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Interval-valued quintuple implication principle of fuzzy reasoning
International Journal of Approximate Reasoning 84 (2017) 23–32
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International Journal of Approximate Reasoning www.elsevier.com/locate/ijar
Interval-valued quintuple implication principle of fuzzy reasoning ✩ Minxia Luo ∗ , Xiaoling Zhou Department of Mathematics, China Jiliang University, Hangzhou 310018, PR China
a r t i c l e
i n f o
Article history: Received 20 April 2016 Received in revised form 7 January 2017 Accepted 18 January 2017 Available online 3 February 2017 Keywords: Interval-valued fuzzy set Interval-valued fuzzy inference Quintuple implication algorithm Robustness
a b s t r a c t In this paper, we propose the quintuple implication principle based on interval-valued fuzzy inference for fuzzy modus ponens and fuzzy modus tollens, which explicitly use the closeness between A ∗ and A (or B ∗ and B) in the process of fuzzy reasoning. We first give the corresponding interval-valued R-type quintuple implication solutions and then investigate the robustness of quintuple implication algorithms based on interval-valued fuzzy inference. The sensitivity of quintuple implication algorithms based on two important interval-valued residuated implication operators is given. © 2017 Elsevier Inc. All rights reserved.
1. Introduction Since the method of Compositional Rule of Inference (C R I ) was put forward in 1973 by Zadeh [23], various methods of fuzzy reasoning have been proposed [10,14,15,21,22,25]. As an alternative for C R I method, Wang [21] proposed triple I method with full inference rule that utilized the implication operator in every step of the reasoning. The unified triple I algorithms based on regular implications and normal implications have been established by Wang and Fu [22]. For all residuated implications induced by left-continuous t-norms, unified full implication triple I algorithms of fuzzy reasoning are constructed by Pei [14,15]. Pei [16,17] conducted a detailed research into the triple I algorithms based on the monoidal t-norms basic logical system MTL setting a sound logic foundation. Luo and Yao [10] investigated triple I algorithms by the combination of Schweizer–Sklar operators and triple I principles for fuzzy reasoning. Liu and Wang [9] discussed continuity problems of triple I methods based on several implications. Dai et al. [3] studied the robustness of full implication inference methods. In both CRI and triple I method, the closeness of A and A ∗ (or B and B ∗ ) is not explicitly used in the process of calculating the consequence, which sometimes makes the computed approximation useless or misleading. Zhou et al. [25] proposed the Quintuple Implication Principle ( Q I P ), which characterizes the approximation A ∗ and A (or B ∗ and B) in the process of fuzzy reasoning. The method efficiently improves triple I method. It may be suitable to be used instead of Wang’s triple I method in some circumstances. Its basic idea is as follows: For known A ∈ F ( X ), B ∈ F (Y ), and A ∗ ∈ F ( X ) (or B ∗ ∈ F (Y )), seek the optimal B ∗ ∈ F (Y ) (or A ∗ ∈ F ( X )) such that
( A (x) → B ( y )) → (( A ∗ (x) → A (x)) → ( A ∗ (x) → B ∗ ( y ))) (or ( A (x) → B ( y )) → (( B ( y ) → B ∗ ( y )) → ( A (x) → A ∗ (x)))) ✩
*
This work is supported by the National Natural Science Foundation of China (No. 61273018). Corresponding author at: School of sciences, China Jiliang University, Hangzhou 310018, PR China E-mail address: [email protected] (M. Luo).
http://dx.doi.org/10.1016/j.ijar.2017.01.010 0888-613X/© 2017 Elsevier Inc. All rights reserved.
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M. Luo, X. Zhou / International Journal of Approximate Reasoning 84 (2017) 23–32
takes the value as large as possible for any x ∈ X and y ∈ Y , where F ( X ) and F (Y ) denote the collections consisting of all fuzzy subsets of X and Y respectively. However, when vague terms appearing in the production rules of a rule-based approximate reasoning system are represented by interval-valued fuzzy sets (I V F S) rather than ordinary fuzzy sets, the approximate inference based on I V F S will become more valuable and more flexible than the general fuzzy reasoning. Recently, it has achieved great attention from many researchers in the fields of interval-valued fuzzy reasoning. In 1975, Zadeh [24] first proposed an approximate reasoning method based on interval-valued fuzzy set by using composition rule. In [18], B. Van Gasse et al. introduced interval-valued monoidal logic (I V M L), and proved that I V M L is sound and complete w.r.t. triangle algebras. The standard completeness of interval-valued monoidal t-norm based logic was proved by B. Van Gasse et al. [19]. In [1], Alcalde et al. gave a constructive method for the definition of interval-valued fuzzy implication operators. Due to the advantages of interval-valued fuzzy set, in recent years, researchers have been investigating the properties as well as the numerous applications of interval-valued fuzzy reasoning in various fields such as neural networks, control systems and so on [2,6]. Li et al. [8] first extended compositional rule of inference to the case of interval-valued fuzzy set, and discussed the robustness of interval-valued fuzzy reasoning. Luo et al. [11,12] extended full implication triple I algorithms and reverse triple I algorithms to the case of interval-valued fuzzy set, and investigated the robustness of full implication algorithms and reverse triple I algorithms based on interval-valued fuzzy inference. In aforementioned algorithms, however, the closeness between the interval-valued fuzzy set A ∗ and A (or B ∗ and B) is not explicitly used in the process of interval-valued fuzzy inference, which sometimes makes the computed approximation useless or misleading. Example 1. Let the universe be X = Y = {1, 2, 3}. Suppose small, medium and large are three fuzzy sets on S I ( X ), which are defined below:
[small] = [0.51,0.6] + [02,0] + [03,0] , [large ] = [01,0] + [0.52,0.6] + [13,1] , [medium] = [0.51,0.6] + [12,1] + [03,0] . For simplicity, we denote
[small] = {[0.5, 0.6], [0, 0], [0, 0]}, [large ] = {[0, 0], [0.5, 0.6], [1, 1]}, [medium] = {[0.5, 0.6], [1, 1], [0, 0]}. The problem of I F M P is stated as in Table 1. Let A (x) denote “x is small”, B ( y ) denote “ y is large” and A ∗ (x) denote “x is medium.” The task is then to compute B ∗ . For interval-valued Gödel and Łukasiewicz implications, the conclusion is B ∗ ( y ) = {[1, 1], [1, 1], [1, 1]}, i.e. the only information we can obtain from the triple I method is the trivial tautology. The problem of I F M T is stated as in Table 2. Let A (x) denote “x is small”, B ( y ) denote “ y is large” and B ∗ ( y ) denote “ y is medium.” The task is then to compute A ∗ . For interval-valued Gödel and Łukasiewicz implications, the conclusion is A ∗ (x) = {[0, 0], [0, 0], [0, 0]}. Note that A ∗ (x) = {[0, 0], [0, 0], [0, 0]} represents the contradiction and this conclusion is not in accordance with human thinking. Table 1 An example of IFMP. Rule Premise
IF x is small, then y is large x is medium
Calculate
y is?
Table 2 An example of IFMT. Rule Premise
IF x is small, then y is large y is medium
Calculate
x is?
After analyzing the weakness of triple I method based on interval-valued fuzzy inference, in this paper, we combine quintuple implication principle and interval-valued fuzzy set, propose the quintuple implication principle based on intervalvalued fuzzy inference for fuzzy modus ponens and fuzzy modus tollens, which explicitly use the closeness between the interval-valued fuzzy set A ∗ and A (or B ∗ and B) in the process of fuzzy reasoning. The reminder of this paper is organized as follows. In Section 2, we review some basic definitions and properties related to this paper. In Section 3, we briefly recall the quintuple implication principle based on general fuzzy sets. In Section 4, we introduce the quintuple implication principle based on interval-valued fuzzy inference and interpret it under two wellknown implication operators. In Section 5, we investigate the robustness of the proposed algorithms with respect to I F M P and I F M T models, a series of corollaries are proved. The last section concludes the paper.
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2. Preliminaries Let S I = {[x, y ]|x ≤ y , x, y ∈ [0, 1]}. An ordering on S I as [a, b] ≤ [c , d] if a ≤ c and b ≤ d is called component-wise order or Kulisch–Miranker order [4]. ‘∧’ is defined as [a, b] ∧ [c , d] = [a ∧ c , b ∧ d]. ‘∨’ is defined as [a, b] ∨ [c , d] = [a ∨ c , b ∨ d]. It is easy to verify that the ordering just defined is a partial ordering on S I . Furthermore, take [a, b] ∧ [c , d] = [a, b] iff [a, b] ≤ [c , d] and [a, b] ∨ [c , d] = [c , d] iff [a, b] ≤ [c , d]. We can verify that the algebraic structure ( S I , ∧, ∨, [0, 0], [1, 1]) is a complete, bounded and distributive lattice. First, we review some basic definitions and theorems related to this paper. Definition 2.1 ([7]). Let T be a t-norm defined on the interval [0, 1], the associated t-norm on S I is defined as follows:
T : S I × S I → S I, where
T ([a, b], [c , d]) = [ T (a, c ), T (b, d)]. The associated t-norm T on S I is called left-continuous, if T is left-continuous t-norm on the interval [0, 1]. The t-norm thus obtained is an extension of the t-norm on [0, 1], since if we identify each element a ∈ [0, 1] with the interval [a, a]:
T ([a, a], [b, b]) = [ T (a, b), T (a, b)] ∀a, b ∈ [0, 1]. Remark 2.1. The associated t-norm T is also called t-representable t-norms [5]. Definition 2.2 ([1]). For every [a, b], [c , d] ∈ S I , an interval-valued residuated implication R is defined by:
R([a, b], [c , d]) =
{[x, y ] ∈ S I | T ([a, b], [x, y ]) ≤ [c , d]}
where T is a t-norm on S I . In this article, we suppose that T is left-continuous t-representable t-norm on S I , R is residuated implication induced by left-continuous t-representable t-norm on S I . Lemma 2.1 ([11]). Suppose that R is an interval-valued residuated implication induced by a left-continuous interval-valued t-norm T on S I . Then
T ([x, y ], [x1 , y 1 ]) ≤ [x2 , y 2 ] ⇔ [x, y ] ≤ R([x1 , y 1 ], [x2 , y 2 ]). Lemma 2.2 ([1]). Given T the t-norm defined on S I associated to the t-norm T , every residuated implication between intervals R associated to the t-norm T has the form:
R([a, b], [c , d]) = [ R T (a, c ) ∧ R T (b, d), R T (b, d)] where R T is the residuated implication associated to the t-norm T on [0, 1]. Example 2 ([11]). (1) The interval-valued Łukasiewicz implication and the corresponding t-norm:
⎧ [(1 − a + c )∧(1 − b + d), ⎪ ⎪ ⎪ if ⎨ 1 − b + d], if R L ([a, b], [c , d]) = [1 − a + c , 1], ⎪ ⎪ 1 , 1 ], if [ ⎪ ⎩ if [1 − b + d, 1 − b + d], T L ([a, b], [c , d]) = [0∨(a + c − 1), 0∨(b + d − 1)]
a>c a>c a≤c a≤c
and and and and
b > d, b≤d, b≤d, b > d.
(2) The interval-valued Gödel implication and the corresponding t-norm:
⎧ ⎪ ⎪ [c , d], ⎨ [c , 1], RG ([a, b], [c , d]) = 1, 1], [ ⎪ ⎪ ⎩ [d, d],
if if if if
a>c a>c a≤c a≤c
and and and and
b > d, b≤d, b≤d, b > d.
TG ([a, b], [c , d]) = [a∧c , b∧d] Lemma 2.3 ([1]). Suppose that R is an interval-valued R-implication induced by a t-norm T on S I , for ∀[a, b], [c , d], [a1 , b1 ], [c 1 , d1 ] ∈ S I , then these properties of the interval-valued R-implication are true:
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(1) (2) (3) (4) (5) (6) (7)
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If [a, b] ≤ [a1 , b1 ], then R([a, b], [c , d]) ≥ R([a1 , b1 ], [c , d]); If [c , d] ≤ [c 1 , d1 ], then R([a, b], [c , d]) ≤ R([a, b], [c 1 , d1 ]); R([0, 0], [a, b]) = [1, 1]; R([a, b], [1, 1]) = [1, 1]; R([1, 1], [a, b]) = [a, b]; R([a, b], [a, b]) = [1, 1]; R([a, b], [c , d]) ≥ [c , d].
Definition 2.3 ([13]). For any [x1 , y 1 ], [x2 , y 2 ] ∈ S I , d([x1 , y 1 ], [x2 , y 2 ]) = max{|x1 − x2 |, | y 1 − y 2 |} is referred as Moore metric. Indeed, d fulfills the following conditions: (1) Positive definiteness. d([x1 , y 1 ], [x2 , y 2 ]) ≥ 0, d([x1 , y 1 ], [x2 , y 2 ]) = 0 if and only if [x1 , y 1 ] = [x2 , y 2 ]; (2) Symmetry. d([x1 , y 1 ], [x2 , y 2 ]) = d([x2 , y 2 ], [x1 , y 1 ]); (3) Triangle inequality. d([x1 , y 1 ], [x3 , y 3 ]) ≤ d([x1 , y 1 ], [x2 , y 2 ]) + d([x2 , y 2 ], [x3 , y 3 ]). Let X and Y be non-empty sets, S I ( X ) and S I (Y ) respectively denote interval-valued fuzzy subsets of non-empty sets X and Y , A (x), A ∗ (x) ∈ S I ( X ) ( A (x) denoted by [ A l (x), A r (x)]) and B ( y ), B ∗ ( y ) ∈ S I (Y ). Generally, they are assumed to be normalized, that is, there exists at least one x ∈ X such that A (x) = [1, 1]. Suppose that R is a residuated implication operator on S I , then we can define the quintuple implication principle based on interval-valued fuzzy inference for fuzzy modus ponens (I F M P ) and fuzzy modus tollens (I F M T ). Theorem 2.1 ([11]). Suppose that R is an interval-valued residuated implication induced by a left-continuous t-norm T on SI. Then the interval-valued R-type triple I solution B ∗ of I F M P is given by the following formula:
B ∗ ( y) =
T (R( A (x), B ( y )), A ∗ (x)), y ∈ Y .
x∈ X
Theorem 2.2 ([11]). Suppose that R is an interval-valued residuated implication induced by a left-continuous t-norm T on SI. Then the interval-valued R-type triple I solution A ∗ of I F M T is given by the following formula:
A ∗ (x) =
R(R( A (x), B ( y )), B ∗ ( y )), x ∈ X .
y ∈Y
3. Quintuple implication principle for fuzzy reasoning After analyzing the weakness of triple I method, Zhou et al. proposed the following principle. Quintuple Implication Principle for F M P ([25]): B ∗ is the smallest fuzzy subset on Y such that ( A (x) → B ( y )) → (( A ∗ (x) → A (x)) → ( A ∗ (x) → B ∗ ( y ))) is a tautology. That is, B ∗ is the smallest fuzzy subset on Y such that the following condition holds for every x in X and every y in Y :
( A (x) → B ( y )) → (( A ∗ (x) → A (x)) → ( A ∗ (x) → B ∗ ( y ))) = 1. Quintuple Implication Principle for F M T ([25]): A ∗ is the smallest fuzzy subset on X such that ( A (x) → B ( y )) → (( B ( y ) → B ∗ ( y )) → ( A (x) → A ∗ (x))) is a tautology. That is, A ∗ is the smallest fuzzy subset on X such that the following condition holds for every x in X and every y in Y :
( A (x) → B ( y )) → (( B ( y ) → B ∗ ( y )) → ( A (x) → A ∗ (x))) = 1. Theorem 3.1 ([25]). Suppose that T is a left-continuous t-norm, R is its residuum. Then the Q I P solutions of F M P and F M T are as follows:
( F M P − Q I P ) B ∗ ( y) =
T ( T ( R ( A (x), B ( y )), R ( A ∗ (x), A (x))), A ∗ (x)), y ∈ Y .
x∈ X
∗
( F M T − Q I P ) A (x) =
T ( T ( R ( A (x), B ( y )), R ( B ( y ), B ∗ ( y ))), A (x)), x ∈ X .
y ∈Y
4. Interval-valued quintuple implication principle of fuzzy reasoning In this section, we extend quintuple implication principle fuzzy inference to the case of interval-valued fuzzy sets. Quintuple Implication Principle for I F M P : B ∗ is the smallest fuzzy subset on S I (Y ) such that ( A (x) → B ( y )) → (( A ∗ (x) → A (x)) → ( A ∗ (x) → B ∗ ( y ))) is a tautology. That is, B ∗ is the smallest fuzzy subset on S I (Y ) such that the fol-
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lowing condition holds for every x in X and every y in Y :
( A (x) → B ( y )) → (( A ∗ (x) → A (x)) → ( A ∗ (x) → B ∗ ( y ))) = [1, 1]. Quintuple Implication Principle for I F M T : A ∗ is the smallest fuzzy subset on S I ( X ) such that ( A (x) → B ( y )) → (( B ( y ) → B ∗ ( y )) → ( A (x) → A ∗ (x))) is a tautology. That is, A ∗ is the smallest fuzzy subset on S I ( X ) such that the following condition holds for every x in X and every y in Y :
( A (x) → B ( y )) → (( B ( y ) → B ∗ ( y )) → ( A (x) → A ∗ (x))) = [1, 1]. Theorem 4.1. Suppose that R is an interval-valued residuated implication induced by a left-continuous interval-valued t-norm T on S I . Then the interval-valued R-type quintuple implication solution of I F M P and I F M T are as follows:
( I F M P − Q I P ) B ∗ ( y) =
T (T (R( A (x), B ( y )), R( A ∗ (x), A (x))), A ∗ (x)), y ∈ Y .
x∈ X
∗
( I F M T − Q I P ) A (x) =
T (T (R( A (x), B ( y )), R( B ( y ), B ∗ ( y ))), A (x)), x ∈ X .
y ∈Y
Proof. We only give the proof for I F M P . First, we shall prove:
R(R( A (x), B ( y )), R(R( A ∗ (x), A (x)), R( A ∗ (x), B ∗ ( y )))) = [1, 1] It follows from the expression of B ∗ ( y ), that T (T (R( A (x), B ( y )), R( A ∗ (x), A (x))), A ∗ (x)) ≤ B ∗ ( y ). Since R is an interval-valued R-implication induced by a left-continuous interval-valued t-norm T , we obtain R( A (x), B ( y )) ≤ R(R( A ∗ (x), A (x)), R( A ∗ (x), B ∗ ( y ))) by the residuation property. Therefore, R(R( A (x), B ( y )), R(R( A ∗ (x), A (x)), R( A ∗ (x), B ∗ ( y )))) = [1, 1]. Second, we shall show that B ∗ ( y ) is the smallest element. Suppose that C is an arbitrary fuzzy subset on Y such that ( A (x) → B ( y )) → (( A ∗ (x) → A (x)) → ( A ∗ (x) → C ( y ))) is a tautology. i.e.
R(R( A (x), B ( y )), R(R( A ∗ (x), A (x)), R( A ∗ (x), C ( y )))) = [1, 1] iff R( A (x), B ( y )) ≤ R(R( A ∗ (x), A (x)), R( A ∗ (x), C ( y ))) for all x; y iff T (T (R( A (x), B ( y )), R( A ∗ (x), A (x))), A ∗ (x)) ≤ C ( y ) for all x; y . It means that C ( y ) is an upper bound of the set {T (T (R( A (x), B ( y )), R( A ∗ (x), A (x))), A ∗ (x))}. Hence B ∗ ( y ) ≤ C ( y ). Therefore, B ∗ ( y ) is the interval-valued R-type quintuple implication solution of I F M P . 2 The proof for I F M T is similar. Remark 4.1. Interval-valued triple I algorithm can be interpreted as a simplified version of interval-valued quintuple implication algorithm for I F M P , that is [11]:
If R( A ∗ , A ) = [1, 1], then B ∗ ( y ) =
T (R( A (x), B ( y )), A ∗ (x)), x ∈ X .
x∈ X
Corollary 4.1. The RG -type quintuple implication solution of I F M P and I F M T are:
B ∗ ( y) =
{[ Al∗ (x)∧ Al (x)∧ B l ( y ), A r∗ (x)∧ A r (x)∧ B r ( y )]} =
x∈ X
∗
A (x) =
{ A ∗ (x)∧ A (x)∧ B ( y )}, y ∈ Y .
x∈ X
∗
∗
{[ B l ( y )∧ Al (x)∧ B l ( y ), B r ( y )∧ A r (x)∧ B r ( y )]} =
y ∈Y
{ B ∗ ( y )∧ A (x)∧ B ( y )}, x ∈ X .
y ∈Y
Corollary 4.2. The R L -type quintuple implication solution of I F M P and I F M T are:
B ∗ ( y) =
{[( Al∗ (x) + (R L )l ( A (x), B ( y )) + (R L )l ( A ∗ (x), A (x)) − 2) ∨ 0, ( A r∗ (x) + (R L )r ( A (x), B ( y ))
x∈ E y
+ (R L )r ( A ∗ (x), A (x)) − 2) ∨ 0]}, y ∈ Y , A ∗ (x) = {[( Al (x) + (R L )l ( A (x), B ( y )) + (R L )l ( B ( y ), B ∗ ( y )) − 2) ∨ 0, ( A r (x) x∈ E x
+ (R L )r ( A (x), B ( y )) + (R L )r ( B ( y ), B ∗ ( y )) − 2) ∨ 0]}, x ∈ X , E y = {x∈ X |[ Al∗ (x) + (R L )l ( A (x), B ( y )) − 1, A r∗ (x) + (R L )r ( A (x), B ( y )) − 1] > [0, 0]}, E x = {x∈ X |[ Al (x) + (R L )l ( A (x), B ( y )) − 1, A r (x) + (R L )r ( A (x), B ( y )) − 1] > [0, 0]}. where
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Example 3. For the dates of Example 1, we apply interval-valued quintuple implication principle to the I F M P problem shown in Table 1 and the I F M T problem shown in Table 2. For the two interval-valued residual implications (Gödel, Łukasiewicz), we have the same conclusion B ∗ ( y ) = {[0, 0], [0.5, 0.6], [0.5, 0.6]}, which, when compared with the solution of interval-valued triple I method, is much closer to the statement that “ y is large” and hence is in accordance with human thinking. For the two interval-valued residual implications (Gödel, Łukasiewicz), we have the same conclusion A ∗ (x) = {[0.5, 0.6], [0, 0], [0, 0]}, which, when compared with the solution of interval-valued triple I method, is much closer to the statement that “x is small” and hence is in accordance with human thinking. Definition 4.1 ([20]). A method of F M P is said recoverable if A ∗ = A implies B ∗ = B. Similarly, a method of F M T is recoverable if B ∗ = B implies A ∗ = A. We have the following result. Proposition 4.1. The Q I P method for I F M P (I F M T , resp.) is recoverable if A (B, resp.) is normal, where a interval-valued fuzzy set F on universe U is normal if there exists u ∈U such that F (u ) = [1, 1]. Proof. We only consider the Q I P solution of I F M P , the proof for Q I P solution of I F M T is similar. Suppose A ∗ = A and there exists an element x0 ∈ X such that A (x0 ) = [1, 1]. Then we have
B ( y) ≥ B ∗ ( y) =
T (T (R( A (x), B ( y )), R( A ∗ (x), A (x))), A ∗ (x))
x∈ X
≥ T (T (R( A (x0 ), B ( y )), R( A ∗ (x0 ), A (x0 ))), A ∗ (x0 )) = T (T (R([1, 1], B ( y )), [1, 1]), [1, 1]) = B ( y ) and therefore B ( y ) = B ∗ ( y ). This shows that the Q I P method of I F M P is recoverable.
2
5. Robustness of the interval-valued quintuple implication algorithms Li et al. [8] first extended compositional rule of inference to the case of interval-valued fuzzy set, and investigated the robustness of C R I method based on interval-valued. In this section, the robustness of interval-valued quintuple implication algorithms are studied by the same measure. Thus several related definitions and lemmas are quoted from [8]. Definition 5.1 ([8]). Let f be an n-tuple mapping from S I n to S I and [xn , yn ]) ∈ S I n , the ε sensitivity of f at point [x, y] is defined by
f ([x, y], ε ) =
ε ∈ [0, 1]. For arbitrary [x, y] = ([x1 , y 1 ], [x2 , y 2 ], . . . ,
{d( f ([x, y]), f ([x , y ]))|[x , y ] ∈ S I n and dn ([x, y], [x , y ]) ≤ ε },
where dn ([x, y], [x , y ]) = max{maxi | xi − x i |, maxi | y i − y i |}. Definition 5.2 ([8]). The maximum
f (ε ) =
ε sensitivity of f is defined as follows:
f ([x, y], ε ).
[x,y]∈ S I n
Definition 5.3 ([8]). Let A and A be two interval-valued fuzzy sets on S I . If A − A ∞ = d( A (x), A (x)) ≤ ε hold for all x ∈ X , then A is called the ε -perturbation of A denoted by A ∈ B ( A , ε ).
x∈ X
Definition 5.4. Let A , A , B , B , A ∗ and A ∗ be interval-valued fuzzy sets on S I . If A − A ∞ ≤ ε , B − B ∞ ≤ ε , A ∗ − A ∗ ∞ ≤ ε , and B ∗ and B ∗ are the interval-valued R-type quintuple implication solutions of I F M P ( A , B , A ∗ ) and I F M P ( A , B , A ∗ ) given by Theorem 4.1 respectively, the sensitivity of the interval-valued R-type quintuple implication solutions of I F M P denoted as B ∗ is defined as follows:
B ∗ = B ∗ − B ∗ ∞ =
d( B ∗ ( y ), B ∗ ( y )).
y ∈Y
Definition 5.5. Let A , A , B , B , B ∗ and B ∗ be interval-valued fuzzy sets on S I . If A − A ∞ ≤ ε , B − B ∞ ≤ ε , B ∗ − B ∗ ∞ ≤ ε , and A ∗ and A ∗ are the interval-valued R-type quintuple implication solutions of I F M T ( A , B , B ∗ ) and
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I F M T ( A , B , B ∗ ) given by Theorem 4.1 respectively, the sensitivity of the interval-valued R-type quintuple implication solutions of I F M T denoted as A ∗ is defined as follows:
A ∗ = A ∗ − A ∗ ∞ =
d( A ∗ (x), A ∗ (x)).
x∈ X
Lemma 5.1 ([8]). For a binary interval-valued fuzzy connective f :S I × S I → S I , we have (i) If f is any t-norm on S I , then
f (([x1 , y 1 ], [x2 , y 2 ]), ε ) = ( f l ([x1 , y 1 ], [x2 , y 2 ]) − f l ([(x1 − ε ) ∨ 0, ( y 1 − ε ) ∨ 0], [(x2 − ε ) ∨ 0, ( y 2 − ε ) ∨ 0])) ∨( f r ([x1 , y 1 ], [x2 , y 2 ]) − f r ([(x1 − ε ) ∨ 0, ( y 1 − ε ) ∨ 0], [(x2 − ε ) ∨ 0, ( y 2 − ε ) ∨ 0])) ∨( f l ([(x1 + ε ) ∧ 1, ( y 1 + ε ) ∧ 1], [(x2 + ε ) ∧ 1, ( y 2 + ε ) ∧ 1]) − f l ([x1 , y 1 ], [x2 , y 2 ])) ∨( f r ([(x1 + ε ) ∧ 1, ( y 1 + ε ) ∧ 1], [(x2 + ε ) ∧ 1, ( y 2 + ε ) ∧ 1]) − f r ([x1 , y 1 ], [x2 , y 2 ])) (ii) If f is any R-implication on S I , then
f (([x1 , y 1 ], [x2 , y 2 ]), ε ) = ( f l ([x1 , y 1 ], [x2 , y 2 ]) − f l ([(x1 + ε ) ∧ 1, ( y 1 + ε ) ∧ 1], [(x2 − ε ) ∨ 0, ( y 2 − ε ) ∨ 0])) ∨( f r ([x1 , y 1 ], [x2 , y 2 ]) − f r ([(x1 + ε ) ∧ 1, ( y 1 + ε ) ∧ 1], [(x2 − ε ) ∨ 0, ( y 2 − ε ) ∨ 0])) ∨( f l ([(x1 − ε ) ∨ 0, ( y 1 − ε ) ∨ 0], [(x2 + ε ) ∧ 1, ( y 2 + ε ) ∧ 1]) − f l ([x1 , y 1 ], [x2 , y 2 ])) ∨( f r ([(x1 − ε ) ∨ 0, ( y 1 − ε ) ∨ 0], [(x2 + ε ) ∧ 1, ( y 2 + ε ) ∧ 1]) − f r ([x1 , y 1 ], [x2 , y 2 ])) Lemma 5.2 ([8]). The Gödel t-norm on S I is the most robust t-norm, and TG (ε ) = ε . Lemma 5.3 ([8]). The Łukasiewicz implication on S I is the most robust R-implication, and RL (ε ) = 2ε ∧ 1. Lemma 5.4 ([8]). Suppose that A − A ∞ ≤ ε , B − B ∞ ≤ ε , and R is an interval-valued residuated implication induced by a left-continuous t-norm T on SI, then
d(R( A (x), B ( y )), R( A (x), B ( y ))) ≤ R (ε ). Lemma 5.5. Suppose that A − A ∞ ≤ ε , B − B ∞ ≤ ε , A ∗ − A ∗ ∞ ≤ ε and R is an interval-valued residuated implication induced by a left-continuous t-norm T on SI, then
d(T (R( A (x), B ( y )), R( A ∗ (x), A (x))), T (R( A (x), B ( y )), R( A ∗ (x), A (x)))) ≤ T (R (ε )). d(T (R( A (x), B ( y )), R( B ( y ), B ∗ ( y ))), T (R( A (x), B ( y )), R( B ( y ), B ∗ ( y )))) ≤ T (R (ε )). Proof. We only prove the first formula, the second can be proved in the same way. Since A − A ∞ ≤ ε and B − B ∞ ≤ ε . Then we have:
d(R( A (x), B ( y )), R( A (x), B ( y ))) ≤ R (ε ) (By Lemma 5.4) Similarly, d(R( A ∗ (x), A (x)), R( A ∗ (x), A (x))) ≤ R (ε ). Thus
d(T (R( A (x), B ( y )), R( A ∗ (x), A (x))), T (R( A (x), B ( y )), R( A ∗ (x), A (x)))) ≤ T (R (ε )).
2
Theorem 5.1. Suppose that A − A ∞ ≤ ε , B − B ∞ ≤ ε , A ∗ − A ∗ ∞ ≤ ε , and B ∗ and B ∗ are the interval-valued R-type quintuple implication solutions of I F M P ( A , B , A ∗ ) and I F M P ( A , B , A ∗ ) given by Theorem 4.1 respectively, then the sensitivity of the interval-valued fuzzy inference I F M P : B ∗ (ε ) = B ∗ − B ∗ ∞ ≤ T (T (R (ε ))).
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M. Luo, X. Zhou / International Journal of Approximate Reasoning 84 (2017) 23–32
Proof.
B ∗ (ε ) = B ∗ − B ∗ ∞ = d( B ∗ ( y ), B ∗ ( y )) y ∈Y
=
d(
y ∈Y
T (T (R( A (x), B ( y )), R( A ∗ (x), A (x))), A ∗ (x)),
x∈ X
T (T (R( A (x), B ( y )), R( A ∗ (x), A (x))), A ∗ (x)))
x∈ X
≤
d(T (T (R( A (x), B ( y )), R( A ∗ (x), A (x))), A ∗ (x)),
y ∈Y x∈ X
T (T (R( A (x), B ( y )), R( A ∗ (x), A (x))), A ∗ (x)))
≤ T (T (R (ε )) ∨ ε ) (by Lemma 5.5) By Lemmas 5.2, 5.3, we can obtain that R (ε ) (2ε ∧ 1) > ε , therefore T (R (ε )) ≥ R (ε ) > ε , i.e. T (R (ε )) ∨ ε =
T (R (ε )). Thus
B ∗ (ε ) = B ∗ − B ∗ ∞ ≤ T (T (R (ε ))) 2 Using Theorem 5.1, we can obtain these following corollaries: Corollary 5.1. If T is interval-valued Minimum t-norm on S I , R is its residuum, then B ∗ (ε ) ≤ R (ε ). Proof. By Theorem 5.1, we have B ∗ (ε )≤T (T (R (ε ))) = ∧ (∧ (R (ε ))), because T is Minimum t-norm. We can use the fact from Lemma 5.2 that ∧ (ε ) = ε , then ∧ (∧ (R (ε ))) = R (ε ). Therefore, B ∗ (ε )≤R (ε ). 2 Corollary 5.2. If T is interval-valued Łukasiewicz t-norm on S I , R is its residuum, then B ∗ (ε ) ≤ ε + 2R (ε ). Proof.
B ∗ (ε ) = B ∗ − B ∗ ∞ = =
d( B ∗ ( y ), B ∗ ( y ))
y ∈Y
∗
{|( Al (x) + (R L )l ( A (x), B ( y )) + (R L )l ( A ∗ (x), A (x)) − 2) ∨ 0 − ( Al ∗ (x) + (R L )l ( A (x), B ( y )) +
y ∈Y x∈ X
(R L )l ( A ∗ (x), A (x)) − 2) ∨ 0| ∨ |( A r∗ (x) + (R L )r ( A (x), B ( y )) + (R L )r ( A ∗ (x), A (x)) − 2) ∨ 0 − ( A r ∗ (x) + (R L )r ( A (x), B ( y )) + (R L )r ( A ∗ (x), A (x)) − 2) ∨ 0|} ≤ ε + 2R (ε ) 2 Theorem 5.2. Suppose that A − A ∞ ≤ ε , B − B ∞ ≤ ε , B ∗ − B ∗ ∞ ≤ ε , and A ∗ and A ∗ are the interval-valued R-type quintuple implication solutions of I F M T ( A , B , B ∗ ) and I F M T ( A , B , B ∗ ) given by Theorem 4.1 respectively, then the sensitivity of the interval-valued fuzzy inference I F M T : A ∗ (ε ) = A ∗ − A ∗ ∞ ≤ T (T (R (ε ))). Proof.
A ∗ (ε ) = A ∗ − A ∗ ∞ = d( A ∗ (x), A ∗ (x)) x∈ X
=
x∈ X
y ∈Y
d(
T (T (R( A (x), B ( y )), R( B ( y ), B ∗ ( y ))), A (x)),
y ∈Y
T (T (R( A (x), B ( y )), R( B ( y ), B ∗ ( y ))), A (x)))
M. Luo, X. Zhou / International Journal of Approximate Reasoning 84 (2017) 23–32
≤
31
d(T (T (R( A (x), B ( y )), R( B ( y ), B ∗ ( y ))), A (x)),
x∈ X y ∈Y
T (T (R( A (x), B ( y )), R( B ( y ), B ∗ ( y ))), A (x)))
≤ T (T (R (ε )) ∨ ε ) (by Lemma 5.5) By Lemmas 5.2, 5.3, we can obtain that R (ε ) (2ε ∧ 1) > ε , therefore T (R (ε )) ≥ R (ε ) > ε , i.e. T (R (ε )) ∨ ε =
T (R (ε )). Thus
A ∗ (ε ) = A ∗ − A ∗ ∞ ≤ T (T (R (ε ))) 2 Using Theorem 5.2, we can obtain these following corollaries: Corollary 5.3. If T is interval-valued Minimum t-norm on S I , R is its residuum, then A ∗ (ε ) ≤ R (ε ). Proof. By Theorem 5.2, we have A ∗ (ε )≤T (T (R (ε ))) = ∧ (∧ (R (ε ))), because T is Minimum t-norm. We can use the fact from Lemma 5.2 that ∧ (ε ) = ε , then ∧ (∧ (R (ε ))) = R (ε ). Therefore, A ∗ (ε )≤R (ε ). 2 Corollary 5.4. If T is interval-valued Łukasiewicz t-norm on S I , R is its residuum, then A ∗ (ε ) ≤ ε + 2R (ε ). Proof.
A ∗ (ε ) = A ∗ − A ∗ ∞ = =
d( A ∗ (x), A ∗ (x))
x∈ X
{|( Al (x) + (R L )l ( A (x), B ( y )) + (R L )l ( B ( y ), B ∗ ( y )) − 2) ∨ 0 − ( Al (x) + (R L )l ( A (x), B ( y )) +
x∈ X y ∈Y
(R L )l ( B ( y ), B ∗ ( y )) − 2) ∨ 0| ∨ |( A r (x) + (R L )r ( A (x), B ( y )) + (R L )r ( B ( y ), B ∗ ( y )) − 2) ∨ 0 − ( A r (x) + (R L )r ( A (x), B ( y )) + (R L )r ( B ∗ ( y ), B ( y )) − 2) ∨ 0|} ≤ ε + 2R (ε ) 2 6. Conclusions In this paper, we have combined quintuple implication principle and interval-valued fuzzy set, proposed the quintuple implication principle based on interval-valued fuzzy inference for fuzzy modus ponens and fuzzy modus tollens. Unlike interval-valued triple I method, we take into account explicitly the similarity between A ∗ and A (or B ∗ and B) when computing the solution to I F M P (or I F M T ). We give the corresponding interval-valued R-type quintuple implication solutions and investigate the robustness of quintuple implication algorithms based on interval-valued fuzzy inference. Finally, the sensitivity of quintuple implication algorithms based on two important interval-valued residuated implication operators is given. Acknowledgements The authors would like to thank Editor-in-Chief Thierry Denoeux, area editor and anonymous reviewers for their valuable comments and suggestions. References [1] C. Alcalde, A. Burusco, R. Fuentes-Gonzalez, A constructive method for the definition of interval-valued fuzzy implication operators, Fuzzy Sets Syst. 153 (2005) 211–227. [2] O. Castillo, P. Melin, A review on interval type-2 fuzzy logic applications in intelligent control, Inf. Sci. 279 (2014) 615–631. [3] S.S. Dai, D.W. Pei, D.H. Guo, Robustness analysis of full implication inference method, Int. J. Approx. Reason. 54 (2013) 653–666. [4] B.A. Davey, H.A. Priestley, Introduction to Lattices and Order, Cambridge University Press, Cambridge, 1990. [5] G. Deschrijver, C. Cornelis, E.E. Kerre, On the representation of intuitionistic fuzzy t-norms and t-conorms, IEEE Trans. Fuzzy Syst. 12 (1) (2004) 45–61. [6] M.Y. Hsiao, T.S. Li, J.-Z. Lee, C.-H. Chao, S.-H. Tsai, Design of interval type-2 fuzzy sliding-mode controller, Inf. Sci. 178 (6) (2008) 1696–1716. [7] S. Jenei, A more efficient method for defining fuzzy connectives, Fuzzy Sets Syst. 90 (1997) 25–35. [8] D.Ch. Li, Y.M. Li, Y.J. Xie, Robustness of interval-valued fuzzy inference, Inf. Sci. 181 (2011) 4754–4764. [9] H.W. Liu, G.J. Wang, Continuity of triple I methods based on several implications, Comput. Math. Appl. 56 (2008) 2079–2087. [10] M.X. Luo, N. Yao, Triple I algorithms based on Schweizer–Sklar operators in fuzzy reasoning, Int. J. Approx. Reason. 54 (2013) 640–652. [11] M.X. Luo, K. Zhang, Robustness of full implication algorithms based on interval-valued fuzzy inference, Int. J. Approx. Reason. 62 (2015) 61–72.
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M.X. Luo, X.L. Zhou, Robustness of reverse triple I algorithms based on interval-valued fuzzy inference, Int. J. Approx. Reason. 66 (2015) 16–26. R.E. Moore, R.B. Kearfott, M.J. Cloud, Introduction to Interval Analysis, SIAM, Philadelphia, 2009. D.W. Pei, The full implication triple I algorithms and their consistency in fuzzy reasoning (in Chinese), J. Math. Res. Expo. 24 (2) (2004) 359–368. D.W. Pei, Unified full implication algorithms of fuzzy reasoning, Inf. Sci. 178 (2008) 520–530. D.W. Pei, On the strict logic foundation of fuzzy reasoning, Soft Comput. 8 (8) (2004) 539–545. D.W. Pei, Formalization of implication based fuzzy reasoning method, Int. J. Approx. Reason. 53 (2012) 837–846. B. Van Gasse, C. Cornelis, G. Deschrijver, E.E. Kerre, Triangle algebras: a formal logic approach to interval-valued residuated lattices, Fuzzy Sets Syst. 159 (2008) 1042–1060. B. Van Gasse, C. Cornelis, G. Deschrijver, E.E. Kerre, The standard completeness of interval-valued monoidal t-norm based logic, Inf. Sci. 189 (2012) 63–76. G.J. Wang, On the logic foundation of fuzzy reasoning, Inf. Sci. 117 (1999) 47–88. G.J. Wang, The full implication triple I method of fuzzy reasoning (in Chinese), Sci. China Ser. E 29 (1999) 43–53. G.J. Wang, L. Fu, Unified forms of triple I method, Comput. Math. Appl. 49 (2005) 923–932. L.A. Zadeh, Outline of a new approach to the analysis of complex systems and decision processes, IEEE Trans. Syst. Man Cybern. 3 (1973) 28–44. L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning-II, Inf. Sci. 8 (1975) 301–357. B.K. Zhou, G.Q. Xu, S.J. Li, The quintuple implication principle of fuzzy reasoning, Inf. Sci. 297 (2015) 202–215.
Contents lists available at ScienceDirect
International Journal of Approximate Reasoning www.elsevier.com/locate/ijar
Interval-valued quintuple implication principle of fuzzy reasoning ✩ Minxia Luo ∗ , Xiaoling Zhou Department of Mathematics, China Jiliang University, Hangzhou 310018, PR China
a r t i c l e
i n f o
Article history: Received 20 April 2016 Received in revised form 7 January 2017 Accepted 18 January 2017 Available online 3 February 2017 Keywords: Interval-valued fuzzy set Interval-valued fuzzy inference Quintuple implication algorithm Robustness
a b s t r a c t In this paper, we propose the quintuple implication principle based on interval-valued fuzzy inference for fuzzy modus ponens and fuzzy modus tollens, which explicitly use the closeness between A ∗ and A (or B ∗ and B) in the process of fuzzy reasoning. We first give the corresponding interval-valued R-type quintuple implication solutions and then investigate the robustness of quintuple implication algorithms based on interval-valued fuzzy inference. The sensitivity of quintuple implication algorithms based on two important interval-valued residuated implication operators is given. © 2017 Elsevier Inc. All rights reserved.
1. Introduction Since the method of Compositional Rule of Inference (C R I ) was put forward in 1973 by Zadeh [23], various methods of fuzzy reasoning have been proposed [10,14,15,21,22,25]. As an alternative for C R I method, Wang [21] proposed triple I method with full inference rule that utilized the implication operator in every step of the reasoning. The unified triple I algorithms based on regular implications and normal implications have been established by Wang and Fu [22]. For all residuated implications induced by left-continuous t-norms, unified full implication triple I algorithms of fuzzy reasoning are constructed by Pei [14,15]. Pei [16,17] conducted a detailed research into the triple I algorithms based on the monoidal t-norms basic logical system MTL setting a sound logic foundation. Luo and Yao [10] investigated triple I algorithms by the combination of Schweizer–Sklar operators and triple I principles for fuzzy reasoning. Liu and Wang [9] discussed continuity problems of triple I methods based on several implications. Dai et al. [3] studied the robustness of full implication inference methods. In both CRI and triple I method, the closeness of A and A ∗ (or B and B ∗ ) is not explicitly used in the process of calculating the consequence, which sometimes makes the computed approximation useless or misleading. Zhou et al. [25] proposed the Quintuple Implication Principle ( Q I P ), which characterizes the approximation A ∗ and A (or B ∗ and B) in the process of fuzzy reasoning. The method efficiently improves triple I method. It may be suitable to be used instead of Wang’s triple I method in some circumstances. Its basic idea is as follows: For known A ∈ F ( X ), B ∈ F (Y ), and A ∗ ∈ F ( X ) (or B ∗ ∈ F (Y )), seek the optimal B ∗ ∈ F (Y ) (or A ∗ ∈ F ( X )) such that
( A (x) → B ( y )) → (( A ∗ (x) → A (x)) → ( A ∗ (x) → B ∗ ( y ))) (or ( A (x) → B ( y )) → (( B ( y ) → B ∗ ( y )) → ( A (x) → A ∗ (x)))) ✩
*
This work is supported by the National Natural Science Foundation of China (No. 61273018). Corresponding author at: School of sciences, China Jiliang University, Hangzhou 310018, PR China E-mail address: [email protected] (M. Luo).
http://dx.doi.org/10.1016/j.ijar.2017.01.010 0888-613X/© 2017 Elsevier Inc. All rights reserved.
24
M. Luo, X. Zhou / International Journal of Approximate Reasoning 84 (2017) 23–32
takes the value as large as possible for any x ∈ X and y ∈ Y , where F ( X ) and F (Y ) denote the collections consisting of all fuzzy subsets of X and Y respectively. However, when vague terms appearing in the production rules of a rule-based approximate reasoning system are represented by interval-valued fuzzy sets (I V F S) rather than ordinary fuzzy sets, the approximate inference based on I V F S will become more valuable and more flexible than the general fuzzy reasoning. Recently, it has achieved great attention from many researchers in the fields of interval-valued fuzzy reasoning. In 1975, Zadeh [24] first proposed an approximate reasoning method based on interval-valued fuzzy set by using composition rule. In [18], B. Van Gasse et al. introduced interval-valued monoidal logic (I V M L), and proved that I V M L is sound and complete w.r.t. triangle algebras. The standard completeness of interval-valued monoidal t-norm based logic was proved by B. Van Gasse et al. [19]. In [1], Alcalde et al. gave a constructive method for the definition of interval-valued fuzzy implication operators. Due to the advantages of interval-valued fuzzy set, in recent years, researchers have been investigating the properties as well as the numerous applications of interval-valued fuzzy reasoning in various fields such as neural networks, control systems and so on [2,6]. Li et al. [8] first extended compositional rule of inference to the case of interval-valued fuzzy set, and discussed the robustness of interval-valued fuzzy reasoning. Luo et al. [11,12] extended full implication triple I algorithms and reverse triple I algorithms to the case of interval-valued fuzzy set, and investigated the robustness of full implication algorithms and reverse triple I algorithms based on interval-valued fuzzy inference. In aforementioned algorithms, however, the closeness between the interval-valued fuzzy set A ∗ and A (or B ∗ and B) is not explicitly used in the process of interval-valued fuzzy inference, which sometimes makes the computed approximation useless or misleading. Example 1. Let the universe be X = Y = {1, 2, 3}. Suppose small, medium and large are three fuzzy sets on S I ( X ), which are defined below:
[small] = [0.51,0.6] + [02,0] + [03,0] , [large ] = [01,0] + [0.52,0.6] + [13,1] , [medium] = [0.51,0.6] + [12,1] + [03,0] . For simplicity, we denote
[small] = {[0.5, 0.6], [0, 0], [0, 0]}, [large ] = {[0, 0], [0.5, 0.6], [1, 1]}, [medium] = {[0.5, 0.6], [1, 1], [0, 0]}. The problem of I F M P is stated as in Table 1. Let A (x) denote “x is small”, B ( y ) denote “ y is large” and A ∗ (x) denote “x is medium.” The task is then to compute B ∗ . For interval-valued Gödel and Łukasiewicz implications, the conclusion is B ∗ ( y ) = {[1, 1], [1, 1], [1, 1]}, i.e. the only information we can obtain from the triple I method is the trivial tautology. The problem of I F M T is stated as in Table 2. Let A (x) denote “x is small”, B ( y ) denote “ y is large” and B ∗ ( y ) denote “ y is medium.” The task is then to compute A ∗ . For interval-valued Gödel and Łukasiewicz implications, the conclusion is A ∗ (x) = {[0, 0], [0, 0], [0, 0]}. Note that A ∗ (x) = {[0, 0], [0, 0], [0, 0]} represents the contradiction and this conclusion is not in accordance with human thinking. Table 1 An example of IFMP. Rule Premise
IF x is small, then y is large x is medium
Calculate
y is?
Table 2 An example of IFMT. Rule Premise
IF x is small, then y is large y is medium
Calculate
x is?
After analyzing the weakness of triple I method based on interval-valued fuzzy inference, in this paper, we combine quintuple implication principle and interval-valued fuzzy set, propose the quintuple implication principle based on intervalvalued fuzzy inference for fuzzy modus ponens and fuzzy modus tollens, which explicitly use the closeness between the interval-valued fuzzy set A ∗ and A (or B ∗ and B) in the process of fuzzy reasoning. The reminder of this paper is organized as follows. In Section 2, we review some basic definitions and properties related to this paper. In Section 3, we briefly recall the quintuple implication principle based on general fuzzy sets. In Section 4, we introduce the quintuple implication principle based on interval-valued fuzzy inference and interpret it under two wellknown implication operators. In Section 5, we investigate the robustness of the proposed algorithms with respect to I F M P and I F M T models, a series of corollaries are proved. The last section concludes the paper.
M. Luo, X. Zhou / International Journal of Approximate Reasoning 84 (2017) 23–32
25
2. Preliminaries Let S I = {[x, y ]|x ≤ y , x, y ∈ [0, 1]}. An ordering on S I as [a, b] ≤ [c , d] if a ≤ c and b ≤ d is called component-wise order or Kulisch–Miranker order [4]. ‘∧’ is defined as [a, b] ∧ [c , d] = [a ∧ c , b ∧ d]. ‘∨’ is defined as [a, b] ∨ [c , d] = [a ∨ c , b ∨ d]. It is easy to verify that the ordering just defined is a partial ordering on S I . Furthermore, take [a, b] ∧ [c , d] = [a, b] iff [a, b] ≤ [c , d] and [a, b] ∨ [c , d] = [c , d] iff [a, b] ≤ [c , d]. We can verify that the algebraic structure ( S I , ∧, ∨, [0, 0], [1, 1]) is a complete, bounded and distributive lattice. First, we review some basic definitions and theorems related to this paper. Definition 2.1 ([7]). Let T be a t-norm defined on the interval [0, 1], the associated t-norm on S I is defined as follows:
T : S I × S I → S I, where
T ([a, b], [c , d]) = [ T (a, c ), T (b, d)]. The associated t-norm T on S I is called left-continuous, if T is left-continuous t-norm on the interval [0, 1]. The t-norm thus obtained is an extension of the t-norm on [0, 1], since if we identify each element a ∈ [0, 1] with the interval [a, a]:
T ([a, a], [b, b]) = [ T (a, b), T (a, b)] ∀a, b ∈ [0, 1]. Remark 2.1. The associated t-norm T is also called t-representable t-norms [5]. Definition 2.2 ([1]). For every [a, b], [c , d] ∈ S I , an interval-valued residuated implication R is defined by:
R([a, b], [c , d]) =
{[x, y ] ∈ S I | T ([a, b], [x, y ]) ≤ [c , d]}
where T is a t-norm on S I . In this article, we suppose that T is left-continuous t-representable t-norm on S I , R is residuated implication induced by left-continuous t-representable t-norm on S I . Lemma 2.1 ([11]). Suppose that R is an interval-valued residuated implication induced by a left-continuous interval-valued t-norm T on S I . Then
T ([x, y ], [x1 , y 1 ]) ≤ [x2 , y 2 ] ⇔ [x, y ] ≤ R([x1 , y 1 ], [x2 , y 2 ]). Lemma 2.2 ([1]). Given T the t-norm defined on S I associated to the t-norm T , every residuated implication between intervals R associated to the t-norm T has the form:
R([a, b], [c , d]) = [ R T (a, c ) ∧ R T (b, d), R T (b, d)] where R T is the residuated implication associated to the t-norm T on [0, 1]. Example 2 ([11]). (1) The interval-valued Łukasiewicz implication and the corresponding t-norm:
⎧ [(1 − a + c )∧(1 − b + d), ⎪ ⎪ ⎪ if ⎨ 1 − b + d], if R L ([a, b], [c , d]) = [1 − a + c , 1], ⎪ ⎪ 1 , 1 ], if [ ⎪ ⎩ if [1 − b + d, 1 − b + d], T L ([a, b], [c , d]) = [0∨(a + c − 1), 0∨(b + d − 1)]
a>c a>c a≤c a≤c
and and and and
b > d, b≤d, b≤d, b > d.
(2) The interval-valued Gödel implication and the corresponding t-norm:
⎧ ⎪ ⎪ [c , d], ⎨ [c , 1], RG ([a, b], [c , d]) = 1, 1], [ ⎪ ⎪ ⎩ [d, d],
if if if if
a>c a>c a≤c a≤c
and and and and
b > d, b≤d, b≤d, b > d.
TG ([a, b], [c , d]) = [a∧c , b∧d] Lemma 2.3 ([1]). Suppose that R is an interval-valued R-implication induced by a t-norm T on S I , for ∀[a, b], [c , d], [a1 , b1 ], [c 1 , d1 ] ∈ S I , then these properties of the interval-valued R-implication are true:
26
(1) (2) (3) (4) (5) (6) (7)
M. Luo, X. Zhou / International Journal of Approximate Reasoning 84 (2017) 23–32
If [a, b] ≤ [a1 , b1 ], then R([a, b], [c , d]) ≥ R([a1 , b1 ], [c , d]); If [c , d] ≤ [c 1 , d1 ], then R([a, b], [c , d]) ≤ R([a, b], [c 1 , d1 ]); R([0, 0], [a, b]) = [1, 1]; R([a, b], [1, 1]) = [1, 1]; R([1, 1], [a, b]) = [a, b]; R([a, b], [a, b]) = [1, 1]; R([a, b], [c , d]) ≥ [c , d].
Definition 2.3 ([13]). For any [x1 , y 1 ], [x2 , y 2 ] ∈ S I , d([x1 , y 1 ], [x2 , y 2 ]) = max{|x1 − x2 |, | y 1 − y 2 |} is referred as Moore metric. Indeed, d fulfills the following conditions: (1) Positive definiteness. d([x1 , y 1 ], [x2 , y 2 ]) ≥ 0, d([x1 , y 1 ], [x2 , y 2 ]) = 0 if and only if [x1 , y 1 ] = [x2 , y 2 ]; (2) Symmetry. d([x1 , y 1 ], [x2 , y 2 ]) = d([x2 , y 2 ], [x1 , y 1 ]); (3) Triangle inequality. d([x1 , y 1 ], [x3 , y 3 ]) ≤ d([x1 , y 1 ], [x2 , y 2 ]) + d([x2 , y 2 ], [x3 , y 3 ]). Let X and Y be non-empty sets, S I ( X ) and S I (Y ) respectively denote interval-valued fuzzy subsets of non-empty sets X and Y , A (x), A ∗ (x) ∈ S I ( X ) ( A (x) denoted by [ A l (x), A r (x)]) and B ( y ), B ∗ ( y ) ∈ S I (Y ). Generally, they are assumed to be normalized, that is, there exists at least one x ∈ X such that A (x) = [1, 1]. Suppose that R is a residuated implication operator on S I , then we can define the quintuple implication principle based on interval-valued fuzzy inference for fuzzy modus ponens (I F M P ) and fuzzy modus tollens (I F M T ). Theorem 2.1 ([11]). Suppose that R is an interval-valued residuated implication induced by a left-continuous t-norm T on SI. Then the interval-valued R-type triple I solution B ∗ of I F M P is given by the following formula:
B ∗ ( y) =
T (R( A (x), B ( y )), A ∗ (x)), y ∈ Y .
x∈ X
Theorem 2.2 ([11]). Suppose that R is an interval-valued residuated implication induced by a left-continuous t-norm T on SI. Then the interval-valued R-type triple I solution A ∗ of I F M T is given by the following formula:
A ∗ (x) =
R(R( A (x), B ( y )), B ∗ ( y )), x ∈ X .
y ∈Y
3. Quintuple implication principle for fuzzy reasoning After analyzing the weakness of triple I method, Zhou et al. proposed the following principle. Quintuple Implication Principle for F M P ([25]): B ∗ is the smallest fuzzy subset on Y such that ( A (x) → B ( y )) → (( A ∗ (x) → A (x)) → ( A ∗ (x) → B ∗ ( y ))) is a tautology. That is, B ∗ is the smallest fuzzy subset on Y such that the following condition holds for every x in X and every y in Y :
( A (x) → B ( y )) → (( A ∗ (x) → A (x)) → ( A ∗ (x) → B ∗ ( y ))) = 1. Quintuple Implication Principle for F M T ([25]): A ∗ is the smallest fuzzy subset on X such that ( A (x) → B ( y )) → (( B ( y ) → B ∗ ( y )) → ( A (x) → A ∗ (x))) is a tautology. That is, A ∗ is the smallest fuzzy subset on X such that the following condition holds for every x in X and every y in Y :
( A (x) → B ( y )) → (( B ( y ) → B ∗ ( y )) → ( A (x) → A ∗ (x))) = 1. Theorem 3.1 ([25]). Suppose that T is a left-continuous t-norm, R is its residuum. Then the Q I P solutions of F M P and F M T are as follows:
( F M P − Q I P ) B ∗ ( y) =
T ( T ( R ( A (x), B ( y )), R ( A ∗ (x), A (x))), A ∗ (x)), y ∈ Y .
x∈ X
∗
( F M T − Q I P ) A (x) =
T ( T ( R ( A (x), B ( y )), R ( B ( y ), B ∗ ( y ))), A (x)), x ∈ X .
y ∈Y
4. Interval-valued quintuple implication principle of fuzzy reasoning In this section, we extend quintuple implication principle fuzzy inference to the case of interval-valued fuzzy sets. Quintuple Implication Principle for I F M P : B ∗ is the smallest fuzzy subset on S I (Y ) such that ( A (x) → B ( y )) → (( A ∗ (x) → A (x)) → ( A ∗ (x) → B ∗ ( y ))) is a tautology. That is, B ∗ is the smallest fuzzy subset on S I (Y ) such that the fol-
M. Luo, X. Zhou / International Journal of Approximate Reasoning 84 (2017) 23–32
27
lowing condition holds for every x in X and every y in Y :
( A (x) → B ( y )) → (( A ∗ (x) → A (x)) → ( A ∗ (x) → B ∗ ( y ))) = [1, 1]. Quintuple Implication Principle for I F M T : A ∗ is the smallest fuzzy subset on S I ( X ) such that ( A (x) → B ( y )) → (( B ( y ) → B ∗ ( y )) → ( A (x) → A ∗ (x))) is a tautology. That is, A ∗ is the smallest fuzzy subset on S I ( X ) such that the following condition holds for every x in X and every y in Y :
( A (x) → B ( y )) → (( B ( y ) → B ∗ ( y )) → ( A (x) → A ∗ (x))) = [1, 1]. Theorem 4.1. Suppose that R is an interval-valued residuated implication induced by a left-continuous interval-valued t-norm T on S I . Then the interval-valued R-type quintuple implication solution of I F M P and I F M T are as follows:
( I F M P − Q I P ) B ∗ ( y) =
T (T (R( A (x), B ( y )), R( A ∗ (x), A (x))), A ∗ (x)), y ∈ Y .
x∈ X
∗
( I F M T − Q I P ) A (x) =
T (T (R( A (x), B ( y )), R( B ( y ), B ∗ ( y ))), A (x)), x ∈ X .
y ∈Y
Proof. We only give the proof for I F M P . First, we shall prove:
R(R( A (x), B ( y )), R(R( A ∗ (x), A (x)), R( A ∗ (x), B ∗ ( y )))) = [1, 1] It follows from the expression of B ∗ ( y ), that T (T (R( A (x), B ( y )), R( A ∗ (x), A (x))), A ∗ (x)) ≤ B ∗ ( y ). Since R is an interval-valued R-implication induced by a left-continuous interval-valued t-norm T , we obtain R( A (x), B ( y )) ≤ R(R( A ∗ (x), A (x)), R( A ∗ (x), B ∗ ( y ))) by the residuation property. Therefore, R(R( A (x), B ( y )), R(R( A ∗ (x), A (x)), R( A ∗ (x), B ∗ ( y )))) = [1, 1]. Second, we shall show that B ∗ ( y ) is the smallest element. Suppose that C is an arbitrary fuzzy subset on Y such that ( A (x) → B ( y )) → (( A ∗ (x) → A (x)) → ( A ∗ (x) → C ( y ))) is a tautology. i.e.
R(R( A (x), B ( y )), R(R( A ∗ (x), A (x)), R( A ∗ (x), C ( y )))) = [1, 1] iff R( A (x), B ( y )) ≤ R(R( A ∗ (x), A (x)), R( A ∗ (x), C ( y ))) for all x; y iff T (T (R( A (x), B ( y )), R( A ∗ (x), A (x))), A ∗ (x)) ≤ C ( y ) for all x; y . It means that C ( y ) is an upper bound of the set {T (T (R( A (x), B ( y )), R( A ∗ (x), A (x))), A ∗ (x))}. Hence B ∗ ( y ) ≤ C ( y ). Therefore, B ∗ ( y ) is the interval-valued R-type quintuple implication solution of I F M P . 2 The proof for I F M T is similar. Remark 4.1. Interval-valued triple I algorithm can be interpreted as a simplified version of interval-valued quintuple implication algorithm for I F M P , that is [11]:
If R( A ∗ , A ) = [1, 1], then B ∗ ( y ) =
T (R( A (x), B ( y )), A ∗ (x)), x ∈ X .
x∈ X
Corollary 4.1. The RG -type quintuple implication solution of I F M P and I F M T are:
B ∗ ( y) =
{[ Al∗ (x)∧ Al (x)∧ B l ( y ), A r∗ (x)∧ A r (x)∧ B r ( y )]} =
x∈ X
∗
A (x) =
{ A ∗ (x)∧ A (x)∧ B ( y )}, y ∈ Y .
x∈ X
∗
∗
{[ B l ( y )∧ Al (x)∧ B l ( y ), B r ( y )∧ A r (x)∧ B r ( y )]} =
y ∈Y
{ B ∗ ( y )∧ A (x)∧ B ( y )}, x ∈ X .
y ∈Y
Corollary 4.2. The R L -type quintuple implication solution of I F M P and I F M T are:
B ∗ ( y) =
{[( Al∗ (x) + (R L )l ( A (x), B ( y )) + (R L )l ( A ∗ (x), A (x)) − 2) ∨ 0, ( A r∗ (x) + (R L )r ( A (x), B ( y ))
x∈ E y
+ (R L )r ( A ∗ (x), A (x)) − 2) ∨ 0]}, y ∈ Y , A ∗ (x) = {[( Al (x) + (R L )l ( A (x), B ( y )) + (R L )l ( B ( y ), B ∗ ( y )) − 2) ∨ 0, ( A r (x) x∈ E x
+ (R L )r ( A (x), B ( y )) + (R L )r ( B ( y ), B ∗ ( y )) − 2) ∨ 0]}, x ∈ X , E y = {x∈ X |[ Al∗ (x) + (R L )l ( A (x), B ( y )) − 1, A r∗ (x) + (R L )r ( A (x), B ( y )) − 1] > [0, 0]}, E x = {x∈ X |[ Al (x) + (R L )l ( A (x), B ( y )) − 1, A r (x) + (R L )r ( A (x), B ( y )) − 1] > [0, 0]}. where
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Example 3. For the dates of Example 1, we apply interval-valued quintuple implication principle to the I F M P problem shown in Table 1 and the I F M T problem shown in Table 2. For the two interval-valued residual implications (Gödel, Łukasiewicz), we have the same conclusion B ∗ ( y ) = {[0, 0], [0.5, 0.6], [0.5, 0.6]}, which, when compared with the solution of interval-valued triple I method, is much closer to the statement that “ y is large” and hence is in accordance with human thinking. For the two interval-valued residual implications (Gödel, Łukasiewicz), we have the same conclusion A ∗ (x) = {[0.5, 0.6], [0, 0], [0, 0]}, which, when compared with the solution of interval-valued triple I method, is much closer to the statement that “x is small” and hence is in accordance with human thinking. Definition 4.1 ([20]). A method of F M P is said recoverable if A ∗ = A implies B ∗ = B. Similarly, a method of F M T is recoverable if B ∗ = B implies A ∗ = A. We have the following result. Proposition 4.1. The Q I P method for I F M P (I F M T , resp.) is recoverable if A (B, resp.) is normal, where a interval-valued fuzzy set F on universe U is normal if there exists u ∈U such that F (u ) = [1, 1]. Proof. We only consider the Q I P solution of I F M P , the proof for Q I P solution of I F M T is similar. Suppose A ∗ = A and there exists an element x0 ∈ X such that A (x0 ) = [1, 1]. Then we have
B ( y) ≥ B ∗ ( y) =
T (T (R( A (x), B ( y )), R( A ∗ (x), A (x))), A ∗ (x))
x∈ X
≥ T (T (R( A (x0 ), B ( y )), R( A ∗ (x0 ), A (x0 ))), A ∗ (x0 )) = T (T (R([1, 1], B ( y )), [1, 1]), [1, 1]) = B ( y ) and therefore B ( y ) = B ∗ ( y ). This shows that the Q I P method of I F M P is recoverable.
2
5. Robustness of the interval-valued quintuple implication algorithms Li et al. [8] first extended compositional rule of inference to the case of interval-valued fuzzy set, and investigated the robustness of C R I method based on interval-valued. In this section, the robustness of interval-valued quintuple implication algorithms are studied by the same measure. Thus several related definitions and lemmas are quoted from [8]. Definition 5.1 ([8]). Let f be an n-tuple mapping from S I n to S I and [xn , yn ]) ∈ S I n , the ε sensitivity of f at point [x, y] is defined by
f ([x, y], ε ) =
ε ∈ [0, 1]. For arbitrary [x, y] = ([x1 , y 1 ], [x2 , y 2 ], . . . ,
{d( f ([x, y]), f ([x , y ]))|[x , y ] ∈ S I n and dn ([x, y], [x , y ]) ≤ ε },
where dn ([x, y], [x , y ]) = max{maxi | xi − x i |, maxi | y i − y i |}. Definition 5.2 ([8]). The maximum
f (ε ) =
ε sensitivity of f is defined as follows:
f ([x, y], ε ).
[x,y]∈ S I n
Definition 5.3 ([8]). Let A and A be two interval-valued fuzzy sets on S I . If A − A ∞ = d( A (x), A (x)) ≤ ε hold for all x ∈ X , then A is called the ε -perturbation of A denoted by A ∈ B ( A , ε ).
x∈ X
Definition 5.4. Let A , A , B , B , A ∗ and A ∗ be interval-valued fuzzy sets on S I . If A − A ∞ ≤ ε , B − B ∞ ≤ ε , A ∗ − A ∗ ∞ ≤ ε , and B ∗ and B ∗ are the interval-valued R-type quintuple implication solutions of I F M P ( A , B , A ∗ ) and I F M P ( A , B , A ∗ ) given by Theorem 4.1 respectively, the sensitivity of the interval-valued R-type quintuple implication solutions of I F M P denoted as B ∗ is defined as follows:
B ∗ = B ∗ − B ∗ ∞ =
d( B ∗ ( y ), B ∗ ( y )).
y ∈Y
Definition 5.5. Let A , A , B , B , B ∗ and B ∗ be interval-valued fuzzy sets on S I . If A − A ∞ ≤ ε , B − B ∞ ≤ ε , B ∗ − B ∗ ∞ ≤ ε , and A ∗ and A ∗ are the interval-valued R-type quintuple implication solutions of I F M T ( A , B , B ∗ ) and
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29
I F M T ( A , B , B ∗ ) given by Theorem 4.1 respectively, the sensitivity of the interval-valued R-type quintuple implication solutions of I F M T denoted as A ∗ is defined as follows:
A ∗ = A ∗ − A ∗ ∞ =
d( A ∗ (x), A ∗ (x)).
x∈ X
Lemma 5.1 ([8]). For a binary interval-valued fuzzy connective f :S I × S I → S I , we have (i) If f is any t-norm on S I , then
f (([x1 , y 1 ], [x2 , y 2 ]), ε ) = ( f l ([x1 , y 1 ], [x2 , y 2 ]) − f l ([(x1 − ε ) ∨ 0, ( y 1 − ε ) ∨ 0], [(x2 − ε ) ∨ 0, ( y 2 − ε ) ∨ 0])) ∨( f r ([x1 , y 1 ], [x2 , y 2 ]) − f r ([(x1 − ε ) ∨ 0, ( y 1 − ε ) ∨ 0], [(x2 − ε ) ∨ 0, ( y 2 − ε ) ∨ 0])) ∨( f l ([(x1 + ε ) ∧ 1, ( y 1 + ε ) ∧ 1], [(x2 + ε ) ∧ 1, ( y 2 + ε ) ∧ 1]) − f l ([x1 , y 1 ], [x2 , y 2 ])) ∨( f r ([(x1 + ε ) ∧ 1, ( y 1 + ε ) ∧ 1], [(x2 + ε ) ∧ 1, ( y 2 + ε ) ∧ 1]) − f r ([x1 , y 1 ], [x2 , y 2 ])) (ii) If f is any R-implication on S I , then
f (([x1 , y 1 ], [x2 , y 2 ]), ε ) = ( f l ([x1 , y 1 ], [x2 , y 2 ]) − f l ([(x1 + ε ) ∧ 1, ( y 1 + ε ) ∧ 1], [(x2 − ε ) ∨ 0, ( y 2 − ε ) ∨ 0])) ∨( f r ([x1 , y 1 ], [x2 , y 2 ]) − f r ([(x1 + ε ) ∧ 1, ( y 1 + ε ) ∧ 1], [(x2 − ε ) ∨ 0, ( y 2 − ε ) ∨ 0])) ∨( f l ([(x1 − ε ) ∨ 0, ( y 1 − ε ) ∨ 0], [(x2 + ε ) ∧ 1, ( y 2 + ε ) ∧ 1]) − f l ([x1 , y 1 ], [x2 , y 2 ])) ∨( f r ([(x1 − ε ) ∨ 0, ( y 1 − ε ) ∨ 0], [(x2 + ε ) ∧ 1, ( y 2 + ε ) ∧ 1]) − f r ([x1 , y 1 ], [x2 , y 2 ])) Lemma 5.2 ([8]). The Gödel t-norm on S I is the most robust t-norm, and TG (ε ) = ε . Lemma 5.3 ([8]). The Łukasiewicz implication on S I is the most robust R-implication, and RL (ε ) = 2ε ∧ 1. Lemma 5.4 ([8]). Suppose that A − A ∞ ≤ ε , B − B ∞ ≤ ε , and R is an interval-valued residuated implication induced by a left-continuous t-norm T on SI, then
d(R( A (x), B ( y )), R( A (x), B ( y ))) ≤ R (ε ). Lemma 5.5. Suppose that A − A ∞ ≤ ε , B − B ∞ ≤ ε , A ∗ − A ∗ ∞ ≤ ε and R is an interval-valued residuated implication induced by a left-continuous t-norm T on SI, then
d(T (R( A (x), B ( y )), R( A ∗ (x), A (x))), T (R( A (x), B ( y )), R( A ∗ (x), A (x)))) ≤ T (R (ε )). d(T (R( A (x), B ( y )), R( B ( y ), B ∗ ( y ))), T (R( A (x), B ( y )), R( B ( y ), B ∗ ( y )))) ≤ T (R (ε )). Proof. We only prove the first formula, the second can be proved in the same way. Since A − A ∞ ≤ ε and B − B ∞ ≤ ε . Then we have:
d(R( A (x), B ( y )), R( A (x), B ( y ))) ≤ R (ε ) (By Lemma 5.4) Similarly, d(R( A ∗ (x), A (x)), R( A ∗ (x), A (x))) ≤ R (ε ). Thus
d(T (R( A (x), B ( y )), R( A ∗ (x), A (x))), T (R( A (x), B ( y )), R( A ∗ (x), A (x)))) ≤ T (R (ε )).
2
Theorem 5.1. Suppose that A − A ∞ ≤ ε , B − B ∞ ≤ ε , A ∗ − A ∗ ∞ ≤ ε , and B ∗ and B ∗ are the interval-valued R-type quintuple implication solutions of I F M P ( A , B , A ∗ ) and I F M P ( A , B , A ∗ ) given by Theorem 4.1 respectively, then the sensitivity of the interval-valued fuzzy inference I F M P : B ∗ (ε ) = B ∗ − B ∗ ∞ ≤ T (T (R (ε ))).
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Proof.
B ∗ (ε ) = B ∗ − B ∗ ∞ = d( B ∗ ( y ), B ∗ ( y )) y ∈Y
=
d(
y ∈Y
T (T (R( A (x), B ( y )), R( A ∗ (x), A (x))), A ∗ (x)),
x∈ X
T (T (R( A (x), B ( y )), R( A ∗ (x), A (x))), A ∗ (x)))
x∈ X
≤
d(T (T (R( A (x), B ( y )), R( A ∗ (x), A (x))), A ∗ (x)),
y ∈Y x∈ X
T (T (R( A (x), B ( y )), R( A ∗ (x), A (x))), A ∗ (x)))
≤ T (T (R (ε )) ∨ ε ) (by Lemma 5.5) By Lemmas 5.2, 5.3, we can obtain that R (ε ) (2ε ∧ 1) > ε , therefore T (R (ε )) ≥ R (ε ) > ε , i.e. T (R (ε )) ∨ ε =
T (R (ε )). Thus
B ∗ (ε ) = B ∗ − B ∗ ∞ ≤ T (T (R (ε ))) 2 Using Theorem 5.1, we can obtain these following corollaries: Corollary 5.1. If T is interval-valued Minimum t-norm on S I , R is its residuum, then B ∗ (ε ) ≤ R (ε ). Proof. By Theorem 5.1, we have B ∗ (ε )≤T (T (R (ε ))) = ∧ (∧ (R (ε ))), because T is Minimum t-norm. We can use the fact from Lemma 5.2 that ∧ (ε ) = ε , then ∧ (∧ (R (ε ))) = R (ε ). Therefore, B ∗ (ε )≤R (ε ). 2 Corollary 5.2. If T is interval-valued Łukasiewicz t-norm on S I , R is its residuum, then B ∗ (ε ) ≤ ε + 2R (ε ). Proof.
B ∗ (ε ) = B ∗ − B ∗ ∞ = =
d( B ∗ ( y ), B ∗ ( y ))
y ∈Y
∗
{|( Al (x) + (R L )l ( A (x), B ( y )) + (R L )l ( A ∗ (x), A (x)) − 2) ∨ 0 − ( Al ∗ (x) + (R L )l ( A (x), B ( y )) +
y ∈Y x∈ X
(R L )l ( A ∗ (x), A (x)) − 2) ∨ 0| ∨ |( A r∗ (x) + (R L )r ( A (x), B ( y )) + (R L )r ( A ∗ (x), A (x)) − 2) ∨ 0 − ( A r ∗ (x) + (R L )r ( A (x), B ( y )) + (R L )r ( A ∗ (x), A (x)) − 2) ∨ 0|} ≤ ε + 2R (ε ) 2 Theorem 5.2. Suppose that A − A ∞ ≤ ε , B − B ∞ ≤ ε , B ∗ − B ∗ ∞ ≤ ε , and A ∗ and A ∗ are the interval-valued R-type quintuple implication solutions of I F M T ( A , B , B ∗ ) and I F M T ( A , B , B ∗ ) given by Theorem 4.1 respectively, then the sensitivity of the interval-valued fuzzy inference I F M T : A ∗ (ε ) = A ∗ − A ∗ ∞ ≤ T (T (R (ε ))). Proof.
A ∗ (ε ) = A ∗ − A ∗ ∞ = d( A ∗ (x), A ∗ (x)) x∈ X
=
x∈ X
y ∈Y
d(
T (T (R( A (x), B ( y )), R( B ( y ), B ∗ ( y ))), A (x)),
y ∈Y
T (T (R( A (x), B ( y )), R( B ( y ), B ∗ ( y ))), A (x)))
M. Luo, X. Zhou / International Journal of Approximate Reasoning 84 (2017) 23–32
≤
31
d(T (T (R( A (x), B ( y )), R( B ( y ), B ∗ ( y ))), A (x)),
x∈ X y ∈Y
T (T (R( A (x), B ( y )), R( B ( y ), B ∗ ( y ))), A (x)))
≤ T (T (R (ε )) ∨ ε ) (by Lemma 5.5) By Lemmas 5.2, 5.3, we can obtain that R (ε ) (2ε ∧ 1) > ε , therefore T (R (ε )) ≥ R (ε ) > ε , i.e. T (R (ε )) ∨ ε =
T (R (ε )). Thus
A ∗ (ε ) = A ∗ − A ∗ ∞ ≤ T (T (R (ε ))) 2 Using Theorem 5.2, we can obtain these following corollaries: Corollary 5.3. If T is interval-valued Minimum t-norm on S I , R is its residuum, then A ∗ (ε ) ≤ R (ε ). Proof. By Theorem 5.2, we have A ∗ (ε )≤T (T (R (ε ))) = ∧ (∧ (R (ε ))), because T is Minimum t-norm. We can use the fact from Lemma 5.2 that ∧ (ε ) = ε , then ∧ (∧ (R (ε ))) = R (ε ). Therefore, A ∗ (ε )≤R (ε ). 2 Corollary 5.4. If T is interval-valued Łukasiewicz t-norm on S I , R is its residuum, then A ∗ (ε ) ≤ ε + 2R (ε ). Proof.
A ∗ (ε ) = A ∗ − A ∗ ∞ = =
d( A ∗ (x), A ∗ (x))
x∈ X
{|( Al (x) + (R L )l ( A (x), B ( y )) + (R L )l ( B ( y ), B ∗ ( y )) − 2) ∨ 0 − ( Al (x) + (R L )l ( A (x), B ( y )) +
x∈ X y ∈Y
(R L )l ( B ( y ), B ∗ ( y )) − 2) ∨ 0| ∨ |( A r (x) + (R L )r ( A (x), B ( y )) + (R L )r ( B ( y ), B ∗ ( y )) − 2) ∨ 0 − ( A r (x) + (R L )r ( A (x), B ( y )) + (R L )r ( B ∗ ( y ), B ( y )) − 2) ∨ 0|} ≤ ε + 2R (ε ) 2 6. Conclusions In this paper, we have combined quintuple implication principle and interval-valued fuzzy set, proposed the quintuple implication principle based on interval-valued fuzzy inference for fuzzy modus ponens and fuzzy modus tollens. Unlike interval-valued triple I method, we take into account explicitly the similarity between A ∗ and A (or B ∗ and B) when computing the solution to I F M P (or I F M T ). We give the corresponding interval-valued R-type quintuple implication solutions and investigate the robustness of quintuple implication algorithms based on interval-valued fuzzy inference. Finally, the sensitivity of quintuple implication algorithms based on two important interval-valued residuated implication operators is given. Acknowledgements The authors would like to thank Editor-in-Chief Thierry Denoeux, area editor and anonymous reviewers for their valuable comments and suggestions. References [1] C. Alcalde, A. Burusco, R. Fuentes-Gonzalez, A constructive method for the definition of interval-valued fuzzy implication operators, Fuzzy Sets Syst. 153 (2005) 211–227. [2] O. Castillo, P. Melin, A review on interval type-2 fuzzy logic applications in intelligent control, Inf. Sci. 279 (2014) 615–631. [3] S.S. Dai, D.W. Pei, D.H. Guo, Robustness analysis of full implication inference method, Int. J. Approx. Reason. 54 (2013) 653–666. [4] B.A. Davey, H.A. Priestley, Introduction to Lattices and Order, Cambridge University Press, Cambridge, 1990. [5] G. Deschrijver, C. Cornelis, E.E. Kerre, On the representation of intuitionistic fuzzy t-norms and t-conorms, IEEE Trans. Fuzzy Syst. 12 (1) (2004) 45–61. [6] M.Y. Hsiao, T.S. Li, J.-Z. Lee, C.-H. Chao, S.-H. Tsai, Design of interval type-2 fuzzy sliding-mode controller, Inf. Sci. 178 (6) (2008) 1696–1716. [7] S. Jenei, A more efficient method for defining fuzzy connectives, Fuzzy Sets Syst. 90 (1997) 25–35. [8] D.Ch. Li, Y.M. Li, Y.J. Xie, Robustness of interval-valued fuzzy inference, Inf. Sci. 181 (2011) 4754–4764. [9] H.W. Liu, G.J. Wang, Continuity of triple I methods based on several implications, Comput. Math. Appl. 56 (2008) 2079–2087. [10] M.X. Luo, N. Yao, Triple I algorithms based on Schweizer–Sklar operators in fuzzy reasoning, Int. J. Approx. Reason. 54 (2013) 640–652. [11] M.X. Luo, K. Zhang, Robustness of full implication algorithms based on interval-valued fuzzy inference, Int. J. Approx. Reason. 62 (2015) 61–72.
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