INTERNATIONAL JOURNAL OF COMPUTERS COMMUNICATIONS & CONTROL Special issue on fuzzy logic dedicated to the centenary of the birth of Lotfi A. Zadeh (1921-2017) Online ISSN 1841-9844, ISSN-L 1841-9836, Volume: 16, Issue: 1, Month: February, Year: 2021 Article Number: 4106, https://doi.org/10.15837/ijccc.2021.1.4106 CCC Publications Information Volume of Fuzzy Membership Function J.X. Deng, Y. Deng Jixiang Deng Institute of Fundamental and Frontier Science University of Electronic Science and Technology of China, Chengdu, 610054, China Yong Deng* 1. Institute of Fundamental and Frontier Science University of Electronic Science and Technology of China, Chengdu, 610054, China 2. School of Education Shannxi Normal University, Xi’an, 710062, China *Corresponding author: dengentropy@uestc.edu.cn; prof.deng@hotmail.com Abstract Fuzzy membership function plays an important role in fuzzy set theory. However, how to measure the information volume of fuzzy membership function is still an open issue. The existing methods to determine the uncertainty of fuzzy membership function only measure the first-order information volume, but do not take higher-order information volume into consideration. To address this issue, a new information volume of fuzzy membership function is presented in this paper, which includes the first-order and the higher-order information volume. By continuously separating the hesitancy degree until convergence, the information volume of the fuzzy membership function can be calculated. In addition, when the hesitancy degree of a fuzzy membership function equals to zero, the information volume of this special fuzzy membership function is identical to Shannon entropy. Two typical fuzzy sets, namely classic fuzzy sets and intuitiontistic fuzzy sets, are studied. Several examples are illustrated to show the efficiency of the proposed information volume of fuzzy membership function. Keywords: fuzzy sets, membership function, information volume, higher-order information volume, entropy. 1 Introduction In the past decades, plenty of theories have been developed for expressing and dealing with the un- certainty in the uncertain environment, such as probability theory [19], fuzzy set theory [51], Dempster- Shafer evidence theory [6, 28], complex evidence theory [37, 39], Z numbers [53], belief structure [47, 49] and D numbers [23, 24]. https://doi.org/10.15837/ijccc.2021.1.4106 2 Since fuzzy set theory was firstly proposed by Zadeh in 1965 [51], uncertain information processing based on fuzzy set has become a heated field [9, 40]. Lots of researchers have been promoting the development of fuzzy set theory. For instance, Zadeh presented type-2 fuzzy set in 1975 [52], which is a generalization of classic fuzzy set. In 1999, Atanassov extended classic fuzzy set into intuitionistic fuzzy set [2], which introduces hesitancy into fuzzy membership function. Yager proposed Pythagorean fuzzy subset in 2013 [46], which can better express the fuzziness under uncertain circumstances. In addition, because of the advantages of modeling vagueness and imprecision in the real world, fuzzy set theory has been applied in many areas, such as data processing [54], approximate reasoning [48], decision making [10, 14, 15, 30, 44], uncertainty measurement [4], risk analysis [27], failure mode and effects analysis [17], and so on [12]. Given a probability distribution, its corresponding information volume can be measured by Shan- non entropy [29]. In evidence theory, mass function is the generalization of probability, whose un- certainty can be measured by Deng entropy [8, 13, 22]. In addition, given a mass function, its corresponding information volume can be calculated [7]. However, how to determine the information volume of fuzzy membership function in fuzzy set theory is still an open issue. Some methods have been proposed to model the uncertainty of fuzzy set theory, such as the uncertainty of the interval type-2 fuzzy set [16], the uncertainty of fuzzy system [21], fuzzy entropy of fuzzy set [5, 25, 31, 50]. Nevertheless, these uncertainty measurements are the first-order information volume of fuzzy set. It is reasonable to take the high-order information volume into consideration [7]. To address the issue mentioned above, an original and novel information volume of fuzzy mem- bership function is proposed in this paper, including the first-order and the higher-order information volume. Since there is extra information volume existing in the hesitancy degree compared with mem- bership function and non-membership function, this extra information volume should be fully utilized to determine the information volume of fuzzy membership function. The major idea of the proposed method is to continuously separate the hesitancy degree until convergence, then the information vol- ume of fuzzy membership function can be calculated. The rest of this paper is organized as follows. Section 2 briefly reviews some preliminaries. Section 3 proposes a new information volume of fuzzy membership function. Some numerical examples are illustrated in Section 4. Section 5 makes a brief conclusion. 2 Preliminaries Several preliminaries are briefly introduced in this section, including fuzzy sets, intuitionistic fuzzy sets and Shannon entropy. 2.1 Fuzzy sets Let X be a universe of discourse. A fuzzy set A based on X can be characterized by the set of pairs which is defined as [51]: Definition 2.1: Fuzzy sets A = {〈x,µ (x)〉 |x ∈ X} (1) where µ(x) : X → [0, 1] is the membership function (MF) of A, which describes the membership degree of each element x to the fuzzy set A. The closer µ (x) is to 1, the more likely x belongs to A. 2.2 Intuitionistic fuzzy sets Given a universe of discourse X, an intuitionistic fuzzy set A is defined as follows [2]: Definition 2.2: Intuitionistic fuzzy sets A = {〈x,µ (x) ,v (x)〉 |x ∈ X} (2) where µ(x) : X → [0, 1] and v(x) : X → [0, 1] are the membership function (MF) and the non- membership function (non-MF) of A, respectively. µ (x) describes the membership degree of x to https://doi.org/10.15837/ijccc.2021.1.4106 3 the set A, and v (x) describes the non-membership degree of x to the set A. For every x ∈ X, 0 ≤ µ (x) + v (x) ≤ 1. The hesitancy degree is defined as: π (x) = 1 −µ (x) −v (x) (3) which represents the hesitancy degree of each element x ∈ X. Besides, intuitionistic fuzzy sets can be represented as mass function, and a lot of works under evidence theory are presented [36, 42, 43, 45]. 2.3 Shannon entropy Entropy function plays an important role in measuring the uncertainty of a system [3, 26, 32, 55] In the field of probability theory, Shannon entropy [29] is often used to measure the uncertainty of a proba- bility distribution. Consider a probability distribution P defined on the set Θ = {H1,H2,H3, · · · ,HN}. Definition 2.3: Shannon entropy Shannon entropy Hs(P) is defined as follows: Hs(P) = − ∑ θ∈Θ P(θ) log P(θ). (4) where ∑ θ∈Θ P(θ) = 1 and P(θ) ∈ [0, 1]. 3 Information volume of fuzzy membership function In this section, firstly, the first-order information volume of fuzzy membership function is proposed. Then, the higher-order information volume of fuzzy membership function is presented. 3.1 First-order information volume of fuzzy membership function In the course of information processing in fuzzy environment, it is important to identify the un- certainty of the data. Taking intuitionistic fuzzy sets for example, intuitively, the uncertainty of fuzzy membership function 〈x1, 0.4, 0.4〉 is larger than that of 〈x2, 0.9, 0.1〉. In fact, this uncertainty of fuzzy membership function is actually the first-order information volume. Hence, the first-order information volume of fuzzy membership function is proposed in this subsection. Definition 3.1: First-order information volume of fuzzy membership function Let the universe discourse be X, and an intuitionistic fuzzy set be {〈x,µ,v〉 |x ∈ X} where µ is membership function, v is non-membership function and π is hesitancy degree. The first-order information volume of fuzzy membership function H1(µ,v) is defined as: H1(µ,v) = −µ log µ−v log v −π log π C (5) where π = 1 −µ−v, and C is the cardinality of fuzzy sets defined as: C = { 2 ( classic fuzzy sets ) 3 ( intuitionistic fuzzy sets ) (6) The hesitancy degree π can be considered as ’hesitant’ which is a hybrid proposition of ’support’ and ’oppose’. Hence, π contains more information volume than µ and v. It is reasonable that π should be divided by the cardinality of the fuzzy sets. The definition of the cardinality C is explained as follows. In classic fuzzy sets, µ and π = 1−µ can be viewed as these two propositions: ’support’ and ’hesitant’, so that the cardinality of classic fuzzy sets is 2. In intuitionistic fuzzy sets, µ, v and π can be respectively seen as these three propositions: ’support’, ’oppose’ and ’hesitant’, so that the cardinality of intuitionistic fuzzy sets is 3. Actually, the classic fuzzy set is a special case of the intuitionistic fuzzy set, which means that the classic fuzzy set satisfies v ≡ 0 and π = 1 − µ. Hence, the first-order information volume of classic fuzzy membership function is that: https://doi.org/10.15837/ijccc.2021.1.4106 4 H1(µ,v) = −µ log µ− (1 −µ) log 1 −µ 2 (7) 3.2 Higher-order information volume of the fuzzy membership function The real world is very complex and complicated [11, 18, 20]. It’s more reasonable to explore a complex thing from different aspects [38]. For example, negation is paid attention recently since it can provides new view from negative side [1, 41]. Another typical modeling is with complex networks [33, 34, 35]. About the information volume, the issue of the first-order information volume is that it divides the hesitancy degree π only once, which cannot make fully use of the information volume of π. Because π can be seen as the proposition of ’hesitant’, which is a hybrid proposition of ’support’ and ’oppose’, π contains extra information volume compared with µ and v. This extra information volume should be fully utilized by continuously separating the hesitancy degree until convergence. Inspired by the idea mentioned above, the higher-order information volume of fuzzy membership function is presented in this subsection. Definition 3.2: Higher-order information volume of fuzzy membership function Let the universe discourse be X, and an intuitionistic fuzzy set be {〈x,µ,v〉 |x ∈ X} where µ is membership function, v is non-membership function and π is the hesitancy degree. The higher-order information volume of fuzzy membership function H∞(µ,v) is defined as follows: H∞ (µ,v) = − ∞∑ i=1 µi log µi − ∞∑ i=1 vi log vi − lim i→∞ πi log πi C (8) where C is the cardinality of the fuzzy sets defined as: C = { 2 ( classic fuzzy sets ) 3 ( intuitionistic fuzzy sets ) (9) µ1 = µ, v1 = v and π1 = π = 1 − µ − v denote the initial states of membership function, non- membership function and hesitancy degree. i denotes the times of separation. µi, vi and πi (i > 1) are the terms derived by separating πi−1 which are defined as:  µi = k1πi−1 vi = k2πi−1 πi = k3πi−1 (10) where k1 = µ, k2 = v and k3 = π = 1 −µ−v are the separating proportions. The reason for the definition of the separating proportion k1, k2 and k3 is that, µ, v and π can be seen as the prior information of fuzzy sets, so that the hesitancy degree can be separated based on these prior information. Besides, it is a convention that the expression 0 log 0 is defined as 0. Hence, when the hesitancy degree π1 = 0, the higher-order information volume of fuzzy membership function H∞(µ,v) can be written as follows: H∞ (µ,v) = −µ1 log µ1 −v1 log v1 (11) Actually, the classic fuzzy set is a special case of the intuitionistic fuzzy set, which means that the classic fuzzy set satisfies vi ≡ 0 and πi = 1 −µi. Thus, the higher-order information volume of classic fuzzy membership function is that: H∞ (µ,v) = − ∞∑ i=1 µi log µi − lim i→∞ (1 −µi) log 1 −µi 2 (12) https://doi.org/10.15837/ijccc.2021.1.4106 5 Algorithm 1 Higher-order information volume of fuzzy membership function H∞(µ,v) Input: : MF µ and non-MF v Output: : Higher-order information volume of fuzzy membership function H∞(µ,v) 1: i = 1 //Initial time of separation 2: µ1 = µ, v1 = v, π1 = 1 −µ−v //Initial states 3: k1 = µ, k2 = v, k3 = 1 −µ−v //Separating proportion 4: repeat 5: i + + 6: µi = k1πi−1, vi = k2πi−1, πi = k3πi−1 //Separating πi−1 7: Hi (µ,v) = − ∑i j=1 µjLOGµj − ∑i j=1 vjLOGvj −πiLOG πi C //xLOGy invokes Algorithm 2 8: until ∆i = Hi(µ,v) −Hi−1(µ,v) < ε //Allowable error ε 9: return H∞(µ,v) = Hi(µ,v) Algorithm 2 Modified logarithm xLOGy Input: : Real numbers x and y Output: : Modified logarithm xLOGy 1: if x == 0 && y == 0 then 2: ans = 0 //The expression 0 log 0 is defined as 0 3: else 4: ans = x∗ log(y) 5: end if 6: return xLOGy = ans In addition, given an allowable error ε, the higher-order information volume of fuzzy membership function H∞(µ,v) can also be calculated by Algorithm 1. It should be noted that, in Algorithm 1, step 7 invokes modified logarithm xLOGy in Algorithm 2 to avoid the computation 0 log 0. It can be derived from Definition 3.2 and Algorithm 1 that, when i = 1, the higher-order in- formation volume of fuzzy membership function H∞(µ,v) degenerates into the first-order information volume of fuzzy membership function H1(µ,v), which can be written as follows: H∞ (µ,v)|i=1 = H1 (µ,v) (13) For better understanding, the calculating procedure of H∞(µ,v) is illustrated in Figure 1. This figure clearly shows that the hesitancy degree is continuously separated based on the proportion k1, k2 and k3. 4 Numerical examples and discussions In this section, some numerical examples are shown to illustrate above conceptions. In the following examples, let the universe discourse be X and the base of the logarithmic function be 2. Example 4.1: Consider that a classic fuzzy membership function is 〈x1, 0.65〉 where x1 ∈ X. The first-order information volume of this classic fuzzy membership function can be calculated as follows: Because this is a classic fuzzy membership function, the cardinality C is 2, and π = 1 −µ = 0.35. Then, the first-order information volume of this classic fuzzy membership function is that: H1(µ,v) = −µ log µ− (1 −µ) log 1 −µ C = −0.65 log2(0.65) − 0.35 log2( 0.35 2 ) = 1.2841 (14) Example 4.2: Let an intuitionistic fuzzy membership function be 〈x2, 0.65, 0.05〉 where x2 ∈ X. The first-order information volume of this intuitionistic fuzzy membership function can be calculated as follows: https://doi.org/10.15837/ijccc.2021.1.4106 6 Figure 1: The calculating procedure of H∞(µ,v) Because this is an intuitionistic fuzzy membership function, the cardinality C is 3, and π = 1−µ−v = 1−0.65−0.05 = 0.3. Then, the first-order information volume of this intuitionistic fuzzy membership function is that: H1(µ,v) = −µ log µ−v log v −π log π C = −0.65 log2(0.65) − 0.05 log2(0.05) − 0.3 log2( 0.3 3 ) = 1.6166 (15) Though the parameter C can be determined by the cardinality of fuzzy sets, for the sake of simple and easy comparison between different types of fuzzy sets, we suggest to use C = 3 in both cases. Then, under the condition of C = 3, the result in Example 4.1 can be calculated below: H1(µ,v) = −µ log µ− (1 −µ) log 1 −µ C = −0.65 log2(0.65) − 0.35 log2( 0.35 3 ) = 1.4888 (16) This result is smaller than the result 1.6166 in Example 4.2, which means that the first-order information volume of 〈x1, 0.65〉 is smaller than that of 〈x2, 0.65, 0.05〉. https://doi.org/10.15837/ijccc.2021.1.4106 7 Example 4.3: Consider that a classic fuzzy membership function is 〈x3, 0.3〉 where x3 ∈ X, and the allowable error ε = 0.00001. Because this is a classic fuzzy membership function, the cardinality C is 2, and π = 1−µ = 1−0.3 = 0.7. Then, based on Algorithm 1, the calculating procedure for the higher-order information volume of this classic fuzzy membership function is shown in Table 1, and the convergence tendency of H∞(µ,v) is illustrated in Figure 2. Table 1: The calculating procedure of H∞(µ,v) in Example 4.3 i Hi(µ,v) i Hi(µ,v) i Hi(µ,v) 1 1.58129 12 2.91082 23 2.93711 2 1.98819 13 2.91886 24 2.93727 3 2.27303 14 2.92449 25 2.93738 4 2.47241 15 2.92844 26 2.93745 5 2.61198 16 2.93120 27 2.93751 6 2.70968 17 2.93313 28 2.93755 7 2.77806 18 2.93448 29 2.93757 8 2.82594 19 2.93543 30 2.93759 9 2.85945 20 2.93609 31 2.93761 10 2.88290 21 2.93655 32 2.93761 11 2.89932 22 2.93688 Figure 2: The convergence tendency of H∞(µ,v) in Example 4.3 According to Table 1 and Figure 2, by continuously separating the hesitancy degree, the higher- order information volume of this classic fuzzy membership function becomes larger and larger, and finally converges to 2.93761. Example 4.4: Let an intuitionistic fuzzy membership function be 〈x4, 0.2, 0.15〉 where x4 ∈ X, and the allowable error ε = 0.00001. Because this is an intuitionistic fuzzy membership function, the cardinality C is 3, and π = 1 −µ−v = 1 − 0.2 − 0.15 = 0.65. Then, based on Algorithm 1, the calculating procedure for the higher-order information volume of this intuitionistic fuzzy membership function is shown in Table 2, and the convergence tendency of H∞(µ,v) is illustrated in Figure 3. https://doi.org/10.15837/ijccc.2021.1.4106 8 Table 2: The calculating procedure of H∞(µ,v) in Example 4.4 i Hi(µ,v) i Hi(µ,v) i Hi(µ,v) 1 2.30912 11 3.63589 21 3.65375 2 2.77983 12 3.64223 22 3.65384 3 3.08579 13 3.64634 23 3.65389 4 3.28466 14 3.64902 24 3.65393 5 3.41393 15 3.65076 25 3.65395 6 3.49795 16 3.65189 26 3.65397 7 3.55257 17 3.65263 27 3.65398 8 3.58807 18 3.65311 28 3.65398 9 3.61114 19 3.65342 10 3.62614 20 3.65362 Figure 3: The convergence tendency of H∞(µ,v) in Example 4.4 According to Table 2 and Figure 3, by continuously separating the hesitancy degree, the higher- order information volume of this intuitionistic fuzzy membership function becomes larger and larger, and finally converges to 3.65398. Although the parameter C can be determined by the cardinality of fuzzy sets, for simply and easily comparing the information volume between different types of fuzzy sets, we suggest to use C = 3 in both cases. Then, under the condition of C = 3, the result of Example 4.3 can be obtained, whoso calculating procedure is shown in Table 3 and the convergence tendency is illustrated in Figure 4. According to Table 3 and Figure 4, when C = 3, the result in Example 4.3 finally converges to 2.93761. This result is smaller than the result 3.65398 in Example 4.4, which means that the higher-order information volume of 〈x3, 0.3〉 is smaller than that of 〈x4, 0.2, 0.15〉. Example 4.5: Consider that an intuitionistic fuzzy membership function is 〈x5, 0.8, 0.2〉 where x5 ∈ X, and the allowable error ε = 0.00001. Because this is an intuitionistic fuzzy membership function, the cardinality C is 3, and π = 1 − µ − v = 1 − 0.8 − 0.2 = 0. Then, based on Algorithm 1, the higher-order information volume of this intuitionistic fuzzy membership function can be calculated as follows. When i = 1, the associated higher-order information volume is that: https://doi.org/10.15837/ijccc.2021.1.4106 9 Table 3: The calculating procedure of H∞(µ,v) in Example 4.3 when C = 3 i Hi(µ,v) i Hi(µ,v) i Hi(µ,v) 1 1.99076 12 2.91891 23 2.93727 2 2.27483 13 2.92453 24 2.93738 3 2.47367 14 2.92846 25 2.93745 4 2.61286 15 2.93121 26 2.93751 5 2.71029 16 2.93314 27 2.93755 6 2.77850 17 2.93449 28 2.93757 7 2.82624 18 2.93543 29 2.93759 8 2.85966 19 2.93609 30 2.93761 9 2.88305 20 2.93656 31 2.93761 10 2.89943 21 2.93688 11 2.91089 22 2.93711 Figure 4: The convergence tendency of H∞(µ,v) in Example 4.3 when C = 3 H∞ (µ,v)|i=1 = − 1∑ j=1 µj log µj − 1∑ j=1 vj log vj −π1 log π1 C = −0.8 log2(0.8) − 0.2 log2(0.2) − 0 = 0.7219 (17) When i = 2, the associated higher-order information volume is that: H∞ (µ,v)|i=2 = − 2∑ j=1 µj log µj − 2∑ j=1 vj log vj −π2 log π2 C = −(0.8 log2(0.8) + 0) − (0.2 log2(0.2) + 0) − 0 = 0.7219 (18) Then, calculate the increment of higher-order information volume: ∆2 = H2(µ,v) −H1(µ,v) = 0.7219 − 0.7219 = 0 < ε (19) https://doi.org/10.15837/ijccc.2021.1.4106 10 Hence, the algorithm is convergent, and the higher-order information volume of this intuitionistic fuzzy membership function is 0.7219. Because the hesitancy degree π = 0, the three propositions of intuitionistic fuzzy membership func- tion, namely ’support’, ’oppose’ and ’hesitant’, degenerate into two propositions, which are ’support’ and ’oppose’. Since these two propositions are exclusive, the intuitionistic fuzzy membership function 〈x5, 0.8, 0.2〉 can be seen as the probability distributions P1 = 0.8,P2 = 0.2 which are also exclusive. Then, the associated Shannon entropy [29] is that Hs(P) = −P1 log P1 −P2 log P2 = −0.8 log2(0.8) − 0.2 log2(0.2) = 0.7219 (20) which is the same as the higher-order information volume H∞ (µ,v). This example shows that, when the hesitancy degree π = 1 −µ−v = 0, the value of H∞ (µ,v) is identical to Shannon entropy. 5 Conclusion In this paper, a novel information volume of fuzzy membership function is proposed. The proposed measure not only takes the first-order information volume, but also the higher-order information volume into consideration. The major contributions and some desirable properties of the proposed method are listed below: (1) The first-order information volume of the fuzzy membership function is presented. (2) The higher-order information volume of the fuzzy membership function is proposed. (3) When the hesitancy degree equals to zero, the information volume of fuzzy membership function is identical to Shannon entropy. (4) When the times of separation equals to one, the higher-order information volume of fuzzy member- ship function degenerates into the first-order information volume of fuzzy membership function. Some examples, especial on classic fuzzy sets and intuitiontistic fuzzy sets, are used to show the efficiency of the proposed measure. Acknowledgments The authors greatly appreciate the editor’s encouragement and the reviews’ suggestions to improve this work. The work is partially supported by National Natural Science Foundation of China (Grant No. 61973332). 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Information Volume of Fuzzy Membership Function, International Journal of Computers Communications & Control, 16(1), 4106, 2021. https://doi.org/10.15837/ijccc.2021.1.4106 Introduction Preliminaries Fuzzy sets Intuitionistic fuzzy sets Shannon entropy Information volume of fuzzy membership function First-order information volume of fuzzy membership function Higher-order information volume of the fuzzy membership function Numerical examples and discussions Conclusion