Fuzzy Sets and Systems 127 (2002) 17–36
www.elsevier.com/locate/fss
Type 2 representation and reasoning for CWW
I. 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 knowledge
representation 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 which
captures varying degrees of membership and hence varying meanings of words or else (2) they are obtained with the
application of modi/ed fuzzy clustering methods, where they all reveal a scatter plot, which captures varying degrees of
meaning for words in a fuzzy cluster. Secondly, it has been shown that the combination of linguistic values with linguistic
operators, “AND”, “OR”, “IMP”, etc., as opposed to crisp connectives that are known as t-norms and t-conorms and standard
negation, lead to the generation of Fuzzy Disjunctive and Conjunctive Canonical Forms, FDCF and FCCF, respectively. In
this paper, we /rst discuss how one captures Type 2 representation. Then we concentrate on Type 2 reasoning that rests on
Type 1 representation. Next, we show how one computes Type 2 reasoning starting with Type 2 representation. It is to be
forecasted that in the new millennium more and more researchers will attempt to capture Type 2 representation and develop
reasoning with Type 2 formulas that reveal the rich information content available in information granules, as well as expose
the risk associated with the graded representation of words and computing with words. This will entail more realistic system
model developments, which will help explore computing with perceptions, and computing with words by exposing graded
c 2002 Published by Elsevier
:exibility as well as uncertainty embedded in meaning representation. Crown Copyright Science B.V. All rights reserved.
1. Introduction
In most current investigations, fuzzy set representations and their logical combinations are based on
Type 1 schema for both the knowledge representation
and approximate reasoning. First, Type 1 representation is a “reductionist” approach for it discards the
spread 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 values. Therefore, Type 1 representation does not provide a good approximation to meaning representation
of words and does not allow computing with words a
richer platform. Secondly, Type 1 approximate reasoning 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” corresponds to a t-norm and linguistic “OR” corresponds
to a t-conorm in a one-to-one mapping. In Type 1 approximate reasoning, “AND”ness is simply mapped
to disjunctive normal form (DNF), and “OR”ness is
c 2002 Published by Elsevier Science B.V. All rights reserved.
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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 habit
from two-valued to fuzzy-valued theory. At least two
intuitive expectations come to mind in the light of
the limitations of Type 1 representation and reasoning
outlined above.
First, intuitively, it is natural to expect that a combination of two imprecise Type 1 memberships should
produce new membership functions that capture the
compounded increase of imprecision, i.e., produce a
Type 2 membership function. Thus, one needs to develop formulas that represent Type 2 uncertainty that
arises out of the combination of Type 1 memberships
with linguistic connectives, “AND”, “OR”, “IMP”,
etc.
Second, again intuitively, we would expect that
the combination of two Type 2 memberships should
produce new membership functions that capture the
compounded increase of imprecision realizable with
additional 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 processes. Furthermore, Zadeh [50] discussed Type 2
fuzzy set representation and its potential in approximate reasoning. A number of researchers began to investigate Type 2 fuzzy sets and their properties after
Zadeh’s introduction of this topic. Some of these are as
follows: Mizumoto and Tanaka [23] (1981); Niemien
[24]; Yager [47]; Hisdal [12]; Wagenkneckt and Hartmann [46]; Roy and Biswas [30]; Burillo and Bustince
[5,6]; Bustince and Burillo [7]; John [13]; John et
al. [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 the
Type 2 fuzzy set memberships are acquired with the
exception of (i) John et al. [14], which indicates that
Type 2 fuzzy sets are obtained via neuro-fuzzy clustering 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, and
use the myopic formulas of Type-1 reasoning in developing inference schemas, e.g., Mendel et al., i.e.,
they do not consider FDCF and FCCF formulas in
reasoning schemas.
In contrast, in our investigations, we have dealt with
interval-valued Type 2 representation and reasoning
starting in the early 80s. First, in experimental studies of measurement of membership functions, it was
shown by Norwich and T(urk*sen [25 – 27] that direct
measurement experiments, conducted in order to extract membership functions for “tall” men, “pleasing”
houses, reveal a Type 2 membership representation
that can be identi/ed with the mean and spread of scatter points, with standard deviations (Figs. 1 and 2).
It should be observed that in these graphs, the spread
of scatter points come from the same subject. It is natural to conjecture that the spread would be larger if
the scatter points come from a number of subjects in a
measurement experiment. It should be noted that the
same individual when asked in an experiment gives
diQerent membership values to the same height of an
individual. The replicated experiments are designed so
that there is no memory. Type 2 fuzziness is a natural
outcome of the measurement theory. Since individuals
give diQerent membership values, the spread of membership values re:ects on the one hand the uncertainty
I.B. T(urks*en / Fuzzy Sets and Systems 127 (2002) 17–36
19
of searching the optimal (m; c) pair (m∗ ; c∗ ), we can
identify a set of m’s, {m} that minimize a c, say
c∗ . Thus having identi/ed c∗ , we can then choose a
lower value, m∗L = minm {m; c∗ } and an upper value
m∗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 a
ground for Type 2 representation. This means that we
can acquire Type 2 membership 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) intervalvalued Type 2 and (ii) full Type 2.
Fig. 2. Direct rating for “pleasing” house (subject #1).
associated in the meaning of a word, say, “tall”, and
robustness association with meaning representation of
linguistic variables.
Secondly, in modi/ed FCM experiments which we
have began to carry out recently, we have observed
that the dilemma concerning the choice of (m; c)
pair, can be resolved in the following way. Instead
(i) Interval-valued Type 2 fuzziness is a special
Type 2 fuzziness where the upper and lower bounds
of membership are identi/ed and the spread of membership, either fuzzy or probabilistic and where the
spread of membership distribution is ignored with the
assumption that membership values between upper
and lower values are uniformly distributed or scattered
with a membership value of “1” on the ((·)) axis
as shown in Fig. 4. Thus, the upper and lower bounds
of interval-valued Type 2 fuzziness specify the range
of uncertainty about the membership values.
(ii) Full Type 2 fuzziness identi/es upper and
lower membership values as well as the spread of
Fig. 3. Type 2 fuzzy sets: (a) a set of membership functions of a cluster; (b) interval-valued membership functions of a cluster.
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I.B. T(urks*en / Fuzzy Sets and Systems 127 (2002) 17–36
we obtain a graded distribution of uncertainty between
the bounds.
3. Interval-valued fuzzy sets
Fig. 4. Interval-valued Type 2.
In conjunction with Type 2 fuzziness found in measurement experiments, interval-valued fuzzy sets were
discovered in combination of concepts and hence approximate reasoning by T(urk*sen [32 – 35] where it was
shown that disjunctive and conjunctive normal forms
(DNF) and (CNF) of Boolean logic are no longer
equivalent due to the fact that the law of excluded
middle (LEM), and its dual the law of contradiction
(LC) do not hold in fuzzy set and logic theory which
were 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 and
they do not directly correspond to one of the t-norms
and t-conorms, which are crisp operators. Linguistic
operators “AND”, “OR”, etc., being inherently fuzzy,
need to be interpreted in a graded manner, whereas
t-norms, T’s, and t-conorms, ’s, are crisp operators that combine singleton fuzzy degrees of information granules and give a singleton fuzzy degree, i.e.,
1 T2 = 3 or 1 2 = 4 .
In fact, Zimmerman and Zysno [53] had shown
that human use of “AND”, “OR” do not directly
correspond to a t-norm, such as min, or algebraic
product and to a t-conorm, such as max or algebraic
sum, respectively. They had proposed “Compensatory
‘AND’”, which either linearly or exponentially combines a t-norm and a t-conorm, such as min–max,
or algebraic product and sum with a compensation
operator, ; i.e.,
(i) Convex linear compensation:
F1 (A ; B ; 1 ) = 1 (A ∧ B )
+(1 − 1 )(A ∨ B )
Fig. 5. Full Type 2.
membership values between these bounds either probabilistically or fuzzily. That is there is a probabilistic
or possibilistic distribution of membership values that
are between upper and lower bound of membership
values in the ((·)) axis as shown in Fig. 5. Thus,
or
(ii) Exponential compensation:
F2 (A ; B ; 2 ) = (A ∧ B )2 (A ∨ B )(1−2 ) ;
where 061 ; 2 61.
Later, T(urk*sen [37] showed that in fact operators of Zimmermann and Zysno [53] were compensation weights that combined DNF and CNF values
I.B. T(urks*en / Fuzzy Sets and Systems 127 (2002) 17–36
of “AND”ness, “OR”ness that T(urk*sen discovered in
1986. That is, the degrees of “AND”ness should be
captured, for the case of complex linear compensation
as
FLAND (A ; B; LAND )
= LAND DNF(A AND B) + (1 − LAND )CNF(A AND B) ;
where “LAND” stands for linear ‘AND’ compensation; the degrees of “OR”ness are captured as
FLOR (A ; B ; LOR )
= LOR DNF(A OR B) + (1 − LOR )CNF(A OR 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(A AND B)
+(1 − L(AND=OR) )DNF(A OR 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(A AND B) )EAND (CNF(A AND B) )(1−EAND ) ;
where ‘EAND’ stands for exponential ‘AND’ compensation; the degrees of “OR”ness are captured as
FEOR (A ; B ; EOR )
(DNF(A OR B) )EOR (CNF(A OR B) )(1−EOR ) ;
where ‘EOR’ stands for exponential ‘OR’ compensation; and the degrees of “AND=OR”ness are captured
as
21
reduced Type-2 uncertainty generated by DNF and
CNF formulae to Type-1 membership functions with
the assignment of operators. In Fig. 6, “exponential
compensatory ‘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 of
view. It was known that Boolean formulae, DNF
(·) ≡ (·) CNF (·) for all 16 possible combinations
of any two two-valued sets. But, it was shown that
Boolean DNF and CNF formulae are no longer equivalent, when used with fuzzy membership values [35].
That is, they give results such that DNF (·) =
CNF (·), where (·) stands for any of the sixteen
combinations of any two linguistic values of any two
linguistic variables (see Table 1). In fact, it is also
shown that DNF ⊆ CNF for the well-known cases
of t-norm and t-conorm based De Morgan Triples
formed with standard negation such as (∧; ∨; −),
(⊗; ⊕; −), (L⊗ ; L⊕ ; −), (D⊗ ; D⊕ ; −), where (∧; ∨)
stand for min–max, (⊗; ⊕) stand for algebraic product
and sum, (L⊗ ; L⊕ ) stand for Lukasiewicz intersection
and union, and (D⊗ ; D⊕ ) stand for drastic intersection
and union operators [40].
The fundamental question at this juncture was
whether or not fuzzy disjunctive and conjunctive
canonical forms can be derived from fuzzy truth tables. In 1970, it was argued that fuzzy truth tables
could not be formed [21] for it would require a table
with in/nite number of rows. In a series of papers
[38 – 42], it was progressively shown that fuzzy truth
tables can be constructed and fuzzy disjunctive and
conjunctive canonical forms (FDCF) and (FCCF)
can be derived from the fuzzy truth tables. More
recently, Resconi and T(urk*sen [29] have shown that
there are deeper reasons why FDCF and FCCF are
realized.
Along the way, other researchers began to investigate such topics “AND=OR” intervals of possible degrees of belief [54], and Fuzzy normal forms [8,9].
FE(AND=OR) (A ; B ; E(AND=OR) )
= (CNF(A AND B) )E(AND=OR) (DNF(A OR 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
4. FDCF and FCCF expressions
The application of the normal form derivation algorithm [see Appendix A] gives us the derivation
of FDCF and FCCF expressions for “A AND B”,
shown in row 6 of Table 1, from the fuzzy truth table,
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I.B. T(urks*en / Fuzzy Sets and Systems 127 (2002) 17–36
Fig. 6. IVFS “A AND B”, “A OR B”, expressed in DNF and CNF and “exponential compensatory AND” using algebraic sum-product De
Morgan 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;
As well, with an appropriate modi/cation of the last
column of Table 5, we can drive FDCF and FCCF
expressions shown in Table 2 for all the remaining 15
meta-linguistic combinations shown in Table 1. For
example for “A OR B”, we get
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 complementbased De Morgan Triples that correspond to symbolic
set operation “∩”, “∪”, “c” in the numerical domain,
respectively.
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:
I.B. T(urks*en / Fuzzy Sets and Systems 127 (2002) 17–36
Table 1
Meta-linguistic expressions of combined concepts for any A and B
No.
Meta-linguistic expressions
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
UNIVERSE
EMPTY SET
A OR B
NOT A AND NOT B
NOT A OR NOT B
A AND B
A IMPLIES B
A AND NOT B
A OR NOT B
NOT A AND B
A IF AND ONLY IF B
A EXCLUSIVE OR B
A
NOT A
B
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, tconorm and standard complementation operations in
the numerical domain that correspond to symbolic set
operations “∩”, “∪”, and “c”, respectively.
It should be noted that for pseudo t-norm, t-conorm
and standard complementation, we need to abide by
the order of A ⊇ B XOR A ⊂ B and adjust all the
formulae accordingly, since they are not commutative
and associative.
5. Impact of FDCF and FCCF
The discovery that we ought to use FDCF and FCCF
in fuzzy combination of linguistic concepts and hence
approximate reasoning, opens up at least three avenues
for the formation of the combination of linguistic con-
23
Table 2
Classical disjunctive normal and fuzzy disjunctive canonical forms
(DNF) and (FDCNF) and classic conjunctive normal and fuzzy
conjunctive canonical forms (CNF) and (FCCF), where ∩ is a
conjunction, ∪ is a disjunctive and c is a complementary operator
No. Fuzzy disjunctive canonical forms=disjunctive normal forms
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
(A ∩ B) ∪ (A ∩ c(B)) ∪ (c(A) ∩ B) ∪ (c(A) ∩ c(B))
(A ∩ B) ∪ (A ∩ c(B)) ∪ (c(A) ∩ B)
(c(A) ∩ c(B))
(A ∩ c(B)) ∪ (c(A) ∩ B) ∪ (c(A) ∩ c(B))
(A ∩ B)
(A ∩ B) ∪
(c(A) ∩ B) ∪ (c(A) ∩ c(B))
(A ∩ c(B))
(A ∩ B) ∪ (A ∩ c(B)) ∪
(c(A) ∩ c(B))
(c(A) ∩ B)
(A ∩ B) ∪
(c(A) ∩ c(B))
(A ∩ c(B)) ∪ (c(A) ∩ B)
(A ∩ B) ∪ (A ∩ c(B))
(c(A) ∩ B) ∪ (c(A) ∩ c(B))
(A ∩ B) ∪
(c(A) ∩ B)
(A ∩ c(B)) ∪
(c(A) ∩ c(B))
No. Fuzzy conjunctive canonical forms=conjunctive normal forms
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
I
(A ∪ B) ∩ (A ∪ c(B)) ∩ (c(A) ∪ B) ∩ (c(A) ∪ c(B))
(A ∪ B)
(A ∪ c(B)) ∩ (c(A) ∪ B) ∩ (c(A) ∪ (c(B))
(c(A) ∪ c(B))
(A ∪ B) ∩ (A ∪ c(B)) ∩ (c(A) ∪ B)
(c(A) ∪ B)
(A ∪ B) ∩ (A ∪ c(B)) ∩
(c(A) ∪ c(B))
(A ∪ c(B))
(A ∪ B) ∩
(c(A) ∪ B) ∩ (c(A) ∪ c(B))
(A ∪ c(B)) ∩ (c(A) ∪ B)
(A ∪ B) ∩
(c(A) ∪ c(B))
(A ∪ B) ∩ (A ∪ c(B))
(c(A) ∪ B) ∩ (c(A) ∪ c(B))
(A ∪ B) ∩
(c(A) ∪ B)
(A ∪ c(B)) ∩
(c(A) ∪ c(B))
cepts and hence approximate reasoning as possible approaches to computing with words:
(i) If we start out with Type 1 membership functions
in knowledge representation, say two fuzzy sets A and
B, then the combination of these two Type 1 fuzzy
sets produce Type 2 fuzzy sets with FDCF ⊆ FCCF. In
Fig. 7, an example of linguistic “OR” combination of
two Type 1 fuzzy membership values are shown with
FDCF(A OR B) (1 ; 2 ) and FCCF(A OR B) (1 ; 2 ).
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I.B. T(urks*en / Fuzzy Sets and Systems 127 (2002) 17–36
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 fuzzy
sets generate 2N −1 membership functions in a binary
tree, where N is the member of linguistic terms. For
example, for the combination of three fuzzy sets with
Type 1 membership functions, we have, 23−1 = 4
membership 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 there
are more than two fuzzy sets with Type 1 membership
functions, the consequent of the combined linguistic
terms 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 2
fuzzy set with eight elements is obtained as follows:
(1) FDCF(1L ; 2L );
(3) FDCF(1U ; 2L );
(5) FCCF(1L ; 2L );
(7) FCCF(1U ; 2L );
(2) FDCF(1L ; 2U );
(4) FDCF(1U ; 2U );
(6) FCCF(1L ; 2U );
(8) FCCF(1U ; 2U );
where the pair (1L ; 1U ) de/nes the interval-valued
Type 2 fuzzy set, say, A, and (2L ; 2U ) de/nes the
interval-valued Type 2 fuzzy set, say, B. Naturally, the
combination of more than two interval-valued fuzzy
sets generate 2N −1 · 2N membership values, where N
is 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 Full
Type 2 fuzzy sets produce Type 3 fuzzy sets with
FDCF = FCCF computed on (1 ) and (2 ). Con-
I.B. T(urks*en / Fuzzy Sets and Systems 127 (2002) 17–36
25
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”ness
and “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 way
to simplify these, after the application of FDCF and
FCCF between any two fuzzy sets with either Type 1
or interval-valued Type 2 or Full Type 2 membership
functions, we may discard the values generated between upper and lower membership values generated
in () space for Type 2 generation and (()) space
for Type 3 generation, and hence reduce the computational complexity to the computation of intervalvalued Type 2 and interval-valued Type 3 membership
generation, if and when, we cannot aQord extended
computing time and eQort. For some applications, this
may be justi/ed. A skeleton of Type 3 membership
generation is shown in Fig. 8.
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I.B. T(urks*en / Fuzzy Sets and Systems 127 (2002) 17–36
Fig. 9. Various representation theorems for membership functions.
5.1. Type-2 knowledge representation
There are two approaches to obtain Type-2 knowledge representation: (i) Membership measurement
experiments conducted with experts and (2) Fuzzy
cluster experiments conducted with FCM algorithm.
Measurement of membership. Membership measurement experiments conducted with experts, i.e.,
subjects, expose the need to represent knowledge with
Type-2 fuzzy sets as depicted in Figs. 1 and 2. We have
presented various aspects of measurement of membership both theoretically and experimentally [1 – 4,25 –
28,31 – 33,36,43].
Brie:y, /rst it is worthwhile to note that measurement theory deals with representation and uniqueness
from empirical relational domain to numerical relational domain. A particular interest in this regard is the
scale strength of the numerical representation depending on which axioms of measurement theory can be
validated for a particular experimental data obtained
from experts. In this regard, it is to be realized that var-
ious representation theorems associated with various
quantitative structures generate algebraic representation 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 order to capture variability of these responses, and their
information granules, one needs to represent membership values in Type-2 schemas (Figs. 1–5).
This becomes particularly important from the perspective of computing with perceptions and words
paradigm [51,52]. In this perspective it is essential that
the meaning of words, linguistic terms of linguistic
variables are represented eQectively with membership
grades of the membership, i.e., variation of membership values.
Type 2 representation exposes the variation of membership values and causes a more richer and robust
meaning representation of words and their computation.
Fuzzy clustering. In applications of fuzzy clustering algorithm, FCM, a major concern is the selection
I.B. T(urks*en / Fuzzy Sets and Systems 127 (2002) 17–36
of (m; c) pairs that provide “good” clusters of linguistic values of linguistic variables. A general heuristic
is to choose m = 2, level of fuzziness, and then determine the optimal c, the number of clusters. Then
one determines in general a Type 1 membership over
these fuzzy clusters with curve /tting techniques.
Another approach is to search through 1¡m¡∞,
the level of fuzziness, and c = 2; : : : ; 10; : : : ; the
number of possible clusters, in order to /nd an optimal pair (m; c) that will give a “good” system
model.
Recently, we have conducted experiments which
show that for certain c-values there are many m-values
that cause minimization of the cluster validity index
for certain c’s. This has led us to choose a c-value
that minimizes the cluster validity index and develops
Type-2 fuzziness around the chosen c-value with all
the m-values that give minimization for that particular
c. In particular, let c∗ be the number of clusters for
which 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 all m ∈{(mL ; c∗ );
: : : ; (mU ; c∗ )}. First /t a curve to the scatter points of
(mL ; c∗ ) to determine the lower bound membership
function and then /t a curve to the scatter points of
(mU ; c∗ ) to determine the upper bound membership
function for an interval-valued Type 2 representation
in -space. For Full Type 2 representation, one needs
to determine the possibilistic or probabilistic distribution function of Type 2 membership in () space.
This also requires that a curve be /tted to all the scatter points obtained from m ∈ {(mL ; c∗ ); : : : ; (mU ; c∗ )}.
Naturally, there are various alternatives. They depend on whether scatter points demonstrate a particular shape or not or whether the analyst needs to
simplify the fuzzy set representation to be triangular,
trapezoidal, S or curves or even some form of exponential. Other simpli/cations can be made by computing mean values and variances with the statistical
methods and the curve may be simpli/ed with statistical methods, etc. These details are application and
context dependent and should be chosen by the system analyst.
In this manner, Type-2 fuzziness is identi/ed
with unsupervised learning and with the application
of FCM. This also provides a more reasonable approach to determine suitable (m; c) pairs. This can be
observed in Fig. 3, as shown before.
27
The determination of Type-2 fuzziness with FCM
algorithm also settles the scale strength issue raised
in measurement theory. That is, if input data that is
used in FCM algorithm is on the absolute scale, then
the resulting membership values in Type-2 representation turn out to be on the absolute scale due to the
computations required in the FCM algorithm.
5.2. Type 2- approximate reasoning
Type 2 approximate reasoning can be computed in
two ways as follows: (1) Interval-valued Type 2 reasoning where we reduce the Type 2 uncertainty by
computing FDCF and FCCF formulas only for the
determination of upper and lower values of Type 2
memberships; and (2) Full Type 2 reasoning where all
available grades between upper and lower membership values are combined in a pairwise combination.
The /rst approach is partially reductionist, but computationally eWcient in comparison to “Full” Type 2
computations. Whereas, the second approach leads to
computational complexity of a higher order as discussed previously. In Table 2, the FDCF and FCCF
formulas are shown. A detailed discussion and derivation of FDCF and FCCF may be found in T(urk*sen [40].
Here, we brie:y review the derivation of FDCF and
FCCF formulae.
In Table 1, meta-linguistic expressions of combined
concepts are shown for any two linguistic concepts
A and B whether they are represented by two or in/nite (fuzzy) valued sets. In Table 2, classical and
fuzzy canonical forms are shown under the headings of
fuzzy disjunctive canonical forms, FDCF=disjunctive
normal forms, DNF, and fuzzy conjunctive canonical forms, FCCF=conjunctive normal forms, CNF, for
fuzzy=classical logic, respectively. The derivation of
classical normal forms are based on Classical Truth
Table, Table 3. The equivalence of classical formula,
i.e., DNF(·) ≡ CNF(·) for all the 16 possible combinations can be shown to hold with the axioms of classical set and logic theory operations. It should be noted
that the set operations, ∩, ∪, are implicitly assumed to
be two-valued, i.e., in {0; 1}, whereas in Table 3, their
truthfullness are explicitly shown to be two-valued,
i.e., {T; F}. Also, in Table 4, “Axioms of classical
set and logic operations”, the set operations, ∩, ∪, are
again implicitly assumed to be two-valued. It should
be pointed out that in Tables 1–4 there is no explicit
28
I.B. T(urks*en / Fuzzy Sets and Systems 127 (2002) 17–36
Table 3
Classical truth table interpretations of “A AND B”
Truth assignments to classical
meta-linguistic variables
Truth assignments to the
meta-linguistic expression
A
B
“A AND B”
T (A)
T (A)
F(A)
F(A)
T (B)
F(B)
T (B)
F(B)
T (A
F(A
F(A
F(A
Table 4
Axioms of classical and set and logic operations
Involution
Commutativity
Associativity
Distributivity
Idempotence
Absorption
Absorption by X and Identity
Law of contradiction
Law of excluded middle
De Morgan’s laws
c(c(A) = A
A∪B=B∪A
A∩B=B∩A
(A ∪ B) ∪ C = A ∪ (B ∪ C)
(A ∩ B) ∩ C = A ∩ (B ∩ C)
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
A∪A=A
A∩A=A
A ∪ (A ∩ B) = A
A ∩ (A ∪ B) = A
A∪X =X
A∩=
A∪=A
A∩X =A
A ∩ c(A) = A ∪ c(A) = X
c(A ∩ B) = c(A) ∪ c(B)
c(A ∪ B) = c(A) ∩ c(B)
indication as to whether sets A and B are two valued
or not.
It is shown, as pointed out previously, that just by
substituting values in [0; 1] as opposed to {0; 1} for
the set membership, we get DNF(·) = CNF(·) [35].
Later, it was shown that /rst, the classical truth table
can be extended to a fuzzy truth table, Table 5, with
the realization that there is an order relationship between the membership values of fuzzy set A, a, and
the 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 now
shown explicitly to be in the interval [0; 1] in contrast
AND
AND
AND
AND
Primary conjunctions
B)
B)
B)
B)
A∩B
A ∩ c(B)
c(A) ∩ B
c(A) ∩ c(B)
Table 5
The “fuzzy truth table” of “A AND B” where a; b 0 [0; 1]
A
(1:1:1)
(1:1:1)
(1:2:1)
(1:2:1)
(1:1:2)
(1:1:2)
(1:2:2)
(1:2:2)
a¿b
a¿b
a¡b
a¡b
a¡b
a¡b
a¿b
a¿b
B
T
T
F
F
T
T
F
F
(2:1:1)
(2:2:1)
(2:1:1)
(2:2:1)
(2:1:2)
(2:2:2)
(2:1:2)
(2:2:2)
A AND B
a¿b
a¡b
a¿b
a¡b
a¡b
a¿b
a¡b
a¿b
T
F
T
F
T
F
T
F
T
F
F
F
T
F
F
F
to Table 3, but the truthfullness of the order relationship 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 expression, but they are equivalent in form only, not in
content. We discuss this next.
6. Containment vs. equivalence
In classical theory, we have DNF(·) ≡ CNF(·) for
all 16 cases shown in Tables 1 and 2. It is shown that
FDNF(·) ⊆ CNF(·) for all 16 cases in fuzzy theory for
the four well-known de Morgan Triples [35,40]. This
is now demonstrated very brie:y. For this purpose, we
will /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)
I.B. T(urks*en / Fuzzy Sets and Systems 127 (2002) 17–36
To show this, we start out with the expressions:
DNF (A AND B) = A ∩ B
CNF (A AND B)
= (A ∪ B) ∩ (A ∪ c(B)) ∩ (c(A) ∪ B)
Next, we apply certain axioms to CNF (·), as
follows:
In a similar manner for the other 15 possible combinations of concepts shown in Table 1, it can be shown
that DNF(·) ≡ CNF(·) in two-valued set and logic.
Containment of FCNF and FCCF
(3)
Next, we show that (1) holds for well-known cases
of t-norms and t-conorms. The following statements
are true in general for the class of t-norms and tconorms. First, it is well known that (4) and (5) are
true:
aD⊗ b 6 aL⊗ b 6 a ⊗ b 6 a ∧ b:
(4)
Next, it is also well known that
In fuzzy theory, we have
a ∨ b 6 a ⊕ b 6 aL⊕ b 6 aD⊕ b:
FDCF(A AND B) ⊆ FCCF(A AND B)
To show this, we start out again with the expressions:
(5)
Consider next the following inequalities, which are all
true:
[(a; b); (a;
X b)] 6 b
FDCF (A AND B) = A ∩ B
FCCF (A AND B)
= (A ∪ B) ∩ (A ∪ c(B)) ∩ (c(A) ∪ B)
(for = V; = ∧; and −);
As indicated, the expressions are the same but in
form only. In general, for the class of t-norms and tconorms and standard complement De Morgan Triples
that correspond to the prepositional set operators, “∩”,
“∪”, “c”, respectively. The axioms of idempotency,
distributivity and the law of contradiction, shown in
Table 4, are not applicable in fuzzy theory. It is because these axioms are relaxed in general in fuzzy theory that we obtain the containment of FDCF in FCCF.
Next, let us demonstrate this containment relationship as an example for the case of “A OR B” combination and this time in the numerical domain where
t-norm, , and t-conorm, operators are applicable.
That is, we need to show that
i:e:;
(A ∩ B) ∪ (c(A) ∩ B) ∪ (A ∩ c(B)) ⊆ A ∪ B:
In the numerical membership domain, we need to
show that
[(a; b); (a;
X b)] 6 b and
(2)
hold in general where aX = 1 − a, bX = 1 − b, and
a; b ∈ [0; 1]. While (2) is true in general for all tnorms, (1) requires a bit of work, but can be shown
to hold for the four well-known De Morgan Triples
as will be done shortly below. If (1) and (2) hold, it
is true that (3) holds.
X (a; b)] 6 (a; b):
[[(a;
X b); (a; b)];
(idempotency and distributivity)
= [A ∪ (B ∩ c(B))] ∩ [B ∪ (A ∩ c(A))]
(law of contradiction) = [A ∪ ∅] ∩ [B ∪ ∅]
(identity; boundary) = A ∩ B
FDCF (A OR B) ⊆ FCCF (A OR B);
X 6a
(a; b)
29
(1)
(6)
[(a; b); (a;
X b)] 6 b
(for = ⊕; = ⊗; and −);
(7)
[(a; b); (a;
X b)] 6 b
(for = L⊕ ; = L⊗ ; and −);
(8)
[(a; b); (a;
X 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, “algebraic”, and “maximum” union operators, respectively.
Since (2) holds in general for all t-norms
and conorms then it is clear that for the wellknown cases discussed above, we have FDCF
(A AND B) ⊆ FCCF (A AND B)!
30
I.B. T(urks*en / Fuzzy Sets and Systems 127 (2002) 17–36
Also, it can be shown that this containment holds
true for all 16 combinations of concepts shown in
Table 1 for the well known-cases of t-norms and
t-conorms discussed above. It is also clear that special
classes of t-norm and t-conorms that are transformed
to one of these well-known t-norms and t-conorms
also have the containment relationship. However, a
general proof for all t-norms and conorms have not
been shown as yet. This is left as an open question
for researchers!
8. FDCF(AND) ⊆ FCCF(→)
Let us show that FDCF (A AND B) is included
in FCCF (A → B) in a similar manner such that
FDCF (A AND B) ⊆ FCCF (A OR B) which is shown
in T(urk*sen [37]. However, we will show the alternate
proof here.
Recall that
FDCF (A AND B) = A ∩ B;
FCCF (A AND B)
7. Comprimized reasoning
= (A ∪ B) ∩ (c(A) ∪ B) ∩ (A ∪ c(B));
Yager and Filev (1994) proposed “Comprimized
Reasoning” as the convex linear combination of the
two extreme reasoning methods that are used in the
current literature as F(y) = (1− )FM (y)+ FGMP (y)
where FM (y) is the consequence of Mamdanitype reasoning and FGMP (y) is the consequence of
GMP-type reasoning. It should be recalled that in
Mamdani-type product reasoning, the implication
A → B = A × B and in GMP-reasoning, the implication A → B = c(A) ∪ B are used.
Naturally, for a singleton observation
A (x) = 1 for x = x0 , we get
FDCF (A → B)
= (A ∩ B) ∪ (c(A) ∩ B) ∪ (c(A) ∩ c(B));
FCCF (A → B) = c(A) ∪ B:
With the general result of subsection previously
discussed, 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
A (x0 ) ◦ (A(x) → B(y)) = A(x0 )TB(y)
FDCF (A AND B) ⊆ FCCF (A → B):
in Mamdani-type reasoning, and
for all t-norms and t-conorms.
Next, it is clear that, for some but not all t-norms
and t-conorms
0
0
A (x ) ◦ (A(x) → B(y)) = c(A(x )B(y)
in GMP-type reasoning.
We observe that
FCCF (A AND B) ⊆ FDCF (A → B):
However, it can be shown that
FDCF (A AND B) = A ∩ B;
FDCF (A AND B) ⊆ FDCF (A → B):
FCCF (A → B) = c(A) ∪ B:
Since,
Therefore, we can write
A ∩ B ⊆ (A ∩ B) ∪ (c(A) ∩ B) ∪ (c(A) ∩ c(B)):
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 general in order to prove that values can be found either experimentally or computationally via supervised
learning.
These arguments are graphically shown in Fig. 10.
Therefore, the “convex-linear-comprimized-reasoning”
A(x0 )TB(y)
6 (1 − )[A(x0 )TB(y)] + [c(A(x0 ))B(y)]
6 c(A(x0 ))B(y);
(10)
I.B. T(urks*en / Fuzzy Sets and Systems 127 (2002) 17–36
31
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 a
t-conorm.
Therefore, inequality (10) represents all gradations between the two boundaries of Mamdani-type
reasoning identi/ed by FDCF (A AND B) and
FCCF (A AND B), and the two boundaries of GMPtype reasoning identi/ed by FDCF (A → B) and
FCCF (A → B).
This “convex-linear-comprimize-reasoning” is
demonstrated graphically in Figs. 11 and 12 for a
32
I.B. T(urks*en / Fuzzy Sets and Systems 127 (2002) 17–36
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 bold
operators.
It is to be observed that the upper and lower bounds
of material and product implications are equal to each
other for bold operators (Fig. 12).
The “exponential-comprimize-reasoning” is demonstrated in Figs. 13 and 14 graphically for particular
values of A(x) and B(y) and for algebraic and bold
operators.
Here, again it should be observed that the upper and
lower bounds of material and product implications are
equal to each other for bold operators (Fig. 14).
9. Exponential-comprimized-reasoning
In analogy to the “linear-comprimized”, we next
proposed 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 of
t-norms and co-norms.
10. Conclusion
It is argued that Type 2 membership functions provide a better and richer grounding for the meaning representation of words and reasoning with robust computations in computing meaning of words in CWW.
The initial step in developing “Full” Type 2 system
modeling is “Interval-Valued” Type 2 schema.
Also, it is suggested that Type-2 knowledge representation and reasoning add further re/nements to
fuzzy system modeling. In Type 2 representation,
second-order gradation for the meaning of words,
linguistic terms of linguistic variables are induced
I.B. T(urks*en / Fuzzy Sets and Systems 127 (2002) 17–36
33
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 representation of words in computing. There are advantages to
Type 2 knowledge representation and reasoning because it does not reduce information granulation due
to reductionist curve /tting techniques in knowledge
representation and it does not reduce the imprecision
expansion due to disjunction and conjunctive canonical forms. The exposition of information content
embedded in information granules allows decision
makers to be more aware of risks associated with
imprecise information availability and its analysis.
It was also shown that for lower and upper bounds
determined with bold operators, the material implication and the product implication types of reasoning
are equivalent to each other. Such results suggest that
there is an extra information content to be discovered
with Type 2 knowledge representation and computation which is executed with Type 2 reasoning formulas.
This further suggests that if we capture knowledge
representation with Type 2 schema and compute with
Type 2 reasoning formulas, we can discover additional
information content. For example, Full Type 2 reasoning produces Type 3 fuzziness which may be useful in
some cases. It should be recalled that Type 1, Type 2
and Type 3 information are analogous to /rst, second
and third moments in statistical calculations but not
the same. We expect to report on some of these insights in future investigations.
Appendix A Canonical form derivation algorithm
(a) First, assign truth values T; F to the metalinguistic values (labels, variables) A and B and then
assign truth values T; F to the meta-linguistic expression of concern, say “A AND B” in order to de/ne its
meaning.
(b) Next, construct primary conjunctions of the set
symbols A; B, corresponding to linguistic values such
that in a given row,
34
I.B. T(urks*en / Fuzzy Sets and Systems 127 (2002) 17–36
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) for
bold operators.
(i) if a T appears, then take the set aWrmation symbol of that meta-linguistic variable; otherwise
(ii) if an F appears, then take the set of complementation symbol of that meta-linguistic variable;
(iii) next, conjunct the two symbols.
(c) Then, construct the disjunctive normal form of
the meta-linguistic expression of concern:
(i) /rst, take the conjunctions corresponding to the
T ’s of the truth assignment made under the column of the meta-linguistic expression, such as
“A AND B”, and
(ii) next, combine these conjunctions with disjunctions.
(d) Next, construct the conjunctive normal form of
the meta-linguistic expression of concern:
(i) /rst, take the conjunctions corresponding to F’s
of the truth assignment made under the column of
the meta-linguistic expression, such as “A AND
B” and
(ii) then, combine these conjunctions with disjunctions, and
(iii) next, take the complement of these disjuncted
conjunctions.
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