The Sigmoidal Approximation of Łukasiewicz Operators

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1 The Sigmoidal Approimation of Łukasiewicz Operators József Dombi, Zsolt Gera Universit of Szeged, Institute of Informatics {dombi Abstract: In this paper we propose an approimation of Łukasiewicz operators b means of sigmoid functions. Łukasiewicz operators pla an important role in fuzz logic. The are widel used due to their good theoretical properties, i.e. the residual and material implications coincide, the law of ecluded middle and the law of non-contradiction both hold. Besides these good theoretical properties this operator famil does not have a continuous gradient. Its approimation is simple and continuousl differentiable. Kewords: approimation, sigmoid function, Łukasiewicz operators Introduction The Łukasiewicz operator class see e.g. [], [2], [3] is commonl used for various purposes. 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] = b a, if b 2 where a, b R and a < b. 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 = 3

2 Figure : The truth tables of the nilpotent conjunction, disjunction and implication Figure 2: Two generalized cut functions. The truth tables of the former three can be seen on fig.. The Łukasiewicz operator famil used above has good theoretical properties. These are: 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.

3 2 Approimation of the Cut Function A solution to above mentioned problem is a continuousl differentiable approimation of the cut function, which can be seen on fig. 3. In this section we ll construct such an approimating function b means of sigmoid functions. The reason for choosing the sigmoid function was that this function has a ver important role in man areas. It is frequentl used in artificial neural networks, optimization methods, economical and biological models Figure 3: The cut function and its approimation 2. The Sigmoid Function The sigmoid function see fig. 4 is defined as d = + e β d 4 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: d = β d d 5 its integral has the following form: d d = β ln d 6 Because the sigmoid function is asmptoticall as tends to infinit, the integral of the sigmoid function is asmptoticall see fig. 6.

4 Figure 4: The sigmoid function, with parameters d = and β = Figure 5: The first derivative of the sigmoid function 2.2 The Squashing Function on [a, b] 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. b a a = b a = b a b d = σ a β d β ln σ a β + β ln b d = b After simplification we get the squashing function on the interval [a, b]:

5 Figure 6: The integral of the sigmoid function, one is shifted b Definition 2.. Let the interval squashing function on [a, b] be S β a,b = b a ln b a /β = + e β a /β b a ln. 7 + e β b 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 2.2. Let a, b R, a < b and β R +. Then and S β a,b is continuous in, a, b and β. lim β Sβ a,b = [] a,b 8 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 limit 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.

6 Case 2 a b: + e β a /β b a ln lim = β + e β b = e β a b a ln e β a + /β lim = β + e β b = b a ln lim β = b a ln e a lim β e a e β a + /β = + e β b /β e β a + /β + e β b /β We transform the nominator so that we can take the e a out of the limes. 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 limit 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 < : + e β a /β b a ln lim = β + e β b = e β a b a ln e β a + /β lim β e β b e β b + = = b a ln e a e β a + /β lim β e b e β b + = /β = b a ln e a e β a + /β e b lim 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 lim β Sβ a,b = b a ln e a e b = b a ln e b a = b a b a =. = b a ln e a b = On fig. 7 the interval squashing function can be seen with various β parameters. The following proposition states some properties of the interval squashing function.

7 Figure 7: On the left image: the interval squashing function with an increasing β parameter a = and b = 2. On the right image: the interval squashing function with a zero and a negative β parameter Figure 8: The approimation of the nilpotent conjunction with β values,4,8 and 32 Proposition 2.3. lim β Sβ a,b = /2 9 S β a,b = S β a,b As an another eample, the nilpotent conjunction is approimated with the interval squashing function on 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 ll use a and δ, 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.

8 Definition 2.4. Let the squashing function be a < δ β = S β a,δ = β σ 2δ ln a+δ /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, 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. Figure 9: The meaning of a < δ β The derivatives of the squashing function can be epressed b itself and sigmoid functions: S β a,δ S β a,δ a S β a,δ δ = 2δ = 2δ = 2δ a δ σβ a+δ a+δ σβ a δ a+δ + σβ a δ 3 4 δ Sβ a,δ 5

9 2.3 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 2.5. Let the approimation error of the squashing function be where β >. ε β = < δ δ = 2δ ln δ δ /β δ 6 δ Because of the smmetr of the squashing function ε β = < δ δ, 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 2.6. Let us fi the value of δ. The following holds for ε β. where c = ln 2 2δ is constant. ε β < c β, 7 Proof. ε β = + e β δ+δ 2δβ ln + e β δ δ = ln 2 2δβ ln + e 2δβ < c 2δβ β = 2δβ ln 2 + e 2δβ = So the error of the approimation can be upper bounded b c β, which means that b increasing parameter β, the error decreases b the same order of magnitude. Conclusion In this paper we have reviewed the cut function, which is the basis of the well known Łukasiewicz operator class. This cut function is piecewise linear, it can not be continuousl differentiated. We have created an approimation of the cut function the squashing function b means of sigmoid functions with good analtical properties, for eample fast convergence and eas calculation.

10 References [] R. Ackermann. An Introduction to Man-Valued Logics. Dover, New York, 967. [2] P. Hájek. Metamathematics of Fuzz Logic. Kluwer, 998. [3] D. Mundici R. Cignoli, I. M. L. D Ottaviano. Algebraic foundations of manvalued reasoning. Trends in Logic, 7, 2.