjunction, ∪ is a disjunctive and c is a complementary operator

No. Fuzzy disjunctive canonical forms=disjunctive normal forms

1 (A∩B)∪ (A∩ c(B))∪ (c(A)∩B)∪ (c(A)∩ c(B))2 �3 (A∩B)∪ (A∩ c(B))∪ (c(A)∩B)4 (c(A)∩ c(B))5 (A∩ c(B))∪ (c(A)∩B)∪ (c(A)∩ c(B))6 (A∩B)7 (A∩B)∪ (c(A)∩B)∪ (c(A)∩ c(B))8 (A∩ c(B))9 (A∩B)∪ (A∩ c(B))∪ (c(A)∩ c(B))10 (c(A)∩B)11 (A∩B)∪ (c(A)∩ c(B))12 (A∩ c(B))∪ (c(A)∩B)13 (A∩B)∪ (A∩ c(B))14 (c(A)∩B)∪ (c(A)∩ c(B))15 (A∩B)∪ (c(A)∩B)16 (A∩ c(B))∪ (c(A)∩ c(B))

No. Fuzzy conjunctive canonical forms=conjunctive normal forms

1 I2 (A∪B)∩ (A∪ c(B))∩ (c(A)∪B)∩ (c(A)∪ c(B))3 (A∪B)4 (A∪ c(B))∩ (c(A)∪B)∩ (c(A)∪ (c(B))5 (c(A)∪ c(B))6 (A∪B)∩ (A∪ c(B))∩ (c(A)∪B)7 (c(A)∪B)8 (A∪B)∩ (A∪ c(B))∩ (c(A)∪ c(B))9 (A∪ c(B))10 (A∪B)∩ (c(A)∪B)∩ (c(A)∪ c(B))11 (A∪ c(B))∩ (c(A)∪B)12 (A∪B)∩ (c(A)∪ c(B))13 (A∪B)∩ (A∪ c(B))14 (c(A)∪B)∩ (c(A)∪ c(B))15 (A∪B)∩ (c(A)∪B)16 (A∪ c(B))∩ (c(A)∪ c(B))

cepts and hence approximate reasoning as possible ap-proaches to computing with words:(i) If we start out with Type 1 membership functions

in knowledge representation, say two fuzzy sets A andB, then the combination of these two Type 1 fuzzysets produce Type 2 fuzzy sets with FDCF⊆FCCF. InFig. 7, an example of linguistic “OR” combination oftwo Type 1 fuzzy membership values are shown with�FDCF(AOR B)(�1; �2) and �FCCF(AOR B)(�1; �2).


Fig. 7. �(�1; �2) for 〈⊕;⊗;−〉 OR combination of two Type 1 memberships say �1 = �A(x), and �2 = �B(y) where �1; �2 ∈ [0; 1],x∈X; y∈ Y , and A; B two fuzzy sets.

Furthermore, combinations of more than two fuzzysets generate 2N−1 membership functions in a binarytree, where N is the member of linguistic terms. Forexample, for the combination of three fuzzy sets withType 1 membership functions, we have, 23−1 = 4membership functions:

(1) FDCF(FDCF(�1; �2); �3);(2) FDCF(FCCF(�1; �2); �3);(3) FCCF(FDCF(�1; �2); �3);(4) FCCF(FCCF(�1; �2); �3); etc:

As it should be appreciated, in this case, when thereare more than two fuzzy sets with Type 1 membershipfunctions, the consequent of the combined linguisticterms approximately becomes Full Type 2 fuzzy sets.(ii) If we start out with interval-valued Type 2 fuzzy

sets in knowledge representation, say two fuzzy sets,A and B then the combination of these two interval-

valued Type 2 fuzzy sets, an approximate Full Type 2fuzzy set with eight elements is obtained as follows:

(1) FDCF(�1L; �2L); (2) FDCF(�1L; �2U);(3) FDCF(�1U; �2L); (4) FDCF(�1U; �2U);(5) FCCF(�1L; �2L); (6) FCCF(�1L; �2U);(7) FCCF(�1U; �2L); (8) FCCF(�1U; �2U);

where the pair (�1L; �1U) de/nes the interval-valuedType 2 fuzzy set, say, A, and (�2L; �2U) de/nes theinterval-valued Type 2 fuzzy set, say, B. Naturally, thecombination of more than two interval-valued fuzzysets generate 2N−1 · 2N membership values, where Nis the number of linguistic terms.(iii) If we start out with Full Type 2 fuzzy sets

in knowledge representation, say for two fuzzy sets,A and B, then the combination of these two FullType 2 fuzzy sets produce Type 3 fuzzy sets withFDCF �=FCCF computed on �(�1) and �(�2). Con-


Fig. 8. Type 3 fuzzy set memberships generated on Type 2 memberships with FDFC and FCCF.

sequently, we would obtain �FDCF(�(�1); �(�2)) and�FCCF(�(�1); �(�2)) as Type 3 fuzzy sets for all points�(�1)∈ [0; 1] and �(�2)∈ [0; 1].It should be noted that Karniak and Mendel

[16,17], Luang and Mendel [20] only compute�FDCF(�(�1); �(�2)) and ignore �FCCF(�(�1); �(�2))in computing “AND”ness, etc. On the other hand,they compute �FCCF(�(�1); �(�2)), but ignore�FDCF(�(�1); �(�2)) in computing “OR”ness. That is,they use the “reductionist”,(myopic) formulas, i.e.,short form of the Boolean formulas for “AND”nessand “OR”ness.It is clear that Type 1–Type 2 and Type 2–Type 3

membership generation created by FDCF and FCCF

formulas create computational complexity. One wayto simplify these, after the application of FDCF andFCCF between any two fuzzy sets with either Type 1or interval-valued Type 2 or Full Type 2 membershipfunctions, we may discard the values generated be-tween upper and lower membership values generatedin �(�) space for Type 2 generation and �(�(�)) spacefor Type 3 generation, and hence reduce the com-putational complexity to the computation of interval-valued Type 2 and interval-valued Type 3 membershipgeneration, if and when, we cannot aQord extendedcomputing time and eQort. For some applications, thismay be justi/ed. A skeleton of Type 3 membershipgeneration is shown in Fig. 8.


Fig. 9. Various representation theorems for membership functions.

5.1. Type-2 knowledge representation

There are two approaches to obtain Type-2 knowl-edge representation: (i) Membership measurementexperiments conducted with experts and (2) Fuzzycluster experiments conducted with FCM algorithm.Measurement of membership. Membership mea-

surement experiments conducted with experts, i.e.,subjects, expose the need to represent knowledge withType-2 fuzzy sets as depicted in Figs. 1 and 2.We havepresented various aspects of measurement of member-ship both theoretically and experimentally [1–4,25–28,31–33,36,43].Brie:y, /rst it is worthwhile to note that measure-

ment theory deals with representation and uniquenessfrom empirical relational domain to numerical rela-tional domain. A particular interest in this regard is thescale strength of the numerical representation depend-ing on which axioms of measurement theory can bevalidated for a particular experimental data obtainedfrom experts. In this regard, it is to be realized that var-

ious representation theorems associated with variousquantitative structures generate algebraic representa-tion from ordinal to absolute scale strengths (Fig. 9).Secondly, issues of the scale strength aside, analysts

generally get a scatter plot of expert responses. In or-der to capture variability of these responses, and theirinformation granules, one needs to represent member-ship values in Type-2 schemas (Figs. 1–5).This becomes particularly important from the per-

spective of computing with perceptions and wordsparadigm [51,52]. In this perspective it is essential thatthe meaning of words, linguistic terms of linguisticvariables are represented eQectively with membershipgrades of the membership, i.e., variation of member-ship values.Type 2 representation exposes the variation of mem-

bership values and causes a more richer and robustmeaning representation of words and their computa-tion.Fuzzy clustering. In applications of fuzzy cluster-

ing algorithm, FCM, a major concern is the selection


of (m; c) pairs that provide “good” clusters of linguis-tic values of linguistic variables. A general heuristicis to choose m=2, level of fuzziness, and then de-termine the optimal c, the number of clusters. Thenone determines in general a Type 1 membership overthese fuzzy clusters with curve /tting techniques.Another approach is to search through 1¡m¡∞,

the level of fuzziness, and c=2; : : : ; 10; : : : ; thenumber of possible clusters, in order to /nd an op-timal pair (m; c) that will give a “good” systemmodel.Recently, we have conducted experiments which

show that for certain c-values there are manym-valuesthat cause minimization of the cluster validity indexfor certain c’s. This has led us to choose a c-valuethat minimizes the cluster validity index and developsType-2 fuzziness around the chosen c-value with allthe m-values that give minimization for that particularc. In particular, let c∗ be the number of clusters forwhich there are a set of m values, {m}. In such a case,choose mL =minm{(m; c∗)} and mU =max{(m; c∗)}.Next, obtain all the scatter plots for allm∈{(mL; c∗);

: : : ; (mU; c∗)}. First /t a curve to the scatter points of(mL; c∗) to determine the lower bound membershipfunction and then /t a curve to the scatter points of(mU; c∗) to determine the upper bound membershipfunction for an interval-valued Type 2 representationin �-space. For Full Type 2 representation, one needsto determine the possibilistic or probabilistic distribu-tion function of Type 2 membership in �(�) space.This also requires that a curve be /tted to all the scat-ter points obtained from m∈{(mL; c∗); : : : ; (mU; c∗)}.Naturally, there are various alternatives. They de-

pend on whether scatter points demonstrate a par-ticular shape or not or whether the analyst needs tosimplify the fuzzy set representation to be triangular,trapezoidal, S or � curves or even some form of ex-ponential. Other simpli/cations can be made by com-puting mean values and variances with the statisticalmethods and the curve may be simpli/ed with statis-tical methods, etc. These details are application andcontext dependent and should be chosen by the sys-tem analyst.In this manner, Type-2 fuzziness is identi/ed

with unsupervised learning and with the applicationof FCM. This also provides a more reasonable ap-proach to determine suitable (m; c) pairs. This can beobserved in Fig. 3, as shown before.

The determination of Type-2 fuzziness with FCMalgorithm also settles the scale strength issue raisedin measurement theory. That is, if input data that isused in FCM algorithm is on the absolute scale, thenthe resulting membership values in Type-2 represen-tation turn out to be on the absolute scale due to thecomputations required in the FCM algorithm.

5.2. Type 2- approximate reasoning

Type 2 approximate reasoning can be computed intwo ways as follows: (1) Interval-valued Type 2 rea-soning where we reduce the Type 2 uncertainty bycomputing FDCF and FCCF formulas only for thedetermination of upper and lower values of Type 2memberships; and (2) Full Type 2 reasoning where allavailable grades between upper and lower member-ship values are combined in a pairwise combination.The /rst approach is partially reductionist, but com-

putationally eWcient in comparison to “Full” Type 2computations. Whereas, the second approach leads tocomputational complexity of a higher order as dis-cussed previously. In Table 2, the FDCF and FCCFformulas are shown. A detailed discussion and deriva-tion of FDCF and FCCFmay be found in T(urk*sen [40].Here, we brie:y review the derivation of FDCF andFCCF formulae.In Table 1, meta-linguistic expressions of combined

concepts are shown for any two linguistic conceptsA and B whether they are represented by two or in-/nite (fuzzy) valued sets. In Table 2, classical andfuzzy canonical forms are shown under the headings offuzzy disjunctive canonical forms, FDCF=disjunctivenormal forms, DNF, and fuzzy conjunctive canoni-cal forms, FCCF=conjunctive normal forms, CNF, forfuzzy=classical logic, respectively. The derivation ofclassical normal forms are based on Classical TruthTable, Table 3. The equivalence of classical formula,i.e., DNF(·)≡CNF(·) for all the 16 possible combina-tions can be shown to hold with the axioms of classi-cal set and logic theory operations. It should be notedthat the set operations, ∩, ∪, are implicitly assumed tobe two-valued, i.e., in {0; 1}, whereas in Table 3, theirtruthfullness are explicitly shown to be two-valued,i.e., {T; F}. Also, in Table 4, “Axioms of classicalset and logic operations”, the set operations, ∩, ∪, areagain implicitly assumed to be two-valued. It shouldbe pointed out that in Tables 1–4 there is no explicit


Table 3Classical truth table interpretations of “A AND B”

Truth assignments to classical Truth assignments to the Primary conjunctionsmeta-linguistic variables meta-linguistic expression

A B “A AND B”

T (A) T (B) T (A AND B) A∩BT (A) F(B) F(A AND B) A∩ c(B)F(A) T (B) F(A AND B) c(A)∩BF(A) F(B) F(A AND B) c(A)∩ c(B)

Table 4Axioms of classical and set and logic operations

Involution c(c(A)=ACommutativity A∪B=B∪A

A∩B=B∩AAssociativity (A∪B)∪C =A∪ (B∪C)

(A∩B)∩C =A∩ (B∩C)Distributivity A∪ (B∩C)= (A∪B)∩ (A∪C)

A∩ (B∪C)= (A∩B)∪ (A∩C)Idempotence A∪A=A

A∩A=AAbsorption A∪ (A∩B)=A

A∩ (A∪B)=AAbsorption by X and � A∪X =X

A∩�=�Identity A∪�=A

A∩X =ALaw of contradiction A∩ c(A)=�Law of excluded middle A∪ c(A)=XDe Morgan’s laws c(A∩B)= c(A)∪ c(B)

c(A∪B)= c(A)∩ c(B)

indication as to whether sets A and B are two valuedor not.It is shown, as pointed out previously, that just by

substituting values in [0; 1] as opposed to {0; 1} forthe set membership, we get DNF(·) �=CNF(·) [35].Later, it was shown that /rst, the classical truth tablecan be extended to a fuzzy truth table, Table 5, withthe realization that there is an order relationship be-tween the membership values of fuzzy set A, a, andthe membership values of fuzzy set B, b, i.e., a¿b,and a¡b, a; b ∈ [0; 1].In the formation of fuzzy truth Table 5, it should be

noted that there, fuzzy set membership values are nowshown explicitly to be in the interval [0; 1] in contrast

Table 5The “fuzzy truth table” of “A AND B” where a; b 0 [0; 1]

A B A AND B

(1:1:1) a¿b T (2:1:1) a¿b T T(1:1:1) a¿b T (2:2:1) a¡b F F(1:2:1) a¡b F (2:1:1) a¿b T F(1:2:1) a¡b F (2:2:1) a¡b F F(1:1:2) a¡b T (2:1:2) a¡b T T(1:1:2) a¡b T (2:2:2) a¿b F F(1:2:2) a¿b F (2:1:2) a¡b T F(1:2:2) a¿b F (2:2:2) a¿b F F

to Table 3, but the truthfullness of the order relation-ship a¿b, and a¡b are shown to be two-valued as{T; F} [38].When we apply the same canonical form derivation

algorithm (see Appendix A), we obtain the same ex-pression, but they are equivalent in form only, not incontent. We discuss this next.

6. Containment vs. equivalence

In classical theory, we have DNF(·)≡CNF(·) forall 16 cases shown in Tables 1 and 2. It is shown thatFDNF(·)⊆CNF(·) for all 16 cases in fuzzy theory forthe four well-known de Morgan Triples [35,40]. Thisis now demonstrated very brie:y. For this purpose, wewill /rst show the equivalence of DNF(·)≡CNF(·),and then FDNF(·)⊆CNF(·).Equivalence of DNF and CNF

In classical theory, we have for example:

CNF(A AND B)≡DNF(A AND B)


To show this, we start out with the expres-sions:

DNF (A AND B)=A∩BCNF (A AND B)= (A∪B)∩ (A∪ c(B))∩ (c(A)∪B)

Next, we apply certain axioms to CNF (·), asfollows:

(idempotency and distributivity)= [A∪ (B∩ c(B))]∩ [B∪ (A∩ c(A))]

(law of contradiction)= [A∪∅]∩ [B∪∅](identity; boundary)=A∩B

In a similar manner for the other 15 possible combi-nations of concepts shown in Table 1, it can be shownthat DNF(·)≡CNF(·) in two-valued set and logic.Containment of FCNF and FCCF

In fuzzy theory, we have

FDCF(A AND B)⊆FCCF(A AND B)

To show this, we start out again with the expres-sions:

FDCF (A AND B)=A∩BFCCF (A AND B)= (A∪B)∩ (A∪ c(B))∩ (c(A)∪B)

As indicated, the expressions are the same but inform only. In general, for the class of t-norms and t-conorms and standard complement DeMorgan Triplesthat correspond to the prepositional set operators, “∩”,“∪”, “c”, respectively. The axioms of idempotency,distributivity and the law of contradiction, shown inTable 4, are not applicable in fuzzy theory. It is be-cause these axioms are relaxed in general in fuzzy the-ory that we obtain the containment of FDCF in FCCF.Next, let us demonstrate this containment relation-

ship as an example for the case of “A OR B” combi-nation and this time in the numerical domain wheret-norm, �, and t-conorm, � operators are applicable.That is, we need to show that

FDCF (A OR B)⊆FCCF (A OR B); i:e:;

(A ∩ B) ∪ (c(A) ∩ B) ∪ (A ∩ c(B)) ⊆ A ∪ B:

In the numerical membership domain, we need toshow that

�[�(a; b); �( Xa; b)]6 b and (1)

�(a; Xb)6 a (2)

hold in general where Xa=1 − a, Xb=1 − b, anda; b∈ [0; 1]. While (2) is true in general for all t-norms, (1) requires a bit of work, but can be shownto hold for the four well-known De Morgan Triplesas will be done shortly below. If (1) and (2) hold, itis true that (3) holds.

�[�[�( Xa; b); �(a; Xb)]; �(a; b)]6 �(a; b): (3)

Next, we show that (1) holds for well-known casesof t-norms and t-conorms. The following statementsare true in general for the class of t-norms and t-conorms. First, it is well known that (4) and (5) aretrue:

aD⊗b6 aL⊗b6 a⊗ b6 a ∧ b: (4)

Next, it is also well known that

a ∨ b6 a⊕ b6 aL⊕b6 aD⊕b: (5)

Consider next the following inequalities, which are alltrue:

�[�(a; b); �( Xa; b)]6 b

(for � = V; � = ∧; and −); (6)

�[�(a; b); �( Xa; b)]6 b

(for � = ⊕; � = ⊗; and −); (7)

�[�(a; b); �( Xa; b)]6 b

(for � = L⊕; � = L⊗; and −); (8)

�[�(a; b); �( Xa; b)]6 b

(for � = D⊕; � = D⊗; and −): (9)

In expressions (3), (4), (6), (7), (8), (9), D⊗, L⊗,⊗, and ∧ are “drastic”, Lukasiewicz, “algebraic” and“minimum” intersection operators, respectively. Also,D⊕, L⊕, ⊕, and V are “drastic”, Lukasiewicz, “alge-braic”, and “maximum” union operators, respectively.Since (2) holds in general for all t-norms

and conorms then it is clear that for the well-known cases discussed above, we have FDCF(A AND B)⊆FCCF (A AND B)!


Also, it can be shown that this containment holdstrue for all 16 combinations of concepts shown inTable 1 for the well known-cases of t-norms andt-conorms discussed above. It is also clear that specialclasses of t-norm and t-conorms that are transformedto one of these well-known t-norms and t-conormsalso have the containment relationship. However, ageneral proof for all t-norms and conorms have notbeen shown as yet. This is left as an open questionfor researchers!

7. Comprimized reasoning

Yager and Filev (1994) proposed “ComprimizedReasoning” as the convex linear combination of thetwo extreme reasoning methods that are used in thecurrent literature as F(y)= (1− )FM(y)+ FGMP(y)where FM(y) is the consequence of Mamdani-type reasoning and FGMP(y) is the consequence ofGMP-type reasoning. It should be recalled that inMamdani-type product reasoning, the implicationA→B=A×B and in GMP-reasoning, the implica-tion A→B= c(A)∪B are used.Naturally, for a singleton observation

A′(x)= 1 for x= x0, we get

A′(x0) ◦ (A(x) → B(y)) = A(x0)TB(y)

in Mamdani-type reasoning, and

A′(x0) ◦ (A(x) → B(y)) = c(A(x0)�B(y)

in GMP-type reasoning.We observe that

FDCF (A AND B) = A ∩ B;

FCCF (A → B) = c(A) ∪ B:

Therefore, we can write

A(x0)TB(y)

6 (1− )[A(x0)�B(y)] + [c(A(x0))�B(y)]

6 c(A(x0))�B(y) for ∈ [0; 1]:

But, we need to show that A∩B⊆ c(A)∪B in gen-eral in order to prove that values can be found ei-ther experimentally or computationally via supervisedlearning.

8. FDCF(AND)⊆ FCCF(→)

Let us show that FDCF (A AND B) is includedin FCCF (A→B) in a similar manner such thatFDCF (A AND B)⊆FCCF (A OR B) which is shownin T(urk*sen [37]. However, we will show the alternateproof here.Recall that

FDCF (A AND B) = A ∩ B;

FCCF (A AND B)

= (A ∪ B) ∩ (c(A) ∪ B) ∩ (A ∪ c(B));

FDCF (A → B)

= (A ∩ B) ∪ (c(A) ∩ B) ∪ (c(A) ∩ c(B));

FCCF (A → B) = c(A) ∪ B:

With the general result of subsection previouslydiscussed, it can be shown that

FDCF (A AND B) ⊆ FCCF (A AND B) and

FDCF (A → B) ⊆ FCCF (A → B)

for the four well-known t-norms and t-conorms.Also, it is clear that

FDCF (A AND B) ⊆ FCCF (A → B):

for all t-norms and t-conorms.Next, it is clear that, for some but not all t-norms

and t-conorms

FCCF (A AND B) ⊆ FDCF (A → B):

However, it can be shown that

FDCF (A AND B) ⊆ FDCF (A → B):

Since,

A ∩ B ⊆ (A ∩ B) ∪ (c(A) ∩ B) ∪ (c(A) ∩ c(B)):

These arguments are graphically shown in Fig. 10.Therefore, the “convex-linear-comprimized-rea-

soning”

A(x0)TB(y)

6 (1− )[A(x0)TB(y)] + [c(A(x0))�B(y)]

6 c(A(x0))�B(y); (10)


Fig. 10. The lattice diagram showing the containment relationship between FDCF(AND), FCCF(AND) and FDCF(→) and FCCF(→).

Fig. 11. Graph of convex linear compromize reasoning �= (1−�) (A(x∗)⊗B(y))+�(c(A(x∗))⊕B(y)) for A(x∗)= (0:8) and B(y)= (0:6)for algebraic operators.

where “T” represents a t-norm and “�” represents at-conorm.Therefore, inequality (10) represents all grada-

tions between the two boundaries of Mamdani-typereasoning identi/ed by FDCF (A AND B) and

FCCF (A AND B), and the two boundaries of GMP-type reasoning identi/ed by FDCF (A→B) andFCCF (A→B).This “convex-linear-comprimize-reasoning” is

demonstrated graphically in Figs. 11 and 12 for a


Fig. 12. Graph of convex linear compromize reasoning �= (1−�) (A(x∗)L⊗B(y))+�(c(A(x∗))L⊕B(y)) for A(x∗)= (0:8) and B(y)= (0:6)for bold operators.

particular A(x) and B(y) and for algebraic and boldoperators.It is to be observed that the upper and lower bounds

of material and product implications are equal to eachother for bold operators (Fig. 12).

9. Exponential-comprimized-reasoning

In analogy to the “linear-comprimized”, we nextproposed the “exponential-comprimized-reasoning”.That is,

F(y) = [FM(y)]1−�[FGMP(y)]�:

It can be shown that

A(x0)TB(y)6 [A(x0)TB(y)]1−�[c(A(x0))�B(y)]�

6 c(A(x0))�B(y):

Since A∩B⊆ c(A)∪B due to the properties oft-norms and co-norms.

The “exponential-comprimize-reasoning” is demo-nstrated in Figs. 13 and 14 graphically for particularvalues of A(x) and B(y) and for algebraic and boldoperators.Here, again it should be observed that the upper and

lower bounds of material and product implications areequal to each other for bold operators (Fig. 14).

10. Conclusion

It is argued that Type 2 membership functions pro-vide a better and richer grounding for the meaning rep-resentation of words and reasoning with robust com-putations in computing meaning of words in CWW.The initial step in developing “Full” Type 2 systemmodeling is “Interval-Valued” Type 2 schema.Also, it is suggested that Type-2 knowledge rep-

resentation and reasoning add further re/nements tofuzzy system modeling. In Type 2 representation,second-order gradation for the meaning of words,linguistic terms of linguistic variables are induced


Fig. 13. Graph of exponential compromize reasoning �= [A(x∗)⊗B(y)](1−�) + [c(A(x∗))⊕B(y)]� for A(x∗)= (0:8) and B(y)= (0:6)for algebraic operators.

to add further re/nements to meaning representa-tion of words in computing. There are advantages toType 2 knowledge representation and reasoning be-cause it does not reduce information granulation dueto reductionist curve /tting techniques in knowledgerepresentation and it does not reduce the imprecisionexpansion due to disjunction and conjunctive canon-ical forms. The exposition of information contentembedded in information granules allows decisionmakers to be more aware of risks associated withimprecise information availability and its analysis.It was also shown that for lower and upper bounds

determined with bold operators, the material implica-tion and the product implication types of reasoningare equivalent to each other. Such results suggest thatthere is an extra information content to be discoveredwith Type 2 knowledge representation and computa-tion which is executed with Type 2 reasoning formu-las.This further suggests that if we capture knowledge

representation with Type 2 schema and compute with

Type 2 reasoning formulas, we can discover additionalinformation content. For example, Full Type 2 reason-ing produces Type 3 fuzziness which may be useful insome cases. It should be recalled that Type 1, Type 2and Type 3 information are analogous to /rst, secondand third moments in statistical calculations but notthe same. We expect to report on some of these in-sights in future investigations.

Appendix A Canonical form derivation algorithm

(a) First, assign truth values T; F to the meta-linguistic values (labels, variables) A and B and thenassign truth values T; F to the meta-linguistic expres-sion of concern, say “A AND B” in order to de/ne itsmeaning.(b) Next, construct primary conjunctions of the set

symbols A; B, corresponding to linguistic values suchthat in a given row,


Fig. 14. Graph of exponential compromize reasoning �= [A(x∗)⊗B(y)]1−� + [c(A(x∗))⊕B(y)]� for A(x∗)= (0:8) and B(y)= (0:6) forbold operators.

(i) if a T appears, then take the set aWrmation sym-bol of that meta-linguistic variable; otherwise

(ii) if an F appears, then take the set of complemen-tation symbol of that meta-linguistic variable;

(iii) next, conjunct the two symbols.

(c) Then, construct the disjunctive normal form ofthe meta-linguistic expression of concern:

(i) /rst, take the conjunctions corresponding to theT ’s of the truth assignment made under the col-umn of the meta-linguistic expression, such as“A AND B”, and

(ii) next, combine these conjunctions with disjunc-tions.

(d) Next, construct the conjunctive normal form ofthe meta-linguistic expression of concern:

(i) /rst, take the conjunctions corresponding to F’sof the truth assignment made under the column ofthe meta-linguistic expression, such as “A ANDB” and

(ii) then, combine these conjunctions with disjunc-tions, and

(iii) next, take the complement of these disjunctedconjunctions.

References

[1] T. Bilgic, I.B. T(urk*sen, Measurement–theoretical justi-/cation of connectives in fuzzy set theory, Fuzzy Sets andSystems 76 (1995) 289–307.

[2] T. Bilgic, I.B. T(urk*sen, Elicitation of membership functions:how far can theory take us?, Proceedings of theFuzzy-IEEE ’97, July 1–5, Barcelona, Spain, Vol. III, 1997,pp. 1321–1325.

[3] T. Bilgic, I.B. T(urk*sen, Measurement–theoretical frame-works in fuzzy theory, in: A. Ralescue, T. Martin (Eds.),Fuzzy Logic in Arti/cial Intelligence, Springer, Berlin, 1997,pp. 552–565.

[4] T. Bilgic, I.B. T(urk*sen, Measurement of membershipfunctions: theoretical and empericial work, in: D. Dubois, H.Prade (Eds.), Handbook of Fuzzy Theory, 2000.

[5] P. Burillo, H. Bustince, Intuitionistic fuzzy relations Part I,Mathware Soft Comput. 2 (1995) 5–38.


[6] P. Burillo, H. Bustince, Entropy on intuitionistic fuzzy setsand on interval valued fuzzy sets, Fuzzy Sets and Systems78 (1996) 305–316.

[7] H. Bustince, P. Burillo, Interval valued fuzzy relations in aset structures, J. Fuzzy Math. 4 (1996) 765–785.

[8] M. Gehrke, C. Walker, E. Walker, Some comments oninterval-valued fuzzy sets, Internat. J. Intell. Systems 11(1996) 751–759.

[9] M. Gehrke, C. Walker, E. Walker, Fuzzy normal forms andtruth tables, Proceedings of the JCIS- 2000, February 27–March 3, Atlantic City, NJ, 2000, pp. 211–214.

[10] M. Gehrke, C. Walker, E. Walker, Some comments on fuzzynormal forms, Proceeding of the FUZZY-IEEE’2000, 7–10May, San Antonio, 2000, pp. 593–598.

[11] M.B. Gorzalczany, A method of inference in approximatereasoning based on interval-valued, Fuzzy Sets and Systems21 (1987) 1–17.

[12] E. Hisdal, The IF THEN ELSE statement and interval-valuedfuzzy sets of higher type, Internat. J. Man-Mach. Stud. 15(1981) 385–455.

[13] R.I. John, Type 2 fuzzy sets: an appraisal of theoryand applications, Internat. J. Uncertainty, FuzzinessKnowledge-based Systems 6 (6) (1998) 563–576.

[14] R.I. John, P.R. Innocent, M.R. Barns, Type 2 fuzzy setsand neuro-fuzzy clustering of radiographic tibia images, 1998IEEE International Conference on Fuzzy Systems, Anchorage,AK., May 1998, pp. 1373–1376.

[15] N.N. Karnik, Type 2 fuzzy logic systems, Ph.D. Dissertation,University of Southern California, Los Angeles, CA, 1998.

[16] N.N. Karnik, J.M. Mendel, Type 2 fuzzy logic systems:type-reduction, Presented at 1998 IEEE SMC Conference,San Diego, CA, October 1998 (CD).

[17] N.N. Karnik, J.M. Mendel, Introduction to type 2 fuzzylogic systems, Presented at 1998 IEEE FUZZ Conference,Anchorage, AK. May 1998, pp. 915–920.

[18] N.N. Karnik, J.M. Mendel, Q. Liang, Type 2 fuzzy logicsystem, IEEE Trans. Fuzzy Systems 7 (1999) 643–658.

[19] Q. Liang, J.M. Mendel, Interval type 2 fuzzy logic system,Proceedings of the Nineth IEEE International Conferenceon Fuzzy Systems, 7–10 May, San Antonio, TX, 2000,pp. 328–333.

[20] Q. Liang, J.M. Mendel, Decision feedback equalizers fornon-linear time varying channels using Type 2 fuzzyadaptive /lters, Proceedings of the Nineth IEEE InternationalConference on Fuzzy Systems, 7–10 May, San Antonio, TX,2000, pp. 883–888.

[21] P.N. Marinos, Fuzzy logic and its application to switchingsystems, IEEE-Trans. Comput. 4 (1969) 343–348.

[22] K. Menger, Statistical metrics, Proceedings of the N.A.S,Vol. 28, 1942.

[23] M. Mizumoto, K. Tanaka, Some properties of fuzzy sets oftype 2, Inform. Control 31 (1976) 312–340.

[24] J. Niemien, On the algebraic structure of fuzzy sets of type2, Kybernetica 13 (4) (1977).

[25] A.M. Norwich. I.B. T(urk*sen, The fundamental measurementof fuzziness, in: R.R. Yager (Ed.), Fuzzy Sets and PossibilityTheory, Pergamon Press, New York, 1982, pp. 49–60.

[26] A.M. Norwich, I.B. T(urk*sen, The construction of membershipfunctions, in: R.R. Yager (Ed.), Fuzzy Sets andPossibility Theory, Pergamon Press, New York, 1982,pp. 61–67.

[27] A.M. Norwich, I.B. T(urk*sen, Stochastic fuzziness, in: M.M.Gupta, E.E. Sanches (Eds.), Fuzzy Information and DecisionProcesses, North-Holland, Amsterdam, 1982, pp. 13–22.

[28] A.M. Norwich, I.B. T(urk*sen, A model for the measurementof membership and the consequences of its empiricalimplementation, Fuzzy Sets and Systems 12 (1984) 1–25.

[29] G. Resconi, I.B. T(urk*sen, Canonical forms of fuzzy truthoodsby meta-theory based upon modal logic, Inform. Sci., in press.

[30] M.K. Roy, R. Biswas, I-v fuzzy relations and Sanchez’sapproach for medical diagnosis, Fuzzy Sets and Systems 47(1992) 35–38.

[31] I.B. T(urk*sen, Measurement of linguistic variables,Proceedings of the 23rd Annual North American Meeting,Society for General Systems Research, January 3–6, Houston,TX, 1979, pp. 278–284.

[32] I.B. T(urk*sen, Stochastic and fuzzy sets, Proceedings of theSecond World Conference on Mathematics at the Service ofMan, June 28–July 3, Las Palmas, Canary Islands, Spain,1982, pp. 649–654.

[33] I.B. T(urk*sen, Inference regions for fuzzy propositions, in:P.P. Wang (Ed.), Advances in Fuzzy Sets, PossibilityTheory and Applications, Plenum Press, New York, 1983,pp. 137–148.

[34] I.B. T(urk*sen, Measurement of fuzziness: an interpretationof the azioms of measurement, Proceedings of theIFAC Symposium, Marseill, France, July 19–21, 1983,pp. 97–102.

[35] I.B. T(urk*sen, Interval-valued fuzzy sets based on normalforms, Fuzzy Sets and Systems 20 (1986) 191–210.

[36] I.B. T(urk*sen, Measurement of membership functions and theiracquisitions, Fuzzy Sets and Systems 40 (1991) 5–38.

[37] I.B. T(urk*sen, Interval-valued fuzzy sets and ‘CompensatoryAND’, Fuzzy Sets and Systems 51 (1992) 295–307.

[38] I.B. T(urk*sen, Fuzzy normal forms, Fuzzy Sets and Systems69 (1995) 319–346.

[39] I.B. T(urk*sen, T. Bilgic, Interval-valued strict preference withZadeh triples, Fuzzy Sets and Systems 78 (1996) 183–195.

[40] I.B. T(urk*sen, Theories of set and logic with crisp or fuzzyinformation granules, J. Adv. Comput. Intell. 3 (4) (2000)264–273.

[41] I.B. T(urk*sen, A. Kandel, Y.Q. Zhang, Universal truth tablesand normal forms, IEEE Trans. Fuzzy Systems 6 (2) (1998)295–303.

[42] I.B. T(urk*sen, A. Kandel, Y.Q. Zhang, Normal forms offuzzy middle and fuzzy contradiction, IEEE-SMC, 29, PartB (Cybernetics) 2, 1999, pp. 237–253.

[43] I.B. T(urk*sen, A.M. Norwich, Measurement of fuzziness,Proceedings of the International Conference on PolicyAnalysis and Inform. Systems, August 19–22, Taipei,Taiwan, Meadea Enterprises Co., Ltd. Taipei, Taiwan, 1981,pp. 745–754.

[44] I.B. T(urk*sen, D.D. Yao, Bounds on fuzzy inference, Pro-ceedings of the Sixth European Meeting on Cybernetics andSystem Research, Vienna, April 13–16, 1982, pp. 729–734.


[45] I.B. T(urk*sen, Z. Zhong, An approximate analogicalreasoning schema based on similarity measures andinterval-valued fuzzy sets, Fuzzy Sets and Systems 34 (1990)323–346.

[46] M. Wagenknecht, K. Hartmann, Application of fuzzy sets ofType 2 to the solution of fuzzy equation systems, Fuzzy Setsand Systems 25 (1988) 183–190.

[47] R.R. Yager, Fuzzy subsets of Type II in decisions,J. Cybernet. 10 (1980) 137–159.

[48] L.A. Zadeh, Fuzzy sets, Inform. Control Systems 8 (1965)338–353.

[49] L.A. Zadeh, Outline of a new approach to the analysisof complex systems and decision processes interval-valuedfuzzy sets, IEEE Trans. Systems Man Cybernet. 3 (1973)28–44.

[50] L.A. Zadeh, The concept of a linguistic variable and itsapplication to approximate reasoning, Part II, Inform. Sci. 8(1975) 301–357.

[51] L.A. Zadeh, Fuzzy logic = computing with words, IEEETrans. Fuzzy Systems 4 (2) (1996) 103–111.

[52] L.A. Zadeh, Computing with perceptions, Keynote Address,IEEE-Fuzzy Theory Conference, San Antonicq, May 7–10,2000.

[53] H.J. Zimmermann, Zysno, Latent connectives in humandecision making, Fuzzy Sets and Systems 4 (1980) 37–51.

[54] Q. Zuo, I.B. T(urk*sen, Hung T. Nguyen, et al., In expertsystems, even if we /x AND=OR operations, a natural answerto a composite query is the interval of possible degrees ofbelief, Reliable Computing, Supplement (Extended Abstractsof APIC ’95: International Workshop on Applications ofInterval Computations), El Paso, TX, February 23–25, 1995,pp. 236–240.

Type2representationandreasoningforCWW - Semantic Scholar · Type2representationandreasoningforCWW...

Documents

Transcript of Type2representationandreasoningforCWW - Semantic Scholar · Type2representationandreasoningforCWW...