The approximation of piecewise linear membership functions and Łukasiewicz operators

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1 Fuzz Sets and Sstems The approimation of piecewise linear membership functions and Łukasiewicz operators József Dombi, Zsolt Gera Universit of Szeged, Institute of Informatics, 67 Pf. 652, Szeged, Hungar Received 8 November 24; received in revised form 22 Februar 25; accepted 22 Februar 25 Available online March 25 Abstract In this paper we propose an approimation of piecewise linear membership functions with the help of sigmoid functions and certain arithmetic operations. The gradient-based tuning of piecewise linear membership functions can be achieved with the proposed efficient approimation because it has simple continuous derivatives. With this construction we can even approimate the Łukasiewicz operator famil which plas an important role in fuzz logic, first of all from the theoretical point of view, although in practice in optimization and learning it is rarel used because the lack of good analtical properties, e.g. a continuous gradient. The proposed approimation enlarges the applicabilit of fuzz methods to the operators and membership functions where the differentiabilit is desirable. 25 Elsevier B.V. All rights reserved. Kewords: Fuzz connectives; Membership function; Approimation; Łukasiewicz operators; Nilpotent operators. Introduction The construction and the interpretation of fuzz membership functions have alwas been a crucial question. Bilgic and Türksen gave a comprehensive overview of the most relevant interpretations in []. For the construction of membership functions Dombi [] had an aiomatic point of view, Civanlar and Trussel [8] used statistical data, Bagis [2], Denna et al. [9], Karaboga [4] applied tabu search. However, most fuzz applications use piecewise linear membership functions because of their eas handling, for eample in embedded fuzz control applications where the ited computational resources does not Corresponding author. Tel.: /383. address: gera@inf.u-szeged.hu Z. Gera. 65-4/$ - see front matter 25 Elsevier B.V. All rights reserved. doi:.6/j.fss

2 276 J. Dombi, Z. Gera / Fuzz Sets and Sstems allow the use of complicated membership functions. In other areas where the model parameters are learned b a gradient-based optimization method, the cannot be used because the lack of continuous derivatives. For eample, to fine tune a fuzz control sstem b a simple gradient-based technique it is required that the membership functions are differentiable for ever input. There are numerous papers dealing with the concept of fuzz set approimation and membership function differentiabilit see for e.g. [3,2,6]. In this paper we give a different solution to the problem of non-differentiabilit of piecewise linear functions b approimating the cut function of the Łukasiewicz operators, and use it to construct continuousl differentiable membership functions which approach the well-known triangular or trapezoidal membership functions. The paper is organized as follows: Section 2 is a brief overview of Łukasiewicz operators, in Section 3 we give the basic properties of the sigmoid function which serves as the basis for the approimation, prove the main approimation theorem, and eamine the derivatives and the convergence of the proposed approimation and in Section 4 we appl the approimation to triangular and trapezoidal membership functions. 2. Łukasiewicz operators The Łukasiewicz operator class see e.g. [,7,3] is commonl used for various purposes, see e.g. [4,5]. In this well-known operator famil the cut function denoted b [ ] plas an important role. We can get the cut function from b taking the maimum of and and then taking the minimum of the result and. Definition. Let the cut function be if, [] =minma,, = if <<, if. Let the generalized cut function be if a, a [] a,b =[ a/b a] = if a<<b, b a if b, where a,b R and a<b. In neural networks terminolog this cut function is called saturating linear transfer function. All nilpotent operators are constructed using the cut function. The formulas of the nilpotent conjunction, disjunction, implication and negation are the following: c, =[ + ], d, =[ + ], i, =[ + ], n =, where, [, ]. The truth tables of the former three can be seen in Fig..

3 J. Dombi, Z. Gera / Fuzz Sets and Sstems Fig.. The truth tables of the nilpotent conjunction, disjunction and implication. Throughout this paper we will refer to triangular and trapezoidal membership functions as piecewise linear membership functions. The are ver common in fuzz control because of their eas handling. The generalized cut function can be used to describe piecewise linear membership functions. Generall, a trapezoidal membership function can be constructed as the conjunction of two generalized cut functions as c[] a,b, [] c,d =[[] a,b + [] c,d ] =[[] a,b [] c,d ], 2 where a, b, c, d are real numbers and a<b c <d. As a special case, if b = c then we get a triangular membership function. For an eample of the general case see Fig. 2. The Łukasiewicz operator famil used above has good theoretical properties. These are, for eample, the law of non-contradiction that is the conjunction of a variable and its negation is alwas zero and the law of ecluded middle that is the disjunction of a variable and its negation is alwas one both hold, and the residual and material implications coincide. These properties make these operators to be widel used in fuzz logic and to be the closest one to classic Boolean logic. Besides these good theoretical properties this operator famil does not have a continuous gradient. So for eample gradient-based optimization techniques are impossible with Łukasiewicz operators. The root of this problem is the shape of the cut function itself. Currentl, we are working on new concept called pliant sstem in which we build up our theoretical framework b the practical application oriented point of view. We are using a new notation sstem too. Some elements of it appear in this work as well. 3. Approimation of the cut function A solution to the above-mentioned problem is a continuousl differentiable approimation of the cut function, which can be seen in Fig. 3. In this section we will construct such an approimating function b means of sigmoid functions. The reason for choosing the sigmoid function was that this function has

4 278 J. Dombi, Z. Gera / Fuzz Sets and Sstems Fig. 2. In the first image: two generalized cut functions. In the second image: a trapezoidal membership function constructed as the conjunction of the former two, with a negation applied to the right one. The parameters were the following: a =, b =, c = 2, d = Fig. 3. The cut function and its approimation. a ver important role in man areas. It is frequentl used in artificial neural networks [6], optimization methods, economical and biological models [5].

5 J. Dombi, Z. Gera / Fuzz Sets and Sstems Fig. 4. The sigmoid function, with parameters d = and β = The sigmoid function The sigmoid function see Fig. 4 is defined as σ β d = + e β d, 3 where the lower inde d is omitted if. Let us eamine some of its properties which will be useful later: Its derivative can be epressed b itself see Fig. 5: σ β d = βσ β d σ β d. 4 Its integral has the following form: σ β d d = β ln σ β d. 5 Because the sigmoid function is asmptoticall as tends to infinit, the integral of the sigmoid function is asmptoticall see Fig The interval [a,b] squashing function In order to get an approimation of the generalized cut function, let us integrate the difference of two sigmoid functions, which are translated b a and b a <b, respectivel. σ a β σβ b b a d = σ a β b a d σ β b d 6 = β b a ln σ β a + β ln σ β b. After simplification we get the interval [a,b] squashing function:

6 28 J. Dombi, Z. Gera / Fuzz Sets and Sstems Fig. 5. The first derivative of the sigmoid function Fig. 6. The integral of the sigmoid function, one is shifted b. Definition 2. Let the interval [a,b] squashing function be S β a,b = b a ln σ β /β b σ β a = b a ln + e β a + e β b /β. The parameters a and b affect the placement of the interval squashing function, while the β parameter drives the precision of the approimation. We need to prove that S β a,b is reall an approimation of the generalized cut function. Theorem 3. Let a,b R, a<band β R +. Then β Sβ a,b =[] a,b and S β a,b is continuous in, a, b and β.

7 J. Dombi, Z. Gera / Fuzz Sets and Sstems Proof. It is eas to see the continuit, because S β a,b is a simple composition of continuous functions and because the sigmoid function has a range of [, ] the quotient is alwas positive. In proving the it we separate three cases, depending on the relation between a, b and. Case : <a<b: Since β a <, so e β a and similarl e β b. Hence the quotient converges to if β, and the logarithm of one is zero. Case 2: a b: b a ln + e β a /β β + e β b = b a ln e β a e β a /β + β + e β b = b a ln e a e β a + /β β + e β b /β = b a ln e a e β a + /β β + e β b /β. We transform the nominator so that we can take the e a out of the es. In the nominator e β a remained which converges to as well as e β b in the denominator so the quotient converges to if β. So as the result, the it of the interval squashing function is a/b a, which b definition equals to the generalized cut function in this case. Case 3: a <b<: b a ln β = b a ln β = b a ln = b a ln + e β a + e β b β e a e b /β e β a e β a /β + e β b e β b + e a e β a + /β e b e β b + /β β e β a + /β e β b + /β We do the same transformations as in the previous case but we take e b from the denominator, too. After these transformations the remaining quotient converges to, so β Sβ a,b = e a b a ln e b = b a lne a b.

8 282 J. Dombi, Z. Gera / Fuzz Sets and Sstems Fig. 7. In the first image: the interval squashing function with an increasing β parameter a = and b = 2. In the second image: the interval squashing function with a zero and a negative β parameter..5.3 Fig. 8. The approimation of the nilpotent conjunction with β values, 4, 8 and 32. = b a lneb a = b a b a =. In Fig. 7 the interval squashing function can be seen with various β parameters. The following proposition states some properties of the interval squashing function. Proposition 4. β Sβ a,b S β a,b = /2, = Sβ a,b. As an another eample, the nilpotent conjunction is approimated with the interval squashing function in Fig. 8. For further use, let us introduce an another form of the interval squashing function s formula. Instead of using parameters a and b which were the bounds on the -ais, from now on we will use a and δ,

9 J. Dombi, Z. Gera / Fuzz Sets and Sstems ε β ε β δ a δ Fig. 9. The meaning of a < δ β. where a gives the center of the squashing function and where δ gives its steepness. Together with the new formula we introduce its pliant notation. Definition 5. Let the squashing function be a < δ β = S β a,δ = β σ 2δ ln a+δ /β σ β a δ, where a R and δ R +. If the a and δ parameters are both /2 we will use the following pliant notation for simplicit: β = S β /2,/2, which is the approimation of the cut function. The inequalit relation in the pliant notation refers to the fact that the squashing function can be interpreted as the truthness of the relation a < with decision level /2, according to a fuzziness parameter δ and an approimation parameter β see Fig. 9. The derivatives of the squashing function can be epressed b itself and sigmoid functions: S β a,δ = 2δ S β a,δ = a 2δ S β a,δ = δ 2δ σ β a δ σβ a+δ, 7 σ β a+δ σβ a δ, 8 σ β a+δ + σβ a δ δ Sβ a,δ. 9

10 284 J. Dombi, Z. Gera / Fuzz Sets and Sstems The error of the approimation The squashing function approimates the cut function with an error. This error can be defined in man was. We have chosen the following definition. Definition 6. Let the approimation error of the squashing function be ε β = < δ δ β = 2δ ln σ β /β δ δ σ β δ δ, where β >. Because of the smmetr of the squashing function ε β = < δ δ β, see Fig. 9. The purpose of measuring the approimation error is the following inverse problem: we want to get the corresponding β parameter for a desired ε β error. We state the following lemma on the relationship between ε β and β. Lemma 7. Let us fi the value of δ. The following holds for ε β : ε β <c β, where c = ln 2/2δ is a constant. Proof. ε β = 2δβ ln + e β δ+δ + e β δ δ = 2δβ ln = ln 2 2δβ ln + e 2δβ <c 2δβ β. 2 + e 2δβ So the error of the approimation can be upper bounded b c parameter β, the error decreases b the same order of magnitude. β, which means that b increasing 4. Approimation of piecewise linear membership functions In fuzz theor triangular and trapezoidal membership functions pla an important role. For eample, fuzz control uses mainl this tpe of membership functions because of their eas handling. The are piecewise linear, hence the cannot be continuousl differentiated. Our motivation was to construct an approimation which has the same properties in the it as the approimated membership function and has a continuous gradient. If we are using approimated piecewise linear membership functions in fuzz control sstems then the can be tuned b a gradient-based optimization method and we can get the optimal parameters of the membership functions.

11 J. Dombi, Z. Gera / Fuzz Sets and Sstems Fig.. The approimation of a trapezoid and a triangular membership function. Piecewise linear membership functions can be constructed b generalized cut functions, so the can be approimated b squashing functions and an appropriate conjunction operator. We have chosen the Łukasiewicz conjunction. But in the conjunction s formula instead of using the cut function we also used the squashing function, so the membership function and the operator are constructed using the same component. To describe a trapezoid membership function using the conjunction operator and two squashing functions four parameters are needed, namel a, δ and a 2, δ 2, where a and a 2 give the positions of its left and right sides, and δ and δ 2 give its left and right slopes. The two β parameters of the squashing functions have to have opposite signs to form a trapezoid or triangle, and of course the equations a <a 2 and a + δ a 2 δ 2 must hold. So the approimation of a trapezoid membership function is the following see Fig. : S β /2,/2 with pliant notation: S β a,δ + S β a 2,δ 2 a < δ β + a 2 < δ2 β β. As a special case of the trapezoid membership function we get the triangular membership function. To describe one, onl two parameters are needed, the center a, and its fuzziness δ see Fig.. Definition 8. Let the approimation of the triangular membership function be defined as in pliant notation δ a β = a δ/2 < δ/2 β + a + δ/2 < δ/2 β β, where a is its center and δ is its fuzziness.

12 286 J. Dombi, Z. Gera / Fuzz Sets and Sstems B this wa we can represent a fuzz number b squashing functions. 5. Conclusion In this paper we approimated piecewise linear membership functions, so that the can be tuned b gradient-based optimizations in a fuzz control sstem to achieve better results. We have reviewed the cut function which is the basis of the Łukasiewicz operator class. This cut function is piecewise linear, it cannot be continuousl differentiated. We have created an approimation of the cut function the squashing function b sigmoid functions with good analtical properties, for eample simple derivatives, fast convergence and eas calculation, and applied this approimation to piecewise linear membership functions. References [] R. Ackermann, An Introduction to Man-Valued Logics, Dover, New York, 967. [2] A. Bagis, Determining fuzz membership functions with tabu search an application to control, Fuzz Sets and Sstems [3] S. Bodjanova, A generalized α-cut, Fuzz Sets and Sstems [4] C. Chen, W. Chen, Fuzz controller design b using neural network techniques, IEEE Fuzz Sstems, Vol. 2. [5] C.-L. Chen, S.-N. Wang, C.-T. Hsieh, F.-Y. Chang, Theoretical analsis of a fuzz-logic controller with unequall spaced triangular membership functions, Fuzz Sets and Sstems [6] P. Chandra, Y. Singh, An activation function adapting training algorithm for sigmoidal feedforward networks, Neurocomputing [7] R. Cignoli, I.M.L. D Ottaviano, D. Mundici, Algebraic foundations of man-valued reasoning, Trends in Logic, Vol. 7. [8] M. Civanlar, H. Trussel, Constructing membership functions using statistical data, Fuzz Sets and Sstems [9] M. Denna, G. Mauri, A.M. Zanaboni, Learning fuzz rules with tabu search an application to control, IEEE Trans. Fuzz Sstems [] J. Dombi, Membership function as an evaluation, Fuzz Sets and Sstems [] D. Dubois, H. Prade Eds., Fundamentals of Fuzz Sets, Kluwer, Dordrecht, 2. [2] L.L.A. Grauel, Construction of differentiable membership functions, Fuzz Sets and Sstems [3] P. Hájek, Metamathematics of Fuzz Logic, Kluwer, Dordrecht, 998. [4] D. Karaboga, Design of fuzz logic controllers using tabu search algorithms, Biennial Conf. North American Fuzz Inf. Processing Societ, 996, pp [5] M. Kodaka, Requirements for generating sigmoidal timecourse aggregation in nucleation-dependent polmerization model, Biophs. Chem [6] J.T.B. Thomas, A. Runkler, Function approimation with polnomial membership functions and alternating cluster information, Fuzz Sets and Sstems