Interval Valued Fuzzy Sets from Continuous Archimedean
Triangular Norms
Taner Bilgic and I. Burhan Turksen
Department of Industrial Engineering
University of Toronto
Toronto, Ontario,
M5S 1A4 Canada
[email protected], [email protected]
1 Introduction
2 Basic operations on the
unit interval
Interval valued fuzzy sets are suggested in (Turksen 1986) to model the situations where linguistic connectives as well as the variables are fuzzy.
They are dened using the discrepancy of conjunctive and disjunctive Boolean Normal Forms
in the fuzzy case. The discrepancy is due to relaxing some of the axioms of classical logic.
In Section 2 we briey investigate the basic operations in the unit interval. Specically the literature on representing the negation functions and
triangular norms is recalled. Archimedean triangular norms are investigated as possible candidates for the logical connective AND. De Morgan
triples are constructed utilizing a general result
for negations. Two broad families of De Morgan
triples are identied; strict and strong, which are
neither distributive nor idempotent.
Section 3 introduces the concept of an interval
valued fuzzy set and presents the main results of
the paper, namely for strict and strong De Morgan triples interval valued fuzzy sets are well dened.
The paper is technical in nature and extends
some results obtained in (Turksen 1986) to more
general settings using generator functions.
This section summarizes some basic operations
on the unit interval.
A continuous, strictly increasing function ' :
[0; 1]2 ! [0; 1] satisfying boundary conditions
'(0) = 0 and '(1) = 1 is called an automorphism of the unit interval. Note that the inverse,
';1, is also increasing.
A continuous, strictly decreasing function n :
[0; 1] ! [0; 1] satisfying boundary conditions
n(0) = 1 and n(1) = 0 is called a strict negation .
A strict negation which satises n(n(x)) = x
for every x 2 [0; 1] is called a strong negation
and is denoted by N. A standard example of a
strong negation is given by N(x) = 1 ; x called
the pseudo-complement .
As for the representation of a negation function
we have the following theorem (Trillas 1979).
Theorem 2.1 n is a negation function if and
only if there exists a continuous strictly increasing function : [0; 1] ! Re such that (0) = 0,
(1) < +1 and n(x) = ;1((1) ; (x)):
The representation of the above theorem can
also be stated in terms of a continuous strictly
Appeared in the Proceedings of FUZZ-IEEE '94, Or- decreasing function for strong negation functions
as in the following
lando, pp. 1142{1147
1
Theorem 2.2 n is a negation function if and Note that since t{norms are associative,
only if there exists a continuous strictly decreasT(x; y; z) = T (T(x; y); z) = T (x; T(y; z))
ing function g : [0; 1] ! Re such that g(1) = 0,
g(0) < +1 and n(x) = g;1 (g(0) ; g(x)):
is well dened.
A similar result gives a representation in terms
of automorphisms (Ovchinnikov and Roubens
1991).
Theorem 2.3 Any strong negation N can be
represented by an automorphism ' of [0; 1] as
N(x) = ';1 (1 ; '(x))
Triangular norms (t{norms) are developed as
tools to use in probabilistic metric spaces (cf.
Schweizer and Sklar (1983)). Weber (1983) proposed to use them as connectives in fuzzy set theory.
Although, in general, t{norms are not necessarily in [0; 1] all continuous t{norms are in [0; 1].
In this study only continuous t{norms are considered.
A continuous t{norm is dened as a symmetric, associative, nondecreasing and continuous
function, T : [0; 1]2 ! [0; 1], satisfying boundary
condition T (1; x) = x for all x 2 [0; 1]:
Denition 2.1 A t{norm T
(a) is Archimedean if T (x; x) < x for all x 2
(0; 1),
(b) has zero divisors if T (x; y) = 0 for some positive x and y,
(c) is strict if it is continuous on [0; 1]2 and
strictly increasing in each place on (0; 1]2.
A typical example of a continuous t{norm with
zero divisors is the Lukasiewicz t{norm or the
bold intersection: TB (x; y) = maxfx+y ; 1; 0g A
typical continuous strict t{norm is the algebraic
product: TA (x; y) = xy
A symmetric, associative and nondecreasing
function S : [0; 1]2 ! [0; 1] is called a t{conorm
if it satises the boundary condition S(0; x) = x
for every x 2 [0; 1].
A t{conorm can be obtained from a t{norm by:
S(x; y) = n;1 (T(n(x); n(y))):
(1)
2.1 Generators of Continuous
Archimedean t{norms
In this section we briey present additive generators of continuous Archimedean t{norms. For
more details see Schweizer and Sklar (1983).
The following is a representation theorem of
Ling (1965). (See Schweizer and Sklar (1983) for
historical comments on this representation.)
Theorem 2.4 A t{norm T is continuous and
Archimedean if and only if there exists a continuous and strictly decreasing function g : [0; 1] !
Re+ with g(1) = 0 and
T(; ) = g;1 (minfg() + g(); g(0)g): (2)
where g[;1] () = g;1 (minf; g(0)g) for 2 Re+
and is called the quasi{inverse of g.
A t{norm, T, which satisfy the hypotheses of
Theorem 2.4 is said to be additively generated by
g and g is called the additive generator of T.
In terms of Theorem 2.4 a t{norm is strict if
and only if g(0) = +1 and has zero divisors if
and only if g(0) < +1.
Since all continuous t{norms are in [0; 1] it is
interesting to nd representations for continuous
Archimedean t{norms in terms of automorphisms
of the unit interval. The following result is proved
in Schweizer and Sklar (1983)
Theorem 2.5 Any continuous, strict t{norm T
can be represented as a '{transform of algebraic
product, TA as
T(x; y) = ';1 ('(x)'(y))
(3)
where ' is an automorphism of the unit interval.
This result shows that any continuous, strict
t{norm is isomorphic to the algebraic product.
A similar result is valid for t{norms with zero divisors, (Ovchinnikov and Roubens 1991), stating
that any continuous t{norm with zero divisors is
isomorphic to the Lukasiewicz t{norm.
2
2.2 De Morgan Triples
DNF representations of the concepts do not coto their CNF representations in many{
Denition 2.2 If T is a continuous t{norm, n incide
valued
logic,
for certain t{norm families DNF
is a strict negation and (1) holds, then the triple is included inand
the
corresponding CNF, Turksen
hT; S; ni is called a De Morgan triple.
(1986) proposed to dene the interval{valued
Denition 2.3 If hT; S; N i is a De Morgan fuzzy set (IVFS) as follows:
triple such that T has zero divisors, N is a strong
IV FS() = [DNF(); CNF()]
negation, then hT; S; N i is called a strong or
Lukasiewicz like De Morgan triple. In this case Assume that a De Morgan triple hT; S; N i is
using Theorems 2.2 and 2.4,
used to model conjunction, disjunction and complement respectively, Table 2 shows the canonical
T (; ) = g;1 (minfg() + g(); g(0)g) (4) Boolean normal forms in the membership domain
S(; ) = g;1 (maxfg() +
for the concepts given in Table 1. The lowercase
g() ; g(0); 0g)
(5) letters denote membership function values in the
N() = g;1 (g(0) ; g())
(6) unit interval (i.e., x = X ()).
Denition 2.4 If hT; S; N i is a De Morgan 3.2 Interval{Valued Fuzzy Sets
triple such that T is strict, N is a strong negafrom Continuous Archimedean
tion and both are generated by the same autot{norms
morphism ', then hT; S; N i is called a strict De
Morgan triple. In this case,
It should be observed that for a given De Mor-
gan triple, the sixteen t{normed representaT(x; y) = ';1 ('(x)'(y))
(7) tions given in Table 2 can be partitioned as
S(x; y) = ';1 ('(x) + '(y) ; '(x)'(y))(8) f1; 2g; f3; :: :; 10g; f11; 12g; f13;: ::; 16g, i.e., (1)
N(x) = ';1 (1 ; '(x))
(9) and (2) are equivalent, (3){(10) are equivalent,
(11) and (12) are equivalent and (13){(16) of Table 2 are equivalent to each other in form.
This fact is rst realized in (Piaget 1949) for
3 Interval{Valued
Fuzzy two{valued
logic. Dubois and Prade (1980) disSets from Boolean Normal cuss it for fuzzy
sets without going into normal
forms.
Turksen
(1984) establishes the equivaForms
lence of (1) and (2), (3){(6) and (13){(14) as
In this section, interval{valued fuzzy sets as de- examples of the equivalence in terms of a group
ned by Turksen (1986) are introduced. Basic structure.
denitions are given in Section 3.1 and then it is Therefore in order to show DNF() CNF()
shown that interval{valued fuzzy sets are well de- for all sixteen combined concepts with a particuned for De Morgan triples built from continuous lar De Morgan triple, it is sucient to show that
the following are satised:
Archimedean t{norms in Section 3.2.
S[T(x; y); T(x; N(y)); T(N(x); y)] 3.1 Basic Denitions
S(x; y)
(10)
The Boolean Disjunctive and Conjunctive normal
S[T(x; y); T (N(x); N(y))] forms (DNF and CNF, respectively) are equivaT[S(x; N(y)); S(N(x); y)]
(11)
lent in classical logic.
S[T(x; y); T(x; N(y))] In many{valued logics however, DNF may not
be equal to CNF in general. Establishing that
T[S(x; y); S(x; N(y))]
(12)
3
Table 1: List of Combined Concepts for X and Y
Number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Concept
Complete Armation
Complete Negation
Disjunction
Conjunctive Negation
Incompatibility
Conjunction
Implication (IF .. THEN)
Non-Implication
Inverse Implication
Non-inverse Implication
Equivalence
Exclusion
Armation
Negation
Armation
Negation
Combination
True
False
X OR Y
NOT X AND NOT Y
NOT X OR NOT Y
X AND Y
NOT X OR Y
X AND NOT Y
X OR NOT Y
NOT X AND Y
X IFF Y
X XOR Y
X
NOT X
Y
NOT Y
Note that for rows (1) and (2) of Table 2, hT; S; N i. First we write (11), (12) and (13) in
DNF() CNF () is always satised for all terms of (4),(5) and (6).
t{norms and it is sucient to show that (Turksen 1986)
S(T(x; y); T (x; N(y))) x
(13) S(T(x;;1y); T (x; N(y))) =
g (maxfminfg(x) + g(y); g(0)g +
holds for all x 2 [0; 1] in order (10) to be satised.
minfg(x) ; g(y); 0g; 0g)
(14)
The main result of this paper is that, for strong S(T(x; y); T (N(x); N(y))) =
and strict De Morgan triples, interval valued
g;1 (maxfminfg(x) + g(y); g(0)g +
fuzzy sets are well dened. Theorem 3 of Turksen
minfg(0) ; g(x) ; g(y); 0g; 0g)
(15)
(1986) establishes the conditions under which the
premise should hold for dierent families of con- T(S(x; N(y)); S(N(x); y)) =
nectives. Here, we establish the result for strict
g;1 (minfmaxfg(x) ; g(y); 0g +
and strong De Morgan triples using their genermaxfg(y) ; g(x); 0g; g(0)g)
(16)
ating functions.
T(S(x; y); S(x; N(y))) =
Theorem 3.1 If hT; S; N i is a strong De Morg;1 (minfmaxfg(x) + g(y) ; g(0); 0g +
gan triple then DNF() CNF() for the sixteen
maxfg(x) ; g(y); 0g; g(0)g)
(17)
combined concepts.
Proof: The proof is given in terms of the gener-
Now our aim is translated to showing (14) x,
ating functions. Recall that the strong De Mor- (14) (17), and (15) (16). There are 4 cases
gan triple, hT; S; N i is given by (4),(5) and (6) to consider:
respectively. We will show that (11), (12) and
(13) are satised for a strong De Morgan triple, 1. g(0) g(x) + g(y) and g(x) ; g(y) > 0.
4
Table 2: T-normed representation of Boolean Normal Forms
No:
1
DNF
S [T (x; y); T (x; N (y)); T (N (x); y); T (N (x); N (y))]
2
0
3
4
5
6
7
8
9
10
11
12
13
14
15
16
S [T (x; y); T (x; N (y)); T (N (x); y)]
T (N (x); N (y))]
S [T (x; N (y)); T (N (x); y); T (N (x); N (y))]
T (x; y)
S [T (x; y); T (N (x); y); T (N (x); N (y))]
T (x; N (y))
S [T (x; y); T (x; N (y)); T (N (x); N (y))]
T (N (x); y)
S [T (x; y); T (N (x); N (y))]
S [T (x; N (y)); T (N (x); y)]
S [T (x; y); T (x; N (y))]
S [T (N (x); y); T (N (x); N (y))]
S [T (x; y); T (N (x); y)]
S [T (x; N (y)); T (N (x); N (y))]
2. 2g(0) g(x) + g(y) > g(0) and
g(y) ; g(x) g(0)
3. 2g(0) g(x) + g(y) > g(0) and
g(y) ; g(x) < 0
4. g(0) g(x) + g(y) and
g(x) ; g(y) 0
The details are straightforward. 2
An immediate consequence is the following
Corollary 3.1 If '(z) = z for all z 2 [0; 1] then
CNF
1
T [S (x; y); S (x; N (y)); S (N (x); y); S (N (x); N (y))]
S (x; y)
T [S (x; N (y)); S (N (x); y); S (N (x); N (y))]
S (N (x); N (y))]
T [S (x; y); S (x; N (y)); S (N (x); y)]
S (N (x); y)
T [S (x; y); S (x; N (y)); S (N (x); N (y))]
S (x; N (y))
T [S (x; y); S (N (x); y); S (N (x); N (y))]
T [S (x; N (y)); S (N (x); y)]
T [S (x; y); S (N (x); N (y))]
T [S (x; y); S (x; N (y))]
T [S (N (x); y); S (N (x); N (y))]
T [S (x; y); S (N (x); y)]
T [S (x; N (y)); S (N (x); N (y))]
S(T (x; y); T(x; N(y))) =
';1('(x)'(y) + '(x)'(y)
;
'(x)2'(y)'(y))
(18)
S(T (x; y); T(N(x); N(y))) =
';1('(x)'(y) + '(x)
'(y)
;
'(x)'(y)'(x)
'(y))
(19)
T(S(x; N(y)); S(N(x); y)) =
';1(('(x) + '(y) ; '(x)'(y))
('(x)'(y)
; '(x)'(y)))
(20)
T(S(x; y); S(x; N(y))) =
';1(('(x) + '(y) ; '(x)'(y))
('(x) + '(y) ; '(x)'(y)))
(21)
Note that '(x)
= 1 ; '(x) is used to simplify
the notation.
Now our aim is translated to showing (18) x,
(18) (21), and (19) (20). The rest of the
proof is straightforward. 2
Note that one does not have to resort to dierentiability in order to prove Theorem 3.2.
An immediate consequence is the following
Corollary 3.2 If '(z) = z for all z 2 [0; 1] then
DNF() CNF() for Lukasiewicz triples.
The same result can be shown for strict De
Morgan triples.
Theorem 3.2 If hT; S; N i is a strict De Morgan triple then DNF() CNF() for the sixteen
combined concepts.
Proof: Recall that a strict De Morgan triple,
hT; S; N i is given by (7),(8) and (9) respectively.
We will show that (11), (12) and (13) are satised for a strict De Morgan triple, hT; S; N i.
First we write (11), (12) and (13) in terms of
(7),(8) and (9).
5
DNF() CNF() for Algebraic triples.
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