Extended Triangular Norms on Gaussian Fuzzy Sets - Semantic Scholar
EUSFLAT - LFA 2005
Extended Triangular Norms on Gaussian Fuzzy Sets
Janusz T. Starczewski
Department of Computer Engineering, Czestochowa University of Technology, Czestochowa, Poland ˛ ˛ Department of Artificial Intelligence, WSHE University in Łód´z, Łód´z, Poland [email protected]
Abstract
So far, computational complextity of the general formula for the extended t-norm does not allow to construct fuzzy logic systems of type-2 other than interval type. In this paper, we derive new formulae for extended t-norms for arguments with Gaussian and piecewise-Gaussian membership functions basing on our original theorems. Keywords: Gaussian Type-2 Fuzzy Sets, Extended T-norms. 1
Introduction
Recently, we have witnessed a rising interest in design of type-2 fuzzy logic systems. Let us recall that type-2 fuzzy sets are characterized by classical fuzzy subsets of [0, 1] as their membership grades. The key to design type-2 fuzzy logic systems is to find efficient formulae for extensions of classical triangular norms, called extended tnorms. In this case, extended t-norms operate on classical fuzzy subsets of [0, 1] called fuzzy truth values. In literature, only interval type-2 fuzzy sets have been used to construct concrete designs of fuzzy logic systems (see e.g. [5]). This has been caused mainly by the lack of exact output formulae for non-interval membership functions of extended tnorms. The unique exception is our design of a triangular type-2 fuzzy logic system [10]. This paper presents new formulae of extended t-norms for Gaussian fuzzy truth values, and efficient methods which calculate approximate extended t-norms for Gaussian and piecewise-
872
Gaussian fuzzy truth values. The derived formulae allow to examine the utility of Gaussian type-2 fuzzy sets in designing fuzzy logic systems. We are positive that there exists a field of applications where Gaussian type-2 fuzzy systems accomplish better performance than traditional (type-1) fuzzy logic systems, since we have demonstrate it for interval type-2 fuzzy logic systems [8]. 1.1
Type-2 Fuzzy Sets Context
Let the set of all fuzzy subsets of the unit interval [0, 1] be denoted by F ([ 0, 1]). The type-2 fuzzy set in the real line R, denoted by A˜, is characterized by the fuzzy membership function (MF) A˜ : R → F ([0, 1]). The values of this function are fuzzy membership grades A˜ (x), i.e., classical fuzzy subsets of the unit interval [0, 1] characterized by fx : [0, 1] → [0, 1]. The function fx of each fuzzy membership grade is called a secondary MF. In this paper only Gaussian and piecewise-Gaussian secondary MFs are considered. Usually F ([0, 1]) comprise a special kind of fuzzy truth values, so-called fuzzy truth numbers. A fuzzy truth number (FTN) is a fuzzy subset of [0, 1] which is also normal for unique element, i.e., ∃!u ∈ [0, 1] : f (u) = 1, and convex, i.e., ∀u1 , u2 , λ ∈ [0, 1], f (λu1 + (1 − λ) u2) min (f (u1 ) , f (u2)). 2
Extended triangular norms
The commonly known Zadeh extension principle extends classical t-norms to operate on fuzzy truth values. Let F and G be fuzzy truth values,
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with the membership functions f and g, respectively. An extension of a t-norm T based on a t-norm T∗ according to the generalized extension principle is called an extended triangular norm and is expressed as a function of w, i.e.,
T˜ ∗ (F, G) (w) = sup T∗ (f (u) , g (v)) . (1) T
T (u,v )=w
This general presentation of the extended t-norm is useless in practise because of a huge computational effort. Note that the resultant membership grade is the maximal value of T∗ (f (u) , g (v)) for all pairs {u, v} which produce the same element w. In this approach, we are forced to discretize domains of u and v, and to work on tabularized functions instead of an explicit parametrized function of w. Only a combination of certain classes of membership functions (in this paper piecewiseGaussian functions are considered) and certain classes of fuzzified t-norms and t-norms being the base for the extension, let us achieve exact analytical formulae for extended t-norms [1, 2, 3, 9]. In [4] it has been proved that the extended tnorm satisfies type-2 t-norm axioms (monotonicity, commutativity, associativity, existence of the unit element) while T∗ is min and participant MFs are convex and upper semicontinuous. Mathematical basics of type-2 fuzzy sets are presented in [11]. Extended t-norms are used to form the intersection of type-2 fuzzy sets, in construction of type2 FLSs. Having type-2 fuzzy sets A˜ (x) = f (u) and B˜ (x) = g (v) (u, v ∈ [0, 1]) the intersection may be calculated ∀x ∈ R as x
3
Extended T-norms on Gaussian FTNs
3.1 Extended Łukasiewicz T-norm Based on Product T-norm In this section we will make use of our theorem concerning strict t-norms and arguments of a specific function form (as in [6]). IF a continuous t-norm Ts is strict the inequality Ts (u, w) < Ts (v, w) for all u < v and w > 0 is satisfied. Its strictly decreasing additive generator φ : [0, 1] → −1 [0, ∞] ensures that Ts (u, v) = φ (φ ◦ u + φ ◦ v).
[−∞, ∞] → [0, ∞) be a continuous convex function such that κ (0) = 0 and κ (x) = κ (−x) for all x ∈ R. Let Ts be a strict t-norm with an additive generator φ. If the arguments F and u−mGare character− 1 and g (v) = ized by f (u) = φ aκ a φ−1 bκ v−b n ; a, b > 0; m, n ∈ [0, 1]. Then the extension of the Łukasiewicz t-norm TL based on a strict t-norm Ts is characterized by
Theorem 1 Let κ :
T˜L Ts (F, G) (w) µ0 φ−1 (a + b) κ w−ma+−bn+1 |
=
if
w = 0,
otherwise
,
where
µ0
=
1
φ−1 (a + b) κ
−m−n 1
a+b
TL (m, n) = 0, otherwise. if
x
A˜ (x) ∩ B˜ (x)
= =
T˜ A˜ (x) , B˜ (x) (w) sup T ∗ (f (u) , g (v )) . x
x
T (u,v)=w
If arguments of T˜ are defined on different domains, the extended t-norm forms a type-2 fuzzy relation, i.e., if B˜ (y) = g (v) then
Since the additive generator
=
=
x
T (u,v)=w
y
(2) Therefore, the proof relies on the evaluation of =
(details will be given in [9]).
873
aκ u−am + bκ v−b n u+v−1=w −n inf aκ u−am + bκ w−u+1 b u∈[w,1] inf
T˜ A˜ (x) , B˜ (y) (w) sup T ∗ (f (u) , g (v)) .
is strictly decreas-
T˜Ts (F, G) = φ−1 T (u,v inf (φ ◦ f (u) + φ ◦ g (v)) )=w
y
R˜ (x, y)
φ
ing, the generalized extension principle is
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Now consider the extended Łukasiewicz t-norm based on the product t-norm on Gaussian FTNs, i.e.,
f (u)
= exp
g (v)
= exp
u − mF 2 , − σF v − mG 2 . − σG
(3) (4)
and let the FTNs F and G be characterized by upper semicontinuous MFs f and g, such that f (mF ) = g (mG) = 1; mF mG. Then the extension of a continuous t-norm T based on TD is characterized by
− − max f TmG (w) , g TmF (w) w ∈ [0,T (mF , mG)] , max f Tm−G (w) , g Tm−F (w) T˜TD (F, G) = − w∈ (T (mF , mG), mF ] , f TmG (w) w ∈ (mF , mG] , 0 [ 1]
[ 1]
( 1)
( 1)
if
The product TP is obviously a strict t-norm. For the additive generator φ = − log x, the function κ must be defined by κ = x2 and the following substitutions must be done: σ2F = a, σ2G = b, mF = m, mG = n; mF , mG ∈ [0, 1]. The use of Theorem 1 leads to the result
T˜L|Ts (F, G) (w) = exp
Theorem 3 Let TD be the drastic product t-norm
if
( 1)
if
otherwise.
By applying the theorem to the two Gaussian 2 w − m − m + 1) given by (3) and (4), MF of the extended F G . FTNs − product based on the drastic product is σ2F + σ2G
(
3.2 Extended Product T-norm Based on Drastic Product
→ [0, 1] be a nonconstant and either non-decreasing or nonincreasing function. A pseudo-inverse of f is defined by { ∈
|
}
f [−1] (φ) = sup u [a, b] f (u) φ
−
˜P TD (F, G
T
Since at the beginning we do not assume the strictness of t-norms, the notion of a pseudoinverse will be very useful.
Definition 2 Let f : [0, 1]
2 w m m F G exp − mGσF ) = max 2 w m m F G exp − mF σG 2 w mF mG
|
,
−
= exp −
−
max(mG σF ,mF σG )
.
Note, that this result have the same form as the approximate result of Karnik and Mendel [3, 5]. 4
Gaussian Approximations for Triangular Norms
4.1 Approximate Product-based Extended Product T-norm
for non-decreasing f , and
{ ∈
| } for non-increasing f , where sup ∅ = a. f [−1] (φ) = sup u [a, b] f (u) φ
Let φ( −1) signify the strict pseudo-inverse when in Definition 2 the operators and are replaced by < and >, respectively. Here we present the formula for extended continuous t-norms based on the weakest t-norm, i.e. the drastic product, TD = {min (x, y) if max (x, y) = 1; and 0 otherwise} (for the proof see [9]).
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Calculations of the extended product are usually complicated for Gaussian operands. Moreover, the exact results of the extended t-norms quite often do not remain Gaussian. Therefore, some Gaussian approximations of extended t-norms may be presented. One known approach for Gaussian approximations of the product-based extended product has been proposed by Karnik and Mendel [3, 5], i.e., ˜P |T (F, G) = exp P
T
−
√(σ wm−m)2F+(mσG m )2 F G G F
2
.
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4.2 Approximate Minimum-based Extended Product T-norm Here we may propose new approximation derived from our theorem [9]. In this theorem, arguments for an extended continuous t-norm will be regarded as upper semicontinuous FTNs, i.e., its all α-cuts are closed subintervals of [0, 1].
Theorem 4 Consider two FTNs, F and G, with
upper semicontinuous MFs f and g, which are normal, f (mF ) = g (mG ) = 1. The extension of continuous t-norm T based on the minimum t-norm can be expressed as
w[−1] (w) T˜TM (F, G) = 1 w[−1] (w)
where
if if if
w ∈ (0, T (mF , mG)) w = T (mF , mG) w ∈ (T (mF , mG) , 1]
w = T f [−1] (µ) , g[−1] (µ) for f [−1] (µ) ∈ [0, mF ) and g[−1] (µ) ∈ [0, mG ) w = T f [−1] (µ) , g[−1] (µ) for f [−1] (µ) ∈ (mF , 1] and g[−1] (µ) ∈ (mG , 1] We may use the Nguyen theorem [7], which states that for a continuous binary operation ∗, its extension can be derived using µcuts, i.e., [F ˜∗TM G]µ = [F ]µ ∗ [G]µ, where ∗ may be any arbitrary continuous t-norm T. Obviously, T˜TM (F, G) 1 = T ([F ]1 , [G]1), i.e., the resultant MF is equal to unity whenw = T (mF , mG ). Otherwise, since T˜TM (F, G) µ = T [F ]µ , [G]µ , Proof.
µ[−1] = T f [−1] (µ) , g[−1] (µ)
separately for w ∈ (T (mF , mG ) , 1] and w ∈ (0, T (mF , mG )). The upper semicontinuity of f and g ensures that w[−1] [−1] = w. Consequently, for w ∈ (T (nF , nG ) , 1], we consider only upper bounds of µ-cuts, i.e., f [−1] (µ) ∈ (mF , 1] and g[−1] (µ) ∈ (nG , 1], and for w ∈ [0, T (mF , mG )), lower bounds of µ-cuts are respected, i.e., f [−1] (µ) ∈ [0, mF ) and g[−1] (µ) ∈ [0, mG ).
The use of Theorem 4 leads to a nice closed-form Gaussian approximations. Firstly, the following equations have to be rearranged according to u and v on separate intervals of invertibility:
2
2
F log µ = − u−σm F
G log µ = − v−σm G
,
. Thus, the inverse functions are given by √− log µ for f −1 ∈ [0, m ] , − σ m F F F − 1 f = √ − 1 mF + σF − log µ for f ∈ [mF , 1] , (5) √ −1 g−1 = mG − σG√− log µ for g−1 ∈ [0, mG] , mG + σG − log µ for g ∈ [mG, 1] . By the product of lower and upper inverse functions, we have √ mF mG − (mGσF + mF σG) − log µ if µ ∈ [0, mF mG ] , f −1g−1 = −σF σG log µ √ mF mG + (mGσF + mF σG) − log µ −σF σG log µ if µ ∈ [mF mG , 1] . (6) At this point, differences between the forms (6) and (5) should be noticed. Since a Gaussian MF is√expected as a result, in (6) the summand σF σG − log µ should be omitted. Thus, the approximating assumption is as follows (mG σF + mF σG ) − log µ |σF σG log µ| = −σ F σG log µ mF + mG − log µ σG σF √ The function √− log µ is decreasing on [0, 1] and its average is 2π . From the other hand, mean values mF and mG are certain numbers in [0, 1]. Therefore the Gaussian approximation is justified for sufficiently small standard deviations σF and σG . The most advantageous case obviously is when the argument MF with a lower mean value has a greater standard deviation value. Then the following approximation is achieved √ mF mG − (mGσF + mF σG) − log µ if µ ∈ [0, mF mG ] , f −1g−1 ∼ = mF mG + (mGσF + mF σG) √− log µ if µ ∈ [mF mG, 1] .
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which has suitable form to obtain the inverse function
T˜P TM (F, G) (w) = exp − |
5
w mF mG 2 . σF mG +σGmF −
(7)
Piecewise-Gaussian approximations for triangular norms
All presented Gaussian approximations of the extended t-norms vary significantly from the usually strongly asymmetric exact results of applying the generalized extension principle. For that simple reason, a better approximation can be accomplished by two monotonic pieces of Gaussian functions with the same mean value and with not necessarily equal standard deviations, i.e., 2 u − m F exp − σF f (u) = 2 u − m F exp − ζ F 2 v − m G exp − σ g (v) = G 2 G exp − v−ζm G
u ∈ [0, mF ] , u ∈ (mF , 1] , v ∈ [0, mG] , v ∈ (mG, 1] .
(8)
(9)
5.1 Approximate Extended Product Based on Minimum Obviously, the extended product of FTNs characterized by (8) and (9) has a MF with the mean value m = mF mG . Here we can assume that the result consists of Gaussian components of the form
2 exp − w−σ m w ∈ [0, m] , T˜P TM (F, G) = w−m 2 w ∈ (m, 1] . exp − ζ |
(10) If we assume that the approximation has an exact value at w = 0, it is sufficient that u = 0 or v = 0. The use of the normality of f and g leads to the expression
Consequently,
2 exp − mσFF , 2 m exp − σ = max
2 exp − mσGG 2 2
= max exp − mσFFmmGG , exp − mσGFmmFG Since both functions under maximum are normal, the greater one has a greater standard deviation 2 2 m m m F G exp − σ = exp − max(σF mG ,σG mF ) . (11) The assumption of the exact value of the result at w = 1 ensures that both u = 1 and v = 1, accordingly, ˜TP (F, G) (1) = min (f (1) , g (1)) T 2 2 1 − m 1−mG F = min exp − ζ F , exp − . ζG
Since exp −x2 is a decreasing function of any positive x, the minimum of the functions goes into the function of the maximum of the arguments suchthat 2 2 . exp − 1−ζm = exp − max 1−ζmF F , 1−ζmG G (12) Combining (11) and (12) we get the following formula for a piecewise-Gaussian approximation ˜P TM (F, G) (w) T 2 w − m m F G exp − max(σF mG ,σG mF ) if w ∈ [0, mF mG ] , 2 = w−mF mG 1−m exp − F mG ,ζ 1−mF mG min ζ F G 1 − m 1 − m F G |
if w ∈ (mF mG, 1] .
Actually this approximation is a rather simple three-point piecewise Gaussian interpolation of the exact membership function. However, it is one exemplary step toward parametrized operations preserving shapes of membership functions.
sup min (f (0) , g (v)) , 5.2 Use of Classical T-norms for [0,1] Approximate Extensions T˜P |TM (F, G) (0) = max v∈sup min (f (u) , g (0)) u∈[0,1] The aim of this subsection is to present a class of approximations of extended t-norms for = max (f (0) , g (0)) .
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.
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piecewise-Gaussian FTNs which use traditional t-norms. Let the arguments be characterized by (8) and (9). The approximate extended tnorm is defined in terms of (10) where the center m = T (mF , mG) and two remaining interpolation points satisfy the following equations:
m−ζ m+σ
Therefore
ζ σ
= =
= =
T (mF − ζ F , mG − ζ G) , T (mF + σF , mG + σG) .
m − T (mF − ζ F , mG − ζ G) , T (mF + σF , mG + σG) − m.
This approach reduces calculations of extended tnorms to computing only the three characteristic variables m, ζ and σ. Arbitrary traditional tnorms here may be used. Detailed justification of this approach will be presented in our future paper. 6
Conclusion
We have proposed new formulae for extended triangular norms. In the context of Gaussian FTNs, the product-based extended Łukasiewicz t-norm, the drastic product-based extended product and the approximation of the product-based extended product have been derived. For piecewiseGaussian FTNs, the approximate minimum-based extended product and simple approximation have been proposed. These efficient results play a pivotal role in the design of type-2 fuzzy logic systems. "Symmetrical" results can be obtained for the extended complementary norms. A tremendously useful feature of the derived formulae is that resultant MFs preserve Gaussian or piecewise-Gaussian shapes of the two arguments, and this way the extended t-norms can be expanded into their multi-argument forms. Generally, the derived class of approximate tnorms belongs to the class of type-2 triangular norms. But this will be demonstrated in a journal extension of this issue. The next step toward designing piecewiseGaussian type-2 fuzzy logic systems is the construction of the efficient type-reduction dedicated to piecewise-Gaussian fuzzy sets of type-2.
References
[1] S. Coupland and R. John, “A new and efficient method for the type-2 meet operation,” Proc. FUZZ-IEEE 2004, Budapest, pp. 959964, 2004. [2] D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and Applications, Mathematics in Science and Engineering, Academic Press, Inc., NY, 1980. [3] N. N. Karnik and J. M. Mendel, "Operations on type-2 fuzzy sets," Fuzzy Sets and Systems, vol. 122, pp. 327—348, 2000. [4] M. F. Kawaguchi and M. Miyakoshi, "Extended Triangular Norms in type-2 Fuzzy Logic," EUFIT’99 7th European Congress on Intelligent Techniques & Soft Computing, Aachen, September, 1999. [5] J. M. Mendel, Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions, Prentice Hall PTR, Upper Saddle River, NJ 2001. [6] R. Mesiar, "Triangular-norm-based addition of fuzzy intervals," Fuzzy Sets and Systems, vol. 91, pp. 231—237, 1997. [7] H. T. Nguyen, "A note on the extension principle for fuzzy sets," J. Math. Anal. Appl., vol. 64, pp. 369—380, 1978. [8] J. Starczewski, "What differs interval type2 FLS from type-1 FLS," in. L. Rutkowski (Eds.) Artificial Intelligence and Soft Computing - ICAISC 2004, Lecture Notes in Computer Science, Springer pp. 381—387, 2004. [9] J. Starczewski, "Extended triangular norms," unpublished. [10] J. Starczewski and L. Rutkowski, "NeuroFuzzy Systems of Type 2," 1st Int’l Conf. on Fuzzy Systems and Knowledge Discovery, vol. 2, Singapore, pp. 458—462, 2002. [11] C. Walker and E. Walker, "The algebra of fuzzy truth values," Fuzzy Sets and Systems, vol. 149, pp. 309-347, 2005.
877
Extended Triangular Norms on Gaussian Fuzzy Sets
Janusz T. Starczewski
Department of Computer Engineering, Czestochowa University of Technology, Czestochowa, Poland ˛ ˛ Department of Artificial Intelligence, WSHE University in Łód´z, Łód´z, Poland [email protected]
Abstract
So far, computational complextity of the general formula for the extended t-norm does not allow to construct fuzzy logic systems of type-2 other than interval type. In this paper, we derive new formulae for extended t-norms for arguments with Gaussian and piecewise-Gaussian membership functions basing on our original theorems. Keywords: Gaussian Type-2 Fuzzy Sets, Extended T-norms. 1
Introduction
Recently, we have witnessed a rising interest in design of type-2 fuzzy logic systems. Let us recall that type-2 fuzzy sets are characterized by classical fuzzy subsets of [0, 1] as their membership grades. The key to design type-2 fuzzy logic systems is to find efficient formulae for extensions of classical triangular norms, called extended tnorms. In this case, extended t-norms operate on classical fuzzy subsets of [0, 1] called fuzzy truth values. In literature, only interval type-2 fuzzy sets have been used to construct concrete designs of fuzzy logic systems (see e.g. [5]). This has been caused mainly by the lack of exact output formulae for non-interval membership functions of extended tnorms. The unique exception is our design of a triangular type-2 fuzzy logic system [10]. This paper presents new formulae of extended t-norms for Gaussian fuzzy truth values, and efficient methods which calculate approximate extended t-norms for Gaussian and piecewise-
872
Gaussian fuzzy truth values. The derived formulae allow to examine the utility of Gaussian type-2 fuzzy sets in designing fuzzy logic systems. We are positive that there exists a field of applications where Gaussian type-2 fuzzy systems accomplish better performance than traditional (type-1) fuzzy logic systems, since we have demonstrate it for interval type-2 fuzzy logic systems [8]. 1.1
Type-2 Fuzzy Sets Context
Let the set of all fuzzy subsets of the unit interval [0, 1] be denoted by F ([ 0, 1]). The type-2 fuzzy set in the real line R, denoted by A˜, is characterized by the fuzzy membership function (MF) A˜ : R → F ([0, 1]). The values of this function are fuzzy membership grades A˜ (x), i.e., classical fuzzy subsets of the unit interval [0, 1] characterized by fx : [0, 1] → [0, 1]. The function fx of each fuzzy membership grade is called a secondary MF. In this paper only Gaussian and piecewise-Gaussian secondary MFs are considered. Usually F ([0, 1]) comprise a special kind of fuzzy truth values, so-called fuzzy truth numbers. A fuzzy truth number (FTN) is a fuzzy subset of [0, 1] which is also normal for unique element, i.e., ∃!u ∈ [0, 1] : f (u) = 1, and convex, i.e., ∀u1 , u2 , λ ∈ [0, 1], f (λu1 + (1 − λ) u2) min (f (u1 ) , f (u2)). 2
Extended triangular norms
The commonly known Zadeh extension principle extends classical t-norms to operate on fuzzy truth values. Let F and G be fuzzy truth values,
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with the membership functions f and g, respectively. An extension of a t-norm T based on a t-norm T∗ according to the generalized extension principle is called an extended triangular norm and is expressed as a function of w, i.e.,
T˜ ∗ (F, G) (w) = sup T∗ (f (u) , g (v)) . (1) T
T (u,v )=w
This general presentation of the extended t-norm is useless in practise because of a huge computational effort. Note that the resultant membership grade is the maximal value of T∗ (f (u) , g (v)) for all pairs {u, v} which produce the same element w. In this approach, we are forced to discretize domains of u and v, and to work on tabularized functions instead of an explicit parametrized function of w. Only a combination of certain classes of membership functions (in this paper piecewiseGaussian functions are considered) and certain classes of fuzzified t-norms and t-norms being the base for the extension, let us achieve exact analytical formulae for extended t-norms [1, 2, 3, 9]. In [4] it has been proved that the extended tnorm satisfies type-2 t-norm axioms (monotonicity, commutativity, associativity, existence of the unit element) while T∗ is min and participant MFs are convex and upper semicontinuous. Mathematical basics of type-2 fuzzy sets are presented in [11]. Extended t-norms are used to form the intersection of type-2 fuzzy sets, in construction of type2 FLSs. Having type-2 fuzzy sets A˜ (x) = f (u) and B˜ (x) = g (v) (u, v ∈ [0, 1]) the intersection may be calculated ∀x ∈ R as x
3
Extended T-norms on Gaussian FTNs
3.1 Extended Łukasiewicz T-norm Based on Product T-norm In this section we will make use of our theorem concerning strict t-norms and arguments of a specific function form (as in [6]). IF a continuous t-norm Ts is strict the inequality Ts (u, w) < Ts (v, w) for all u < v and w > 0 is satisfied. Its strictly decreasing additive generator φ : [0, 1] → −1 [0, ∞] ensures that Ts (u, v) = φ (φ ◦ u + φ ◦ v).
[−∞, ∞] → [0, ∞) be a continuous convex function such that κ (0) = 0 and κ (x) = κ (−x) for all x ∈ R. Let Ts be a strict t-norm with an additive generator φ. If the arguments F and u−mGare character− 1 and g (v) = ized by f (u) = φ aκ a φ−1 bκ v−b n ; a, b > 0; m, n ∈ [0, 1]. Then the extension of the Łukasiewicz t-norm TL based on a strict t-norm Ts is characterized by
Theorem 1 Let κ :
T˜L Ts (F, G) (w) µ0 φ−1 (a + b) κ w−ma+−bn+1 |
=
if
w = 0,
otherwise
,
where
µ0
=
1
φ−1 (a + b) κ
−m−n 1
a+b
TL (m, n) = 0, otherwise. if
x
A˜ (x) ∩ B˜ (x)
= =
T˜ A˜ (x) , B˜ (x) (w) sup T ∗ (f (u) , g (v )) . x
x
T (u,v)=w
If arguments of T˜ are defined on different domains, the extended t-norm forms a type-2 fuzzy relation, i.e., if B˜ (y) = g (v) then
Since the additive generator
=
=
x
T (u,v)=w
y
(2) Therefore, the proof relies on the evaluation of =
(details will be given in [9]).
873
aκ u−am + bκ v−b n u+v−1=w −n inf aκ u−am + bκ w−u+1 b u∈[w,1] inf
T˜ A˜ (x) , B˜ (y) (w) sup T ∗ (f (u) , g (v)) .
is strictly decreas-
T˜Ts (F, G) = φ−1 T (u,v inf (φ ◦ f (u) + φ ◦ g (v)) )=w
y
R˜ (x, y)
φ
ing, the generalized extension principle is
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Now consider the extended Łukasiewicz t-norm based on the product t-norm on Gaussian FTNs, i.e.,
f (u)
= exp
g (v)
= exp
u − mF 2 , − σF v − mG 2 . − σG
(3) (4)
and let the FTNs F and G be characterized by upper semicontinuous MFs f and g, such that f (mF ) = g (mG) = 1; mF mG. Then the extension of a continuous t-norm T based on TD is characterized by
− − max f TmG (w) , g TmF (w) w ∈ [0,T (mF , mG)] , max f Tm−G (w) , g Tm−F (w) T˜TD (F, G) = − w∈ (T (mF , mG), mF ] , f TmG (w) w ∈ (mF , mG] , 0 [ 1]
[ 1]
( 1)
( 1)
if
The product TP is obviously a strict t-norm. For the additive generator φ = − log x, the function κ must be defined by κ = x2 and the following substitutions must be done: σ2F = a, σ2G = b, mF = m, mG = n; mF , mG ∈ [0, 1]. The use of Theorem 1 leads to the result
T˜L|Ts (F, G) (w) = exp
Theorem 3 Let TD be the drastic product t-norm
if
( 1)
if
otherwise.
By applying the theorem to the two Gaussian 2 w − m − m + 1) given by (3) and (4), MF of the extended F G . FTNs − product based on the drastic product is σ2F + σ2G
(
3.2 Extended Product T-norm Based on Drastic Product
→ [0, 1] be a nonconstant and either non-decreasing or nonincreasing function. A pseudo-inverse of f is defined by { ∈
|
}
f [−1] (φ) = sup u [a, b] f (u) φ
−
˜P TD (F, G
T
Since at the beginning we do not assume the strictness of t-norms, the notion of a pseudoinverse will be very useful.
Definition 2 Let f : [0, 1]
2 w m m F G exp − mGσF ) = max 2 w m m F G exp − mF σG 2 w mF mG
|
,
−
= exp −
−
max(mG σF ,mF σG )
.
Note, that this result have the same form as the approximate result of Karnik and Mendel [3, 5]. 4
Gaussian Approximations for Triangular Norms
4.1 Approximate Product-based Extended Product T-norm
for non-decreasing f , and
{ ∈
| } for non-increasing f , where sup ∅ = a. f [−1] (φ) = sup u [a, b] f (u) φ
Let φ( −1) signify the strict pseudo-inverse when in Definition 2 the operators and are replaced by < and >, respectively. Here we present the formula for extended continuous t-norms based on the weakest t-norm, i.e. the drastic product, TD = {min (x, y) if max (x, y) = 1; and 0 otherwise} (for the proof see [9]).
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Calculations of the extended product are usually complicated for Gaussian operands. Moreover, the exact results of the extended t-norms quite often do not remain Gaussian. Therefore, some Gaussian approximations of extended t-norms may be presented. One known approach for Gaussian approximations of the product-based extended product has been proposed by Karnik and Mendel [3, 5], i.e., ˜P |T (F, G) = exp P
T
−
√(σ wm−m)2F+(mσG m )2 F G G F
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4.2 Approximate Minimum-based Extended Product T-norm Here we may propose new approximation derived from our theorem [9]. In this theorem, arguments for an extended continuous t-norm will be regarded as upper semicontinuous FTNs, i.e., its all α-cuts are closed subintervals of [0, 1].
Theorem 4 Consider two FTNs, F and G, with
upper semicontinuous MFs f and g, which are normal, f (mF ) = g (mG ) = 1. The extension of continuous t-norm T based on the minimum t-norm can be expressed as
w[−1] (w) T˜TM (F, G) = 1 w[−1] (w)
where
if if if
w ∈ (0, T (mF , mG)) w = T (mF , mG) w ∈ (T (mF , mG) , 1]
w = T f [−1] (µ) , g[−1] (µ) for f [−1] (µ) ∈ [0, mF ) and g[−1] (µ) ∈ [0, mG ) w = T f [−1] (µ) , g[−1] (µ) for f [−1] (µ) ∈ (mF , 1] and g[−1] (µ) ∈ (mG , 1] We may use the Nguyen theorem [7], which states that for a continuous binary operation ∗, its extension can be derived using µcuts, i.e., [F ˜∗TM G]µ = [F ]µ ∗ [G]µ, where ∗ may be any arbitrary continuous t-norm T. Obviously, T˜TM (F, G) 1 = T ([F ]1 , [G]1), i.e., the resultant MF is equal to unity whenw = T (mF , mG ). Otherwise, since T˜TM (F, G) µ = T [F ]µ , [G]µ , Proof.
µ[−1] = T f [−1] (µ) , g[−1] (µ)
separately for w ∈ (T (mF , mG ) , 1] and w ∈ (0, T (mF , mG )). The upper semicontinuity of f and g ensures that w[−1] [−1] = w. Consequently, for w ∈ (T (nF , nG ) , 1], we consider only upper bounds of µ-cuts, i.e., f [−1] (µ) ∈ (mF , 1] and g[−1] (µ) ∈ (nG , 1], and for w ∈ [0, T (mF , mG )), lower bounds of µ-cuts are respected, i.e., f [−1] (µ) ∈ [0, mF ) and g[−1] (µ) ∈ [0, mG ).
The use of Theorem 4 leads to a nice closed-form Gaussian approximations. Firstly, the following equations have to be rearranged according to u and v on separate intervals of invertibility:
2
2
F log µ = − u−σm F
G log µ = − v−σm G
,
. Thus, the inverse functions are given by √− log µ for f −1 ∈ [0, m ] , − σ m F F F − 1 f = √ − 1 mF + σF − log µ for f ∈ [mF , 1] , (5) √ −1 g−1 = mG − σG√− log µ for g−1 ∈ [0, mG] , mG + σG − log µ for g ∈ [mG, 1] . By the product of lower and upper inverse functions, we have √ mF mG − (mGσF + mF σG) − log µ if µ ∈ [0, mF mG ] , f −1g−1 = −σF σG log µ √ mF mG + (mGσF + mF σG) − log µ −σF σG log µ if µ ∈ [mF mG , 1] . (6) At this point, differences between the forms (6) and (5) should be noticed. Since a Gaussian MF is√expected as a result, in (6) the summand σF σG − log µ should be omitted. Thus, the approximating assumption is as follows (mG σF + mF σG ) − log µ |σF σG log µ| = −σ F σG log µ mF + mG − log µ σG σF √ The function √− log µ is decreasing on [0, 1] and its average is 2π . From the other hand, mean values mF and mG are certain numbers in [0, 1]. Therefore the Gaussian approximation is justified for sufficiently small standard deviations σF and σG . The most advantageous case obviously is when the argument MF with a lower mean value has a greater standard deviation value. Then the following approximation is achieved √ mF mG − (mGσF + mF σG) − log µ if µ ∈ [0, mF mG ] , f −1g−1 ∼ = mF mG + (mGσF + mF σG) √− log µ if µ ∈ [mF mG, 1] .
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which has suitable form to obtain the inverse function
T˜P TM (F, G) (w) = exp − |
5
w mF mG 2 . σF mG +σGmF −
(7)
Piecewise-Gaussian approximations for triangular norms
All presented Gaussian approximations of the extended t-norms vary significantly from the usually strongly asymmetric exact results of applying the generalized extension principle. For that simple reason, a better approximation can be accomplished by two monotonic pieces of Gaussian functions with the same mean value and with not necessarily equal standard deviations, i.e., 2 u − m F exp − σF f (u) = 2 u − m F exp − ζ F 2 v − m G exp − σ g (v) = G 2 G exp − v−ζm G
u ∈ [0, mF ] , u ∈ (mF , 1] , v ∈ [0, mG] , v ∈ (mG, 1] .
(8)
(9)
5.1 Approximate Extended Product Based on Minimum Obviously, the extended product of FTNs characterized by (8) and (9) has a MF with the mean value m = mF mG . Here we can assume that the result consists of Gaussian components of the form
2 exp − w−σ m w ∈ [0, m] , T˜P TM (F, G) = w−m 2 w ∈ (m, 1] . exp − ζ |
(10) If we assume that the approximation has an exact value at w = 0, it is sufficient that u = 0 or v = 0. The use of the normality of f and g leads to the expression
Consequently,
2 exp − mσFF , 2 m exp − σ = max
2 exp − mσGG 2 2
= max exp − mσFFmmGG , exp − mσGFmmFG Since both functions under maximum are normal, the greater one has a greater standard deviation 2 2 m m m F G exp − σ = exp − max(σF mG ,σG mF ) . (11) The assumption of the exact value of the result at w = 1 ensures that both u = 1 and v = 1, accordingly, ˜TP (F, G) (1) = min (f (1) , g (1)) T 2 2 1 − m 1−mG F = min exp − ζ F , exp − . ζG
Since exp −x2 is a decreasing function of any positive x, the minimum of the functions goes into the function of the maximum of the arguments suchthat 2 2 . exp − 1−ζm = exp − max 1−ζmF F , 1−ζmG G (12) Combining (11) and (12) we get the following formula for a piecewise-Gaussian approximation ˜P TM (F, G) (w) T 2 w − m m F G exp − max(σF mG ,σG mF ) if w ∈ [0, mF mG ] , 2 = w−mF mG 1−m exp − F mG ,ζ 1−mF mG min ζ F G 1 − m 1 − m F G |
if w ∈ (mF mG, 1] .
Actually this approximation is a rather simple three-point piecewise Gaussian interpolation of the exact membership function. However, it is one exemplary step toward parametrized operations preserving shapes of membership functions.
sup min (f (0) , g (v)) , 5.2 Use of Classical T-norms for [0,1] Approximate Extensions T˜P |TM (F, G) (0) = max v∈sup min (f (u) , g (0)) u∈[0,1] The aim of this subsection is to present a class of approximations of extended t-norms for = max (f (0) , g (0)) .
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piecewise-Gaussian FTNs which use traditional t-norms. Let the arguments be characterized by (8) and (9). The approximate extended tnorm is defined in terms of (10) where the center m = T (mF , mG) and two remaining interpolation points satisfy the following equations:
m−ζ m+σ
Therefore
ζ σ
= =
= =
T (mF − ζ F , mG − ζ G) , T (mF + σF , mG + σG) .
m − T (mF − ζ F , mG − ζ G) , T (mF + σF , mG + σG) − m.
This approach reduces calculations of extended tnorms to computing only the three characteristic variables m, ζ and σ. Arbitrary traditional tnorms here may be used. Detailed justification of this approach will be presented in our future paper. 6
Conclusion
We have proposed new formulae for extended triangular norms. In the context of Gaussian FTNs, the product-based extended Łukasiewicz t-norm, the drastic product-based extended product and the approximation of the product-based extended product have been derived. For piecewiseGaussian FTNs, the approximate minimum-based extended product and simple approximation have been proposed. These efficient results play a pivotal role in the design of type-2 fuzzy logic systems. "Symmetrical" results can be obtained for the extended complementary norms. A tremendously useful feature of the derived formulae is that resultant MFs preserve Gaussian or piecewise-Gaussian shapes of the two arguments, and this way the extended t-norms can be expanded into their multi-argument forms. Generally, the derived class of approximate tnorms belongs to the class of type-2 triangular norms. But this will be demonstrated in a journal extension of this issue. The next step toward designing piecewiseGaussian type-2 fuzzy logic systems is the construction of the efficient type-reduction dedicated to piecewise-Gaussian fuzzy sets of type-2.
References
[1] S. Coupland and R. John, “A new and efficient method for the type-2 meet operation,” Proc. FUZZ-IEEE 2004, Budapest, pp. 959964, 2004. [2] D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and Applications, Mathematics in Science and Engineering, Academic Press, Inc., NY, 1980. [3] N. N. Karnik and J. M. Mendel, "Operations on type-2 fuzzy sets," Fuzzy Sets and Systems, vol. 122, pp. 327—348, 2000. [4] M. F. Kawaguchi and M. Miyakoshi, "Extended Triangular Norms in type-2 Fuzzy Logic," EUFIT’99 7th European Congress on Intelligent Techniques & Soft Computing, Aachen, September, 1999. [5] J. M. Mendel, Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions, Prentice Hall PTR, Upper Saddle River, NJ 2001. [6] R. Mesiar, "Triangular-norm-based addition of fuzzy intervals," Fuzzy Sets and Systems, vol. 91, pp. 231—237, 1997. [7] H. T. Nguyen, "A note on the extension principle for fuzzy sets," J. Math. Anal. Appl., vol. 64, pp. 369—380, 1978. [8] J. Starczewski, "What differs interval type2 FLS from type-1 FLS," in. L. Rutkowski (Eds.) Artificial Intelligence and Soft Computing - ICAISC 2004, Lecture Notes in Computer Science, Springer pp. 381—387, 2004. [9] J. Starczewski, "Extended triangular norms," unpublished. [10] J. Starczewski and L. Rutkowski, "NeuroFuzzy Systems of Type 2," 1st Int’l Conf. on Fuzzy Systems and Knowledge Discovery, vol. 2, Singapore, pp. 458—462, 2002. [11] C. Walker and E. Walker, "The algebra of fuzzy truth values," Fuzzy Sets and Systems, vol. 149, pp. 309-347, 2005.
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