Extended Triangular Norms on Gaussian Fuzzy Sets

EUSFLAT - LFA 005 Extended Triangular Norms on Gaussian Fuzzy Sets Janusz T Starczewski Department of Computer Engineering, Częstochowa University of Technology, Częstochowa, Poland Department of Artificial Intelligence, WSHE University in Łódź, Łódź, Poland jasio@kikpczczestpl Abstract So far, computational complextity of the general formula for the extended t-norm does not allow to construct fuzzy logic systems of type- 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- Fuzzy Sets, Extended T-norms 1 Introduction Recently, we have witnessed a rising interest in design of type- fuzzy logic systems Let us recall that type- fuzzy sets are characterized by classical fuzzy subsets of 0, 1 as their membership grades The key to design type- fuzzy logic systems is to find efficient formulae for extensions of classical triangular norms, called extended t- norms In this case, extended t-norms operate on classical fuzzy subsets of 0, 1 called fuzzy truth values In literature, only interval type- fuzzy sets have been used to construct concrete designs of fuzzy logic systems see eg 5 This has been caused mainly by the lack of exact output formulae for non-interval membership functions of extended t- norms The unique exception is our design of a triangular type- 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- Gaussian fuzzy truth values The derived formulae allow to examine the utility of Gaussian type- fuzzy sets in designing fuzzy logic systems We are positive that there exists a field of applications where Gaussian type- fuzzy systems accomplish better performance than traditional type-1 fuzzy logic systems, since we have demonstrate it for interval type- fuzzy logic systems 8 11 Type- Fuzzy Sets Context Let the set of all fuzzy subsets of the unit interval 0, 1 be denoted by F 0, 1 The type- fuzzy set in the real line R, denotedby Ã, ischaracterized by the fuzzy membership function MF à : R F0, 1 The values of this function are fuzzy membership grades à x, ie, classical fuzzy subsets of the unit interval 0, 1 characterized by f x : 0, 1 0, 1 The function f x of each fuzzy membership grade is called a secondary MF In this paper only Gaussian and piecewise-gaussian secondary MFs are considered Usually F0, 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, ie,!u 0, 1 : f u 1, and convex, ie, u 1,u,λ 0, 1, f λu 1 +1 λ u min f u 1,fu 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, 87

EUSFLAT - LFA 005 with the membership functions f and g, respectively An extension of a t-norm T basedona t-norm T according to the generalized extension principle is called an extended triangular norm and is ressed as a function of w, ie, T T F, Gw sup Tu,vw T f u,gv 1 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,gv 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 licit parametrized function of w Only a combination of certain classes of membership functions in this paper piecewise- Gaussian 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,, 3, 9 In 4 it has been proved that the extended t- norm satisfies type- 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- fuzzy sets are presented in 11 Extended t-norms are used to form the intersection of type- fuzzy sets, in construction of type- FLSs Having type- fuzzy sets à x f x u and B x g x v u, v 0, 1 the intersection may be calculated x R as à x B x T à x, B x w sup Tu,vw T f x u,g x v If arguments of T are defined on different domains, the extended t-norm forms a type- fuzzy relation, ie, if B y g y v then R x, y T sup à x, B y w Tu,vw T f x u,g y v 3 Extended T-norms on Gaussian FTNs 31 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 T s is strict the inequality T s u, w < T s v, w for all u<vand w>0 is satisfied Its strictly decreasing additive generator φ: 0, 1 0, ensures that T s u, v φ 1 φ u + φ v Theorem 1 Let κ:, 0, be a continuous convex function such that κ 0 0 and κ x κ x for all x R Let T s be a strict t-norm with an additive generator φ If the arguments F and G are characterized by f u φ 1 aκ u m a and g v φ 1 bκ v n b ; a, b > 0; m, n 0, 1 Then the extension of the Łukasiewicz t-norm T L based on astrictt-normt s is characterized by where T L T s F, Gw µ0 if w 0, φ a 1 + b κ µ 0 w m n+1 a+b otherwise, 1 if T L m, n 0, φ a 1 1 m n + b κ otherwise a+b Since the additive generator φ is strictly decreasing, the generalized extension principle is T Ts F, G φ 1 inf φ f u +φ g v T u,vw Therefore, the proof relies on the evaluation of inf aκ u m u+v 1w a inf u w,1 aκ u m a detailswillbegivenin9 + bκ v n b + bκ w u+1 n b 873

EUSFLAT - LFA 005 Now consider the extended Łukasiewicz t-norm based on the product t-norm on Gaussian FTNs, ie, f u g v u m F F v m G G, 3 4 The product T P is obviously a strict t-norm For theadditivegeneratorφ log x, the function κ must be defined by κ x and the following substitutions must be done: F a, G b, m F m, m G n; m F,m G 0, 1 The use of Theorem 1 leads to the result T L Ts F, Gw w m F mg +1 F + G 3 Extended Product T-norm Based on Drastic Product Since at the beginning we do not assume the strictness of t-norms, the notion of a pseudoinverse will be very useful Definition Let f : 0, 1 0, 1 be a nonconstant and either non-decreasing or nonincreasing function A pseudo-inverse of f is defined by f 1 φ supu a, b f u φ} for non-decreasing f,and f 1 φ supu a, b f u φ} for non-increasing f,wheresup a Let φ 1 signify the strict pseudo-inverse when in Definition the operators and are replaced by < and >, respectively Here we present the formula for extended continuous t-norms based on the weakest t-norm, ie the drastic product, TD min x, y if max x, y 1;and0 otherwise} for the proof see 9 Theorem 3 Let TD be the drastic product t-norm and let the FTNs F and G be characterized by upper semicontinuous MFs f and g, such that f m F g m G 1; m F m G Then the extension of a continuous t-norm T based on T D is characterized by max f T m 1 G w,g m F T 1 w if w 0,T mf,mg, max f T m 1 G w,g m F T T D F, G f T m 1 G w T 1 w if w T mf,mg,mf, if w mf,mg, 0 otherwise By applying the theorem to the two Gaussian FTNs given by 3 and 4, MF of the extended product based on the drastic product is T P TD F, G max w mf mg, mgf w mf mg mf G w mf mg maxmgf,mf G Note,thatthisresulthavethesameformasthe approximate result of Karnik and Mendel 3, 5 4 Gaussian Approximations for Triangular Norms 41 Approximate Product-based Extended Product T-norm 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, ie, T P T P F, G w mf mg F mg +GmF 874

EUSFLAT - LFA 005 4 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, ie, 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 m F g m G 1 The extension of continuous t-norm T based on the minimum t-norm can be ressed as T T M F, G where w T w 1 w if w 0,T mf,mg 1 if w T mf,mg w 1 w if w T mf,mg, 1 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 Proof We may use the Nguyen theorem 7, which states that for a continuous binary operation, its extension can be derived using µ- cuts, ie, F T M G µ F µ G µ,where may be any arbitrary continuous t-norm T Obviously, T T M F, G 1 T F 1, G 1, ie, the resultant MF is equal to unity when w T m F,m G Otherwise, since T T M F F, G T µ, G µ, µ µ 1 T f 1 µ,g 1 µ separately for w T m F,m G, 1 and w 0,T m F,m G The upper semicontinuity of f and g ensures that 1 w 1 w Consequently, for w T n F,n G, 1, we consider only upper bounds of µ-cuts, ie, f 1 µ m F, 1 and g 1 µ n G, 1, and for w 0,T m F,m G, lower bounds of µ-cuts are respected, ie, f 1 µ 0,m F and g 1 µ 0,m G 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:, log µ u mf F log µ v mg G Thus, the inverse functions are given by f 1 g 1 mf F log µ for f 1 0,mF, mf + F log µ for f 1 mf, 1, mg G log µ for g 1 0,mG, mg + G log µ for g 1 mg, 1 5 By the product of lower and upper inverse functions, we have f 1 g 1 mf mg mgf + mf G log µ F G log µ if µ 0,mFmG, mf mg +mgf + mf G log µ F G 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 ected as a result, in 6 the summand F G log µ should be omitted Thus, the approximating assumption is as follows mgf + mf G log µ F G log µ F G log µ m F m + G G F log µ The function log µ is decreasing on 0, 1 and its average is π 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 F and G 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 m F m G m G F + m F G log µ f 1 g 1 if µ 0,m F m G, m F m G +m G F + m F G log µ if µ m F m G, 1 875

EUSFLAT - LFA 005 which has suitable form to obtain the inverse function T P TM F, w mf Gw mg F mg+gmf 7 5 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, ie, u m F u F 0,m F, f u 8 u m F, 1, g v u m F ζ F v mg G v mg ζg v 0,m G, v m G, 1 51 Approximate Extended Product BasedonMinimum 9 Consequently, m max max m F, F m G G T Obviously, the extended product of FTNs characterizedby8and9hasamfwiththemean P TM F, Gw w m F mg value m m F m G Here we can assume that maxf mg,gmf the result consists of Gaussian components of the if w 0,m F m G, form w m w mf mg w 0,m, min ζf 1 m F mg 1 mf,ζg 1 m F mg 1 mg T P TM F, G w m ζ w if w m, 1 mf mg, 1 Actually this approximation is a rather simple 10 three-point piecewise Gaussian interpolation of If we assume that the approximation has an exact the exact membership function However, it is value at w 0,itissufficientthatu 0or v 0 one exemplary step toward parametrized operations preserving shapes of membership functions The use of the normality of f and g leads to the ression sup min f 0,gv, 5 Use of Classical T-norms for v 0,1 T P TM F, G 0 max Approximate Extensions sup min f u,g0 u 0,1 The aim of this subsection is to present a maxf 0,g0 class of approximations of extended t-norms for m F m G F m, G } m F m G G m F Since both functions under maximum are normal, the greater one has a greater standard deviation m m F m G max F m G, G m F 11 The assumption of the exact value of the result at w 1 ensures that both u 1 and v 1, accordingly, T TP F, G1 minf 1,g1 min 1 m F, ζ F 1 mg ζg } Since x is a decreasing function of any positive x, the minimum of the functions goes into the function of the maximum of the arguments such that 1 m ζ } max 1 mf, ζf 1 m G ζg 1 Combining 11 and 1 we get the following formula for a piecewise-gaussian approximation 876

EUSFLAT - LFA 005 piecewise-gaussian FTNs which use traditional t-norms Let the arguments be characterized by 8 and 9 The approximate extended t- norm is defined in terms of 10 where the center m T mf,mg and two remaining interpolation points satisfy the following equations: Therefore m ζ T mf ζf,m G ζg, m + T mf + F,mG + G ζ m T mf ζf,m G ζg, T mf + F,mG + G m This approach reduces calculations of extended t- norms to computing only the three characteristic variables m, ζ and Arbitrary traditional t- norms 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 piecewise- Gaussian 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- 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 anded into their multi-argument forms Generally, the derived class of approximate t- norms belongs to the class of type- triangular norms But this will be demonstrated in a journal extension of this issue The next step toward designing piecewise- Gaussian type- fuzzy logic systems is the construction of the efficient type-reduction dedicated to piecewise-gaussian fuzzy sets of type- References 1 S Coupland and R John, A new and efficient method for the type- meet operation, Proc FUZZ-IEEE 004, Budapest, pp 959-964, 004 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- fuzzy sets," Fuzzy Sets and Systems, vol 1, pp 37 348, 000 4 M F Kawaguchi and M Miyakoshi, "Extended Triangular Norms in type- Fuzzy Logic," EUFIT 99 7th European Congress on Intelligent Techniques & Soft Computing, Aachen, September, 1999 5JMMendel,Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions, Prentice Hall PTR, Upper Saddle River, NJ 001 6 R Mesiar, "Triangular-norm-based addition of fuzzy intervals," Fuzzy Sets and Systems, vol 91, pp 31 37, 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 type- FLSfromtype-1FLS,"inLRutkowski Eds Artificial Intelligence and Soft Computing - ICAISC 004, Lecture Notes in Computer Science, Springer pp 381 387, 004 9 J Starczewski, "Extended triangular norms," unpublished 10 J Starczewski and L Rutkowski, "Neuro- Fuzzy Systems of Type," 1st Int l Conf on Fuzzy Systems and Knowledge Discovery, vol, Singapore, pp 458 46, 00 11 C Walker and E Walker, "The algebra of fuzzy truth values," Fuzzy Sets and Systems, vol 149, pp 309-347, 005 877