Abstract
Fuzzy Inf. Eng. (2011) 2: 157-167 DOI 10.1007/s12543-011-0074-9 ORIGINAL ARTICLE Reverse Triple I Reasoning Methods Based on the Łukasiewicz Implication Hua-wen Liu Received: 11 January 2010/ Revised: 25 March 2011/ Accepted: 5 May 2011/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract The reverse andα-reverse triple I reasoning methods based on Łukasiewi -cz implication I are established in new manners which correct the mistakes in the existing literature. Furthermore, the α-reverse triple I reasoning method is extended to a new form, called α(u, v)-reverse triple I reasoning method, which can contain the reverse triple I reasoning method as its particular case. This is another improved point to the existing results. Keywords Fuzzy reasoning · Reverse triple I method · Łukasiewicz implication · Fuzzy modus ponens· Fuzzy modus tollens ·α(u, v)-reverse triple I method 1. Introduction It is well known that Zadeh’s CRI (Compositional Rule of Inference) method [1] is the most widespread means for solving the following fuzzy modus ponens (FMP for short) and fuzzy modus tollens (FMT for short) problems: ∗ ∗ FMP : for given A → B (rule) and A (input), to compute B (output), (1) ∗ ∗ FMT : for given A → B (rule) and B (input), to compute A (output), (2) ∗ ∗ where A, A ∈ F(U) (the set of all fuzzy subsets of universe U) and B, B ∈ F(V) (the set of all fuzzy subsets of universe V). To improve the CRI method, Wang analyzed ∗ ∗ in [2] that, the fact that A (u) implies B (v) should be fully supported by the major premise A(u) → B(v) in solving FMP and FMT problems, i.e., ∗ ∗ (A(u) → B(v)) → (A (u) → B (v)) should take its maximum whenever u ∈ U and v ∈ V. Then the so-called triple I (the abbreviation of triple implications) method for solving the above problems was Hua-wen Liu () School of Mathematics, Shandong University, Jinan, Shandong 250100, China email: hw.liu@sdu.edu.cn 158 Hua-wen Liu (2011) proposed in [2] (or, see [3-10]). Furthermore, by means of the theory of sustaining degrees introduced by Wang in [2], the triple I method was generalized to theα-triple I method (see [2-10]). Later, Song and Wu considered the reverse sustention problem and proposed a new method for solving FMP and FMT problems in [11] called reverse triple I method, ∗ ∗ which is to seek optimal B for FMP (respectively, A for FMT), such that ∗ ∗ (A (u) → B (v)) → (A(u) → B(v)) (3) takes its maximum whenever u ∈ U and v ∈ V. Furthermore, the α-reverse triple I method as an generalization of the reverse triple I method was also proposed in [11]. ∗ ∗ Its basic idea is to seek optimal B for FMP (respectively, A for FMT) such that the following inequality holds for any u ∈ U, v ∈ V and a givenα ∈ [0, 1], ∗ ∗ (A (u) → B (v)) → (A(u) → B(v)) ≥ α. (4) Meanwhile, the expressions of the reverse andα-reverse triple I methods were estab- lished in [11] based on the residual implication generated by the nilpotent minimum with the standard negation n on [0,1] (see [12]). Recently, Zhao and Li [13], Peng and Hou et al [14] and Qin and Pei [15] respectively and independently established the reverse and α-reverse triple I methods based on Łukasiewicz implication I de- ﬁned by I (x, y) = (x + y) ∧ 1 for any x, y ∈ [0, 1] (where x = 1 − x). But, the established formulas focusing on a same problem are diﬀerent. Hence, the natural questions arise: which is right? why? or, what relations do they have? In the present paper, we will reestablish the expressions of the reverse and α-reverse triple I meth- ods in new manners based on I , and will give answers to the above questions. At the same time, the way used in our proofs greatly simpliﬁes the corresponding proofs of the existing results. Noticing that the α-reverse triple I method cannot contain the reverse triple I method as its particular case, we will make a further extension to the α-reverse triple I methods and propose the α(u, v)-reverse triple I methods for FMP and FMT. In the next section, we establish the reverse triple I methods for FMP and FMT based on I . In Sections 3, we establish the α-reverse triple I methods for FMP and FMT based on I . At the same time, we discuss the existing formulas established in [13-15]. In Sections 4, the α-reverse triple I methods are extended to α(u, v)-reverse triple I methods. The ﬁnal section is the conclusion. 2. Reverse Triple I Methods for FMP and FMT Based on I At ﬁrst, we list some properties of the Łukasiewicz implication I for our usage. (P ) I (y, z) = 1 ⇐⇒ y ≤ z; 1 L (P ) x ≤ I (y, z) ⇐⇒ y ≤ I (x, z); 2 L L (P ) I is nonincreasing in its ﬁrst and nondecreasing in its second variable, 3 L where x, y, z ∈ [0, 1]. In our discussion, we also write I as→. 2.1. Reverse Triple I Method for FMP Based on I Reverse triple I principle for FMP: Fuzzy Inf. Eng. (2011) 2: 157-167 159 The solution B (∈ F(V)) of FMP (1) is the largest fuzzy subset of V making ex- pression (3) take its maximum at any u ∈ U and v ∈ V. This B is called reverse triple I solution to FMP. According to the above principle, we establish the corresponding triple I algorithm based on I as follows. Theorem 2.1 (Reverse triple I method for FMP) Suppose that Łukasiewicz impli- cation I is used in FMP, then the reverse triple I solution to FMP can be expressed as follows, for v ∈ V: ∗ ∗ inf{I (A(u), B(v))+ A (u)− 1}, if (A (u)) ≤ I (A(u), B(v)) < 1, L L ⎪ u∈E B (v) = (5) 1, if I (A(u), B(v)) = 1, or I (A(u), B(v)) < 1 and A (u) = 0, L L ⎩ ∗ ∗ 0, if I (A(u), B(v)) < (A (u)) and A (u) > 0, where E = {u ∈ U|(A (u)) ≤ I (A(u), B(v)) < 1}. v L Proof For any v ∈ V, since Łukasiewicz implication I is nonincreasing in its ﬁrst and nondecreasing in its second variable, the maximum of (3) for given A (u) and A(u) → B(v) equals M(u, v)= (A (u) → 0) → (A(u) → B(v)) ⎪ ∗ 1, if (A (u)) ≤ I (A(u), B(v)), (6) ⎨ L ⎪ ∗ ∗ A (u)+ I (A(u), B(v)), if (A (u)) > I (A(u), B(v)). L L (i) If I (A(u), B(v)) = 1, then M(u, v) = 1. So the maximum of B (v) s making (3) take its maximum 1 is B (v) = 1. ∗ ∗ (ii) If I (A(u), B(v)) < 1 and (A (u) → B (v)) → (A(u) → B(v)) = M(u, v), then we have the following discussion. (a) If (A (u)) ≤ I (A(u), B(v)), then ∗ ∗ (A (u) → B (v)) → (A(u) → B(v)) = 1. (7) ∗ ∗ Formula (7) is equivalent to I (A (u), B (v)) ≤ I (A(u), B(v)) due to property (P ) L L 1 ∗ ∗ of I . Again, A (u) > B (v) holds since I (A(u), B(v)) < 1. It follows that (7) is L L ∗ ∗ ∗ ∗ equivalent to 1− A (u)+ B (v) ≤ I (A(u), B(v)), i.e., B (v) ≤ I (A(u), B(v))+ A (u)−1. L L From the fact that B is the largest fuzzy subset satisfying (7), we get ∗ ∗ B (v) = inf{I (A(u), B(v))+ A (u)− 1}, u∈E where E = {u ∈ U|I (A(u), B(v)) < 1, and (A (u)) ≤ I (A(u), B(v))}. v L L (b) If (A (u)) > I (A(u), B(v)), then ∗ ∗ ∗ (A (u) → B (v)) → (A(u) → B(v)) = A (u)+ I (A(u), B(v)) (8) ∗ ∗ ∗ and (A (u) → B (v)) → (A(u) → B(v)) = A (u)+ I (A(u), B(v)) < 1. This means that ∗ ∗ (A (u) → B (v)) > (A(u) → B(v)) holds from (P ). So Formula (8) is equivalent to 1 160 Hua-wen Liu (2011) ∗ ∗ ∗ ∗ ∗ 1− I (A (u), B (v))+ I (A(u), B(v)) = A (u)+ I (A(u), B(v)), i.e., I (A (u), B (v)) = L L L L 1− A (u). ∗ ∗ ∗ ∗ If A (u) = 0, then I (0, B (v)) = 1. From the largestness of B ,weget B (v) = 1. ∗ ∗ ∗ ∗ ∗ If A (u) 0, then from I (A (u), B (v)) = 1 − A (u) 1, it follows that A (u) > ∗ ∗ ∗ ∗ ∗ ∗ B (v). So, formula I (A (u), B (v)) = 1− A (u) is equivalent to 1− A (u)+ B (v) = ∗ ∗ 1− A (u). We get B (v) = 0. Summarizing above (i) and (ii), Formula (5) follows. Remark 2.1 Zhao and Li [13], Peng and Hou et al [14] and Qin and Pei [15] re- spectively and independently established the following reverse triple I solutions for FMP problem based on Łukasiewicz implication I , for any v ∈ V: ∗ ∗ B (v) = inf{(I (A(u), B(v))+ A (u)− 1)∨ 0|I (A(u), B(v)) < 1, u ∈ U}; L L ∗ ∗ ∗ B (v) = inf{I (A(u), B(v))+ A (u)− 1|I (A(u), B(v)) > (A (u)) , u ∈ U}; L L ∗ ∗ ∗ B (v) = inf{(I (A(u), B(v))+ A (u)− 1)∨ 0|(A (u)) ∨ I (A(u), B(v)) < 1, u ∈ U}. L L ∗ ∗ ∗ Obviously, the above expressions of B , B and B are diﬀerent. Comparing them Z P Q ∗ ∗ ∗ with Formula (5) we know that B is equivalent to the B given by (5). B was estab- Q Q lished in [15] in another way. The following counterexample will show that B and B are incorrect. u+ 1 Example 2.1 Let U = V = [0, 1], A(u) = , B(v) = 1− v and A (u) = 1− u for any u, v ∈ [0, 1]. 1 1 (i) For v = , it follows from (5) that B = 1 since there exists u = 1 such that 2 2 1 5 I (A(1), B ) = < 1 and A (u) = 0. But, 2 6 1 1 1 ∗ ∗ B = inf{(I (A(u), B )+ A (u)− 1)∨ 0|I (A(u), B ) < 1, u ∈ U} L L 2 2 2 1 5 ≤ (I (A(1), B )+ A (1)− 1)∨ 0 = + 0− 1 ∨ 0 = 0, 2 6 i.e., B = 0. 1 1 2 (ii) For v = , it follows from (5) that B = 1 since there exists u = such 3 3 3 5 2 1 that I (A(u), B(v)) = I , = 1 and A (u) = > 0. But, L L 9 3 3 1 1 1 ∗ ∗ ∗ B = inf{I (A(u), B )+ A (u)− 1|I (A(u), B ) > (A (u)) , u ∈ U} L L 3 3 3 2 1 2 1 ≤ I (A , B )+ A − 1 = . 3 3 3 3 2.2. Reverse Triple I Methods for FMT Based on I L Fuzzy Inf. Eng. (2011) 2: 157-167 161 Reverse triple I principle for FMT: The solution A (∈ F(U)) of FMT(2) is the smallest fuzzy subset of U making expression (3) take its maximum at any u ∈ U and v ∈ V. This A is called reverse triple I solution of FMT. According to the above principle, we establish the corresponding triple I algorithm based on I as follows. Theorem 2.2 (Reverse triple I method for FMT) Suppose that Łukasiewicz impli- cation I is used in FMT, then the reverse triple I solution to FMT can be expressed as follows, for u ∈ U: ∗ ∗ ⎪ sup{1+ B (v)− I (A(u), B(v))}, if B (v) ≤ I (A(u), B(v)) < 1, L L ⎪ v∈E A (u) = ∗ (9) ⎪ 1, if I (A(u), B(v)) < B (v) < 1, 0, if I (A(u), B(v)) = 1 or I (A(u), B(v)) < B (v) = 1, L L where E = {v ∈ V|B (v) ≤ I (A(u), B(v)) < 1}. u L Proof For any u ∈ U, since I is nonincreasing in its ﬁrst and nondecreasing in its second variable, the maximum of (3) for given B (v) and A(u) → B(v) is as follows: ∗ ∗ N(u, v)= (1 → B (v)) → (A(u) → B(v)) = B (v) → (A(u) → B(v)) ⎪ ∗ (10) 1, if B (v) ≤ I (A(u), B(v)), ⎨ L ⎪ ∗ ∗ (B (v)) + I (A(u), B(v)), if B (v) > I (A(u), B(v)). L L (i) If I (A(u), B(v)) = 1, then N(u, v) = 1. So if the expression (3) takes its maxi- mum 1, then the minimum of A (u)is0. ∗ ∗ (ii) If I (A(u), B(v)) < 1 and (A (u) → B (v)) → (A(u) → B(v)) = N(u, v), then we have the following discussion. (a) If B (v) ≤ I (A(u), B(v)), then ∗ ∗ (A (u) → B (v)) → (A(u) → B(v)) = 1. (11) ∗ ∗ Formula (11) is equivalent to I (A (u), B (v)) ≤ I (A(u), B(v)) owing to the property L L ∗ ∗ (P )of I . Again, A (u) > B (v) holds since I (A(u), B(v)) < 1. It follows that (11) is 1 L L ∗ ∗ ∗ ∗ equivalent to 1− A (u)+ B (v) ≤ I (A(u), B(v)), i.e., A (u) ≥ 1+ B (v)− I (A(u), B(v)). L L From the smallestness of A ,weget ∗ ∗ A (u) = sup{1+ B (v)− I (A(u), B(v))}, v∈E where E = {v ∈ V|I (A(u), B(v)) < 1, and B (v) ≤ I (A(u), B(v))}. u L L (b) If B (v) > I (A(u), B(v)), then ∗ ∗ ∗ (A (u) → B (v)) → (A(u) → B(v)) = 1− B (v)+ I (A(u), B(v)) (12) ∗ ∗ ∗ and 1 − B (v) + I (A(u), B(v)) < 1. This means that (A (u) → B (v)) > (A(u) → ∗ ∗ B(v)) holds from (P ). Thus Formula (12) is equivalent to 1 − I (A (u), B (v)) + 1 L ∗ ∗ ∗ ∗ I (A(u), B(v)) = 1− B (v)+ I (A(u), B(v)), i.e., I (A (u), B (v)) = B (v). L L L 162 Hua-wen Liu (2011) ∗ ∗ ∗ ∗ If B (v) = 1, then I (A (u), 1) = 1. From the smallestness of A , we get A (u) = 0. ∗ ∗ ∗ ∗ ∗ ∗ If B (v) < 1, then from I (A (u), B (v)) = B (v) < 1, it follows that A (u) > B (v). ∗ ∗ ∗ ∗ ∗ ∗ Thus, formula I (A (u), B (v)) = B (v) is equivalent to 1− A (u)+ B (v) = B (v). We get A (u) = 1. Summarizing above (i) and (ii), Formula (9) follows. Remark 2.2 Zhao and Li [13], Peng and Hou et al [14] and Qin and Pei [15] re- spectively and independently established the following reverse triple I solutions to FMT problem based on Łukasiewicz implication I , for any u ∈ U: ∗ ∗ A (u) = sup{1+ B (v)− I (A(u), B(v))|I (A(u), B(v)) < 1, v ∈ V}; L L ∗ ∗ ∗ A (u) = sup{1+ B (v)− I (A(u), B(v))|B (v) < I (A(u), B(v)), v ∈ V}; L L ∗ ∗ ∗ A (u) = sup{(1+ B (v)− I (A(u), B(v)))∧ 1|B (v)∨ I (A(u), B(v)) < 1, v ∈ V}. L L ∗ ∗ ∗ Obviously, the above expressions of A , A and A are diﬀerent from each other. Z P Q ∗ ∗ Comparing them with the Formula (9) we know that A is equivalent to the A given by (9). A was established in [15] in another way. The following counterexample ∗ ∗ will show that A and A are incorrect. Z P u+ 1 2 Example 2.2 Let U = V = [0, 1], A(u) = , B(v) = 1− v and B (v) = for any 3 3 u, v ∈ [0, 1]. (i) For u = 1, we have ∗ ∗ A (1)= sup{1+ B (v)− I (A(1), B(v))|I (A(1), B(v)) < 1, v ∈ V} L L 5 5 2 1 7 ≥ 1+ B − I (A(1), B ) = 1+ − = > 1. 6 6 3 2 6 Obviously, A (1) > 1 is not true. 2 2 1 (ii) For u = , it follows from (9) that A = 0, since there exists v = such 3 3 3 2 1 5 2 2 that I (A , B ) = I , = 1 and B (v) = < 1. But, L L 3 3 9 3 3 2 2 2 ∗ ∗ ∗ A = sup{1+ B (v)− I (A , B(v))|B (v) < I (A , B(v)), v ∈ V} L L 3 3 3 1 2 1 2 2 ≥ 1+ B − I (A , B ) = 1+ − 1 = > 0. 3 3 3 3 3 3.α-reverse Triple I Methods for FMP and FMT Based on I 3.1.α-reverse Triple I Method for FMP Based on I α-reverse triple I principle for FMP: The conclusion B (∈ F(V)) of FMP(1) is the largest fuzzy subset of V satisfying Inequality (4). This B is calledα-reverse triple I solution to FMP. Fuzzy Inf. Eng. (2011) 2: 157-167 163 If Łukasiewicz implication I is employed in FMP, then the maximum of (3) at (u, v) ∈ U × V is M(u, v) given by (6). So, in order to guarantee the Inequality (4) holds for any u ∈ U and v ∈ V, we need the following assumption: 0<α ≤ inf {(A (u) → 0) → (A(u) → B(v))}. (13) u∈U,v∈V According to the above principle, we establish the corresponding α-reverse triple I algorithm based on I as follows. Theorem 3.1 (α-reverse triple I method for FMP) Suppose that Łukasiewicz impli- cation I is used in FMP, then theα-reverse triple I solution to FMP can be expressed as follows: ∗ ∗ B (v) = inf {(I (A(u), B(v))+ A (u)−α)∨ 0}, v ∈ V, (14) u∈E αv where E = {u ∈ U|I (A(u), B(v))<α}. αv L Proof For any v ∈ V, from (P ), it follows that inequality (4) is equivalent to the following ∗ ∗ A (u) → B (v) ≤ α → (A(u) → B(v)). (15) ∗ ∗ (i) If I (A(u), B(v)) ≥ α, then (15) shows as follows: A (u) → B (v) ≤ 1. From the ∗ ∗ largestness of B ,weget B (v) = 1. (ii) If I (A(u), B(v))<α, then it follows from (15) that ∗ ∗ A (u) → B (v) ≤ 1−α+ I (A(u), B(v)) < 1. ∗ ∗ So we get A (u) > B (v) from (P ). Then, (15) is equivalent to ∗ ∗ 1− A (u)+ B (v) ≤ 1−α+ I (A(u), B(v)), i.e., ∗ ∗ B (v) ≤ (I (A(u), B(v))+ A (u)−α)∨ 0. From the largestness of B , it follows that ∗ ∗ B (v) = inf {(I (A(u), B(v))+ A (u)−α)∨ 0}, u∈E αv where E = {u ∈ U|I (A(u), B(v))<α}. αv L Taking into account infφ = 1, (14) follows from the above. Remark 3.1 Zhao and Li [13], Peng and Hou et al [14] and Qin and Pei [15] re- spectively and independently established the followingα-reverse triple I solutions for FMP problem based on Łukasiewicz implication I : for any v ∈ V, ∗ ∗ B (v) = inf{(I (A(u), B(v))+ A (u)−α)∨ 0|I (A(u), B(v))<α, u ∈ U}; L L αZ ∗ ∗ ∗ B (v) = inf{(I (A(u), B(v))+ A (u)−α)∨ 0|I (A(u), B(v))<α, A (u) > 0, u ∈ U}; L L αP ∗ ∗ B (v) = inf{I (A(u), B(v))+ A (u)−α|I (A(u), B(v))<α, u ∈ U}. L L αQ 164 Hua-wen Liu (2011) ∗ ∗ ∗ Although the forms of B , B and B are diﬀerent, it is easy to verify that they αZ αP αQ ∗ ∗ are equivalent to each other and they are all equivalent to the B given by (14). B αP and B were respectively given in [14] and [15] without any proof, and the proof of αQ B in [13] was complex. αZ 3.2.α-reverse Triple I Method for FMT Based on I α-reverse triple I principle for FMT: The conclusion A (∈ F(U)) of FMT(2) is the smallest fuzzy subset of U satisfying inequality (4). This B is called α-reverse triple I solution to FMT. If Łukasiewicz implication I is employed in FMT, then the maximum of (3) at (u, v) ∈ U × V is N(u, v) given by (10). So, in order to guarantee the Inequality (4) holds for any u ∈ U and v ∈ V, we need the following assumption: 0<α ≤ inf {(1 → B (v)) → (A(u) → B(v))}. (16) u∈U,v∈V According to the above principle, we establish the corresponding α-reverse triple I algorithm based on I as follows. Theorem 3.2 (α-reverse triple I method for FMT) Suppose that Łukasiewicz impli- cation I is used in FMT, then its solution to FMT can be expressed as follows: ∗ ∗ A (u) = sup{α+ B (v)− I (A(u), B(v))}, u ∈ U, (17) v∈E αu where E = {v ∈ V|I (A(u), B(v))<α}. αu L Proof For any u ∈ U, from (P ), it follows that Inequality (4) is equivalent to (15). ∗ ∗ (i) If I (A(u), B(v)) ≥ α, then (15) is as follows: A (u) → B (v) ≤ 1. From the ∗ ∗ smallestness of A ,weget A (u) = 0. (ii) If I (A(u), B(v)) <α, then from the proof of Theorem 3.1, we know that (15) is equivalent to ∗ ∗ 1− A (u)+ B (v) ≤ 1−α+ I (A(u), B(v)), i.e., ∗ ∗ A (u) ≥ α+ B (v)− I (A(u), B(v)). From the smallestness of A , it follows that ∗ ∗ A (u) = sup{α+ B (v)− I (A(u), B(v))}, v∈E αu where E = {v ∈ V|I (A(u), B(v))<α}. αu L Taking into account supφ = 0, (17) follows from above (i) and (ii). Remark 3.2 Zhao and Li [13], Peng and Hou et al [14] and Qin and Pei [15] re- spectively and independently established the following α-reverse triple I solutions to FMT based on Łukasiewicz implication I : for any u ∈ U, ∗ ∗ A (u) = sup{α+ B (v)− I (A(u), B(v))|I (A(u), B(v))<α, v ∈ V}; L L αZ ∗ ∗ A (u) = sup{(α+ B (v)− I (A(u), B(v)))∧ 1|I (A(u), B(v))<α, v ∈ V}; L L αP ∗ ∗ A (u) = A (u). αQ αZ Fuzzy Inf. Eng. (2011) 2: 157-167 165 ∗ ∗ ∗ The above expressions of A (i.e., A ) and A are diﬀerent, but it can be easily αZ αQ αP ∗ ∗ ∗ veriﬁed that they are equal and they are all equal to A given by (17). A and A αP αQ were respectively given in [14] and [15] without any proof, and the proof of A in αZ [13] was complex. 4.α(u, v)-reverse Triple I Methods for FMP and FMT Based on I Noticing that above α-reverse triple I method cannot contain the reverse triple I method as its particular case since the α in (4) is a constant in [0,1], we now make a further extension to theα-reverse triple I methods for FMP and FMT. It is noticed that theα in (4) is a constant. As a general extension, we now replace it by α(u, v) for u ∈ U and v ∈ V, where α(u, v) ∈ [0, 1], and thus (4) becomes as follows ∗ ∗ (A (u) → B (v)) → (A(u) → B(v)) ≥ α(u, v), u ∈ U, v ∈ V. (18) For the sake of making Inequality (18) hold, we always assume in this subsection that α(u, v) ≤ M(u, v) (the maximum of (3) in FMP) for FMP and α(u, v) ≤ N(u, v) (the maximum of (3) in FMT) for FMT for all u ∈ U and v ∈ V. Under above assumption, we now generalize aboveα−reverse triple I principles for FMP and FMT as follows. α(u, v)-reverse triple I principle for FMP: The conclusion B (∈ F(V)) to FMP problem (1) is the largest fuzzy subset of V satisfying (18). This B is called α(u, v)-reverse triple I solution to FMP. Especially, if α(u, v) = M(u, v) for any u ∈ U and v ∈ V, then B is the reverse triple I solution to FMP. α(u, v)-reverse triple I principle for FMT: The conclusion A (∈ F(U)) of FMT problem (2) is the smallest fuzzy subset of U satisfying (18). This A is called α(u, v)-reverse triple I solution to FMT. Especially, if α(u, v) = N(u, v) for any u ∈ U and v ∈ V, then A is the reverse triple I solution to FMT. Completely similar to the proofs of Theorem 3.1 and 3.2, we can have the follow- ing results. Theorem 4.1 (α(u, v)-reverse triple I method to FMP) Suppose that I is used in FMP, then the α(u, v)-reverse triple I solution to FMP can be expressed as follows: ∗ ∗ B (v) = inf {I (A(u), B(v))+ A (u)−α(u, v)}, v ∈ V, (19) u∈E (α)v where E = {u ∈ U|I (A(u), B(v))<α(u, v)}. (α)v L It is easy to check that the reverse triple I solution (5) to FMP is the particular case of (19) when α(u, v) = M(u, v). 166 Hua-wen Liu (2011) Theorem 4.2 (α(u, v)-reverse triple I method to FMT) Suppose that I is used in FMT, then the α(u, v)-reverse triple I solution to FMT can be expressed as follows: ∗ ∗ A (u) = sup {α(u, v)+ B (v)− I (A(u), B(v))}, u ∈ U, (20) v∈E (α)u where E = {v ∈ V|I (A(u), B(v))<α(u, v)}. (α)u L We can easily check that the reverse triple I solution (9) to FMT is the particular case of (20) at α(u, v) = N(u, v). 5. Conclusion Aiming at the existing diﬀerent formulas of the reverse andα-reverse triple I methods for fuzzy reasoning based on I , we have discussed their relations and corrected some of them. At the same time, new expressions of the reverse and α-reverse triple I methods based on I have been established in new manners. These new manners can also be used to establish reverse orα-reverse triple I methods based on other common implications satisfying (P ). Furthermore, the α-reverse triple I method has been extended to α(u, v)-reverse triple I method. Thus, we have solved such a problem: α-reverse triple I method as an extension of the reverse triple I method cannot contain the latter as its particular case. Acknowledgments This work was supported by the National Natural Science Foundation of China (No. 60774100) and the Natural Science Foundation of Shandong Province of China (No. Y2007A15). References 1. 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Journal
Fuzzy Information and Engineering
– Taylor & Francis
Published: Jun 1, 2011
Keywords: Fuzzy reasoning; Reverse triple I method; Łukasiewicz implication; Fuzzy modus ponens; Fuzzy modus tollens; α( u, v )-reverse triple I method