- Email: [email protected]
Fuzzy approximations of fuzzy relational structures
JID:IJA AID:8202 /FLA
[m3G; v1.235; Prn:6/04/2018; 13:56] P.1 (1-10)
International Journal of Approximate Reasoning ••• (••••) •••–•••
1
Contents lists available at ScienceDirect
2
1 2
3
International Journal of Approximate Reasoning
4 5
3 4 5
6
6
www.elsevier.com/locate/ijar
7
7
8
8
9
9
10
10
11 12 13 14 15
Fuzzy approximations of fuzzy relational structures a b
17
c
14
Automation School, Beijing University of Posts and Telecommunications, 100876 Beijing, China University of Foreign Languages, 471003 Luoyang, Henan, China School of Science, Beijing University of Posts and Telecommunications, 100876 Beijing, China
18
23 24 25
i n f o
28 29 30 31 32
17
a b s t r a c t
20 21
Article history: Received 6 January 2017 Received in revised form 31 January 2018 Accepted 4 April 2018 Available online xxxx
26 27
16
19
a r t i c l e
21 22
15
18
19 20
12 13
Yibin Du a,b,∗ , Ping Zhu c
16
11
Keywords: Fuzzy relational structure Bisimulation Approximation method Rough set Fuzzy set Complete residuated lattice
In social network analysis, relational structures play a crucial role in the processing of complex data. This paper proposes a fuzzy relational structure which consists of a non-empty universal set and a set of fuzzy relations of finite arity. In order to deduce knowledge hidden in a fuzzy relational structure we present the concept of bisimulations as indiscernibility with respect to the fuzzy relational structure. Furthermore, we give a method of computing the largest bisimulations over fuzzy relational structures and present computational examples of the method. Finally, the fuzzy rough set analysis of fuzzy relational structures is discussed. © 2018 Elsevier Inc. All rights reserved.
22 23 24 25 26 27 28 29 30 31 32
33
33
34
34
35 36
1. Introduction
37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54
59 60 61
38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55
56 58
36 37
The theory of rough sets was firstly proposed by Pawlak in 1982 [31–33]. It is a useful mathematical tool for dealing with attribute data. In this theory, attribute data can be represented by a data table which comprises a set of objects and a set of attributes. The objects of a given data table can be identified within the limits of imprecise data by an indiscernibility relation induced by different values of attributes characterizing these objects. In other words, the indiscernibility relation induced by different values of attributes enables us to characterize a set of objects by a pair of sets, called the lower and upper approximation of the set of objects. Although databases only contain attribute information about objects in many practical cases, data about the relationships between objects has become increasingly important in decision analysis in recent years [17]. A remarkable example can be found in social network analysis, where the principal types of data are attribute data and relational data [40]. Liau et al. [23] studied granulation based on relational information between objects from the viewpoint of modal logic. More recently, Fan [17] further investigated relational information systems and extended rough set analysis from attribute information systems to relational structures which consists of a non-empty universal set and a set of relations. In this paper we pay attention to the generalization of relational structures in fuzzy environment and present bisimulations as indiscernibility with respect to fuzzy relational structures. Milner [25] and Park [34] introduced bisimulation as a concept that provides an effective method for reducing the complexity of concurrent processes and studying the equivalence of automata. In particular, bisimulation is exploited to reduce the state space of a system by combining bisimilar states. In the last thirty years, it has been applied to computer science, modal logic and set theory [16,19,24,26,27,37,41,14,18]. Recently, bisimulations have been introduced to fuzzy systems by
55 57
35
56
*
Corresponding author at: Automation School, Beijing University of Posts and Telecommunications, 100876 Beijing, China. E-mail addresses: [email protected] (Y. Du), [email protected] (P. Zhu).
https://doi.org/10.1016/j.ijar.2018.04.003 0888-613X/© 2018 Elsevier Inc. All rights reserved.
57 58 59 60 61
JID:IJA AID:8202 /FLA
[m3G; v1.235; Prn:6/04/2018; 13:56] P.2 (1-10)
Y. Du, P. Zhu / International Journal of Approximate Reasoning ••• (••••) •••–•••
2
1 2 3 4 5 6 7 8 9 10 11
two general approaches. One is based on a binary relation on the state space of a fuzzy system such that related states have exactly the same possibility degree of making a transition into every class of related states [36,2,42,5,6,48,7,15]. The other ´ c´ and his colleagues [8,21,9,10,39,11] proposed two types of simulations is based on a fuzzy relation on the state space. Ciri and four types of bisimulations for fuzzy automata. The purpose of the paper is to discuss the fuzzy rough set analysis of fuzzy relational structures by bisimulations which are similar to forward bisimulations considered in [10]. The rest of this paper is structured as follows. In Section 2 we introduce necessary terminology and notions of residuated lattices, fuzzy sets, and n-ary fuzzy relations and extend the concept of composition from binary fuzzy relations to n-ary fuzzy relations. Sections 3 presents a fuzzy relational structure which consists of a non-empty universal set and a set of fuzzy relations. In Section 4 we give main results on the computing of the largest bisimulations over fuzzy relational structures. Examples presented in Section 5 demonstrate the application of our method. Section 6 concludes this work.
12 13
16
19 20
25 26 27 28 29
5 6 7 8 9 10 11
15 16 17
(1) ( L , ∧, ∨, 0, 1) is a lattice with the least element 0 and the greatest element 1, (2) ( L , ⊗, 1) is a commutative monoid with the unit 1, (3) ⊗ and → form an adjoint pair, i.e., they satisfy the adjunction property: for all a, b, c ∈ L,
18 19 20 21
a ⊗ b ≤ c ⇔ a ≤ b → c.
22 24
4
14
For the terminology and basic notions in this section we refer to [10,3,4,22]. A residuated lattice is an algebra L = ( L , ∧, ∨, ⊗, →, 0, 1) such that
21 23
3
13
17 18
2
12
2. Preliminaries
14 15
1
22
In addition, a residuated lattice is called complete if ( L , ∧, ∨, 0, 1) is a complete lattice. If any finitely generated subalgebra of residuated lattice L is finite, then L is called locally finite. For any family ai , i ∈ I , of elements of L, we write ∨ ai for the supremum of {ai }i ∈ I , and ∧ ai for the infimum. An operai∈ I
i∈ I
tion ↔ is defined by a ↔ b = (a → b) ∧ (b → a). It can be verified that ⊗ is isotonic in both arguments and for all a, b, c ∈ L and {ai }i ∈ I , {b i }i ∈ I ⊆ L the following hold:
23 24 25 26 27 28 29
( ∨ ai ) ⊗ a = ∨ (ai ⊗ a),
(1)
32
(a ↔ a) = 1, (a ↔ b) = (b ↔ a), (a ↔ b) ⊗ (b ↔ c ) ≤ (a ↔ c ),
(2)
32
33
a ↔ b ≤ a ⊗ c ↔ b ⊗ c,
(3)
33
∧ (ai ↔ bi ) ≤ ( ∨ ai ) ↔ ( ∨ bi ).
(4)
30 31
34 35 36 37 38 39 40 41 42 43
i∈I
i∈I
46 47 48 49 50 51 52 53 54 55 56 57
i∈I
A=
A (a1 ) a1
+
A (a2 ) a2
i∈ I
of { A i }i ∈ I is defined as ( ∧ A i )(a) = ∧ A i (a), for all a ∈ U , the join ∨ A i of { A i }i ∈ I is defined as ( ∨ A i )(a) = ∨ A i (a), for all i∈ I
i∈ I
i∈ I
a ∈ U , and the product A ⊗ B is a fuzzy set defined by ( A ⊗ B )(a) = A (a) ⊗ B (a), for all a ∈ U . A fuzzy set in U 1 × · · · × U n is called an n-ary fuzzy relation between U 1 , U 2 , · · · , and U n . For simplicity, we call an n-ary fuzzy relation as a fuzzy relation. If U 1 = · · · = U n = U , we speak of n-ary fuzzy relation on a set U . The set of all n-ary n fuzzy relations on U will be denoted by L U . 2 2 For ϕ , ψ ∈ L U , the composition ϕ ◦ ψ is a fuzzy relation from L U defined by
38 39 40 41 42 43
47 48 49 50 51 52 53 54 55 56 57 58
(ϕ ◦ ψ)(a, c ) = ∨ ϕ (a, b) ⊗ ψ(b, c ), b ∈U
60 61
37
46
A (a )
i∈ I
36
45
an
where the term a i and i = 1, · · · , n, signifies that A (ai ) is the degree of membership of ai in A and the plus sign i represents the union. Let A, B and A i ∈ L U (i ∈ I ). A and B are said to be equal, which is denoted by A = B, if A (a) = B (a) for all a ∈ U . The inclusion of A and B is defined by A ≤ B if A (a) ≤ B (a) for all a ∈ U . Further, given this partial order the meet ∧ A i i∈ I
35
44
A (an )
58 59
34
i∈I
+ ··· +
30 31
Let L be the real unit interval [0, 1], a ∧ b = min{a, b} and a ∨ b = max{a, b}. Suppose that a ⊗ b = min{a, b} and a → b = 1 if a ≤ b, and = b otherwise. Then the structure of a residuated lattice with assumptions above is called Gödel structure. The structure where a ⊗ b = a · b and a → b = 1 if a ≤ b, and = b/a otherwise is the Goguen (product) structure. From now on, L = ( L , ∧, ∨, ⊗, →, 0, 1) will be a complete residuated lattice. Let U be a non-empty set. A fuzzy set A in U is a mapping A : U → L. The set U is called a universe. A (a) is called the degree of membership of a in A. The set of all fuzzy sets in U will be denoted by L U . The support of a fuzzy set A is a set defined as supp ( A ) = {a ∈ U : A (a) > 0}. If supp ( A ) = {a1 , · · · , an } is a finite set, then we often use Zadeh’s notation
44 45
i∈I
for all (a, c ) ∈ U 2 .
59 60 61
JID:IJA AID:8202 /FLA
[m3G; v1.235; Prn:6/04/2018; 13:56] P.3 (1-10)
Y. Du, P. Zhu / International Journal of Approximate Reasoning ••• (••••) •••–•••
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
2
For ϕ ∈ L U , the inverse of ϕ is a fuzzy relation 2 for all ϕ1 , ϕ2 , and ϕi ∈ L U (i ∈ I ), we have that
25 26 27 28
ϕ −1 ∈ L U defined by ϕ −1 (b, a) = ϕ (a, b), for all a, b ∈ U . Furthermore, 2
3
(5)
( ∨ ϕi )−1 = ∨ ϕi−1 .
(6)
i∈I
i∈I
A binary fuzzy relation
31 32 33
10 11 12
k1 +k
a (k1 + k)-ary fuzzy relation where ϕ ∈ L U , and let ψ be a (k + k2 )-ary fuzzy relation where ψ ∈ L U k-composition of ϕ and ψ is a (k1 + k2 )-ary fuzzy relation defined by
∨
b=(b1 ,b2 ,··· ,bk )∈U k
ϕ (a, b) ⊗ ψ(b, c),
k+k2
. Then, the
38 39 40 41
46 47 48 49
52 53 54 55 56 57 58 59 60 61
16 17 18 19 21
(7)
(8)
The details are as follows. By the definition of compositions, for all (a, d) ∈ U k1 +k4 where a = (a1 , a2 , · · · , ak1 ) and d = (d1 , d2 , · · · , dk4 ) we have
22 24 25 26 27 28
= = = = =
∨
c=(c 1 ,c 2 ,··· ,ck3 )∈U k3
∨
c=(c 1 ,c 2 ,··· ,ck3 )∈U
∨
k3
31 32 33 35
(ϕ1 ◦k2 ϕ2 )(a, c) ⊗ ϕ3 (c, d)
36
{
38
∨
b=(b1 ,b2 ,··· ,bk2 )∈U
{
∨
k2
c=(c 1 ,c 2 ,··· ,ck3 )∈U k3 b=(b1 ,b2 ,··· ,bk2 )∈U k2
ϕ1 (a, b) ⊗ ϕ2 (b, c)} ⊗ ϕ3 (c, d)
40 41 42
ϕ1 (a, b) ⊗ {
∨
ϕ1 (a, b) ⊗ (ϕ2 ◦k3 ϕ3 )(b, d)
b=(b1 ,b2 ,··· ,bk2 )∈U k2
39
ϕ1 (a, b) ⊗ ϕ2 (b, c) ⊗ ϕ3 (c, d)}
∨
b=(b1 ,b2 ,··· ,bk2 )∈U k2
37
∨
c=(c 1 ,c 2 ,··· ,ck3 )∈U k3
ϕ2 (b, c) ⊗ ϕ3 (c, d)}
43 44 45 46 47
= (ϕ1 ◦k2 (ϕ2 ◦k3 ϕ3 ))(a, d). Next, let
30
34
((ϕ1 ◦k2 ϕ2 ) ◦k3 ϕ3 )(a, d)
48
ϕ be a binary fuzzy relation, and let ψ be a k-ary fuzzy relation. Then we have
49
50 51
15
29
44 45
14
23
(ϕ1 ◦k2 ϕ2 ) ◦k3 ϕ3 = ϕ1 ◦k2 (ϕ2 ◦k3 ϕ3 ).
42 43
13
20
for all (a, c) ∈ U k1 +k2 where a = (a1 , a2 , · · · , ak1 ) and c = (c 1 , c 2 , · · · , ck2 ). When k = 1, omit the subscript and write the composition as ϕ ◦ ψ . For the concept of k-composition, we have the following properties. Let ϕ1 be a (k1 + k2 )-ary fuzzy relation, let ϕ2 be a (k2 + k3 )-ary fuzzy relation, and let ϕ3 be a (k3 + k4 )-ary fuzzy relation. Then, by the definition of k-composition we have
35 37
6
9
If ϕ is reflexive and transitive, then ϕ ◦ ϕ = ϕ . A binary fuzzy relation on U which is reflexive, symmetric and transitive is called a fuzzy equivalence relation. The set of all fuzzy equivalence relations on U is denoted by E (U ). Given the ordering of binary fuzzy relations, E (U ) is a complete lattice, in which the meet coincide with the ordinary meet of binary fuzzy relations. For later need, we extend the concept of composition from binary fuzzy relations to n-ary fuzzy relations. Let ϕ be
34 36
5
8
29 30
4
7
ϕ on U is
(1) reflexive if ϕ (a, a) = 1, for every a ∈ U ; (2) symmetric if ϕ (a, b) = ϕ (b, a), for all a, b ∈ U ; (3) transitive if ϕ (a, b) ⊗ ϕ (b, c ) ≤ ϕ (a, c ), for all a, b, c ∈ U .
(ϕ ◦k ψ)(a, c) =
1 2
(ϕ1 ◦ ϕ2 )−1 = ϕ2−1 ◦ ϕ1−1 ,
23 24
3
50
(ϕ ◦ ψ) ◦ ϕ = ϕ ◦ (ψ ◦ ϕ ).
(9)
In fact, Equation (9) is a special case of Equation (8). Further, let ϕ0 be a (k1 + k2 )-ary fuzzy relation, let k3 )-ary fuzzy relations, and let ϕ3 be a (k3 + k4 )-ary fuzzy relation. Then we have that
ϕ1 ≤ ϕ2 implies ϕ0 ◦k2 ϕ1 ≤ ϕ0 ◦k2 ϕ2 and ϕ1 ◦k3 ϕ3 ≤ ϕ2 ◦k3 ϕ3 .
ϕ1 , ϕ2 be (k2 +
i∈I
i∈I
i∈I
i∈I
52 53 54
(10)
Finally, let ϕ , ϕi (i ∈ I ) be (k1 + k2 )-ary fuzzy relations, and let ψ , ψi (i ∈ I ) be (k2 + k3 )-ary fuzzy relations. Then we can easily verify that
ϕ ◦k2 ( ∨ ψi ) = ∨ (ϕ ◦k2 ψi ), ( ∨ ϕi ) ◦k2 ψ = ∨ (ϕi ◦k2 ψ).
51
(11)
55 56 57 58 59 60 61
JID:IJA AID:8202 /FLA
[m3G; v1.235; Prn:6/04/2018; 13:56] P.4 (1-10)
Y. Du, P. Zhu / International Journal of Approximate Reasoning ••• (••••) •••–•••
4
1
3. Fuzzy relational structures
1
2 3 4 5
2
In order to deal with relational data, Fan [17] considers the rough set analysis of relational structures. In this section, we attempt to conduct a further study along this line. We present fuzzy relational structures as basic models for some fuzzy approximation spaces and state the key definition of bisimulations for fuzzy relational structures.
6 7 8 9
12 13 14
Definition 3.1. A fuzzy relational structure is a tuple A = (U , (ϕi )i ∈ I ), where U is the universe of the structure; I is an index set; and for each i ∈ I , ϕi is a fuzzy relation on U .
17 18
k
ϕi for each i ∈ I are finite. For each fuzzy relation ϕi there exists a natural
number k in N such that ϕi ∈ L U . A fuzzy relational structure whose universe and set of fuzzy relations (ϕi )i ∈ I are finite is called a fuzzy finite relational structure. Example 3.1. Let L be the Gödel structure, A = (U , (ϕ1 , ϕ2 )) a fuzzy relational structure over L with U = {a, b}, and the binary fuzzy relation ϕ1 , the ternary fuzzy relation ϕ2 given by
ϕ1 =
20 21
0.5
(a, a)
+
0.5
(b, a)
,
ϕ2 =
0.8
(a, a, b)
+
0.6
(a, b, b)
+
0.5
(b, a, a)
+
0.5
25 26 27 28 29 30 31 32
(b, b, a)
Definition 3.2. Let A = (U , (ϕi )i ∈ I ) be a fuzzy relational structure. A binary fuzzy relation ψ is called a bisimulation on U if for every ϕi , the following hold:
43 44
47 48
51
18
24 25 26 27 28 29
33
I U (a, b) =
34 35
1 if a = b 0 other wise .
36 37 38
Thus, the set of all bisimulations on U is non-empty. The concept of bisimulations is similar to the concept of forward bisimulations which is considered in [10]. For any bisimulation ψ , (U , ψ) is a fuzzy approximation space. Then based on bisimulation the lower and upper fuzzy approximations of a target fuzzy set over (U , ψ) are given, which can be used to deduce the unavailable knowledge represented by fuzzy sets with respect to fuzzy relational structures. Now we are ready to state and prove some properties of bisimulations.
39 40 41 42 43 44 45
Proposition 3.1. Let A = (U , (ϕi )i ∈ I ) be a fuzzy relational structure. Suppose that ψ1 , ψ2 ∈ L
U2
2
are bisimulations on U and
{ψ j } j ∈ J ⊆ L U is a family of bisimulations on U . Then the fuzzy relations ψ1 ◦ ψ2 and ∨ ψ j are bisimulations on U . j∈ J
Proof. According to Definition 3.2 and Equations (5), (8) and (10), for each i ∈ I we have that
52
(ψ1 ◦ ψ2 ) ◦ ϕi = ψ1 ◦ (ψ2 ◦ ϕi )
53 54 55 56 57 58 59 60 61
17
32
It is easy to see that ψ −1 is also a bisimulation on U . A simple example of a bisimulation is the equality relation on U
49 50
16
31
45 46
14
30
(1) ψ −1 ◦ ϕi ≤ ϕi ◦ ψ −1 ; (2) ψ ◦ ϕi ≤ ϕi ◦ ψ .
38
42
13
22
The fuzzy approximation space is the basic construction of fuzzy rough set theory. Given a fuzzy relational structure A = (U , (ϕi )i ∈ I ), we can derive its fuzzy approximation spaces based on bisimulations over A .
37
41
12
23
36
40
11
21
respectively.
35
39
9
20
33 34
8
19
,
23 24
7
15
19
22
5
10
This paper assumes that the index set I and
15 16
4 6
10 11
3
and
46 47 48 49 50 51 52 53
≤ ψ1 ◦ (ϕi ◦ ψ2 )
54
= (ψ1 ◦ ϕi ) ◦ ψ2
56
55
≤ (ϕi ◦ ψ1 ) ◦ ψ2
57
= ϕi ◦ (ψ1 ◦ ψ2 ),
59
58 60 61
JID:IJA AID:8202 /FLA
[m3G; v1.235; Prn:6/04/2018; 13:56] P.5 (1-10)
Y. Du, P. Zhu / International Journal of Approximate Reasoning ••• (••••) •••–•••
1 2
(ψ1 ◦ ψ2 )−1 ◦ ϕi = (ψ2−1 ◦ ψ1−1 ) ◦ ϕi
1 2
= ψ2−1 ◦ (ψ1−1 ◦ ϕi )
3
3
≤ ψ2−1 ◦ (ϕi ◦ ψ1−1 )
4 5
4 5
= (ψ2−1 ◦ ϕi ) ◦ ψ1−1
6
6
≤ (ϕi ◦ ψ2−1 ) ◦ ψ1−1
7 8
7 8
= ϕi ◦ (ψ2−1 ◦ ψ1−1 )
9
−1
= ϕi ◦ (ψ1 ◦ ψ2 )
10 11
5
9 10
.
12
Thus ψ1 ◦ ψ2 is a bisimulation on U . Analogously, by Definition 3.2 and Equations (6) and (11) it holds that ∨ ψ j is a
13
bisimulation on U .
2
14 15 16 17 18 19 20 21 22 23 24 25
j∈ J
28
Since the set of all bisimulations on U is non-empty, we have that the union of all bisimulations gives rise to the largest bisimulation. Furthermore, it can be proved that the largest bisimulation is a fuzzy equivalence relation. Proposition 3.2. Let A = (U , (ϕi )i ∈ I ) be a fuzzy relational structure. Then the largest bisimulation on U is a fuzzy equivalence relation. 2
Proof. First, let {ψ j } j ∈ J ⊆ L U be the set of all bisimulations on U . Then we have that the largest bisimulation is ∨ ψ j . j∈ J
1 −1 Third, by ψ − j ∈ {ψ j } j ∈ J and Proposition 3.1, we derive ∨ ψ j = ∨ (ψ j ∨ ψ j ). Then for all a, b ∈ U , j∈ J
33
41 42 43 44 45 46 47 48 49
52 53
35
∨ ψ j ))(b, a)
36 37 38 j∈ J
j∈ J
j∈ J
j∈ J
j∈ J
( ∨ ψ j ) ◦ ( ∨ ψ j )(a, c ) = ∨ (( ∨ ψ j )(a, b) ⊗ ( ∨ ψ j )(b, c )) ≤ ( ∨ ψ j )(a, c ) j∈ J
j∈ J
b ∈U
j∈ J
j∈ J
58
j∈ J
j∈ J
61
41 42
45 46 47
2
48 49 50
Remark 3.1. Because of symmetry, we have that a fuzzy equivalence relation ψ on U is a bisimulation on U if and only if ψ ◦ ϕi ≤ ϕi ◦ ψ for every ϕi . It is worthy to note that the largest bisimulation as an indiscernibility relation reflects the classification ability of the fuzzy relational structure to the greatest extent.
51 52 53 54
4. Computation of the largest bisimulation
55 56
We will provide a method for computing the largest bisimulations over fuzzy relational structures adapting the method developed in [9]. Now preparing for the method, we are ready to prove the following results.
59 60
40
44
( ∨ ψ j )(a, b) ⊗ ( ∨ ψ j )(b, c ) ≤ ( ∨ ψ j )(a, c ) for all a, b, c ∈ U . Thus we obtain Proposition 3.2.
39
43
for all a, c ∈ U . Then we have
56 57
25
34
54 55
24
33
50 51
23
32
Last, according to Proposition 3.1, we have ( ∨ ψ j ) ◦ ( ∨ ψ j ) ∈ {ψ j } j ∈ J . Then ( ∨ ψ j ) ◦ ( ∨ ψ j ) ≤ ∨ ψ j . It means that
j∈ J
22
1 (b, a) ∨ ψ j (b, a)) = ∨ (ψ − j
j∈ J
j∈ J
21
30
= ( ∨ ψ j )(b, a).
38
20
31
j∈ J
37
19
= ∨ (ψ j (a, b) ∨ ψ j (b, a))
= ( ∨ (ψ j
36
18
29
−1
35
17
= ∨ (ψ j (a, b) ∨ ψ j (a, b))
j∈ J
34
16
28
j∈ J
32
15
27
j∈ J
j∈ J
31
13
26
−1
30
40
j∈ J
1 ( ∨ ψ j )(a, b) = ( ∨ (ψ j ∨ ψ − j ))(a, b )
29
39
j∈ J
Second, from I U ∈ {ψ j } j ∈ J , we can easily conclude that ( ∨ ψ j )(a, a) = 1 for every a ∈ U .
j∈ J
12 14
26 27
11
57 58 59
Theorem 4.1. Let A = (U , (ϕi )i ∈ I ) be a fuzzy relational structure and ψ a fuzzy equivalence relation on U . Then the following conditions are equivalent:
60 61
JID:IJA AID:8202 /FLA
[m3G; v1.235; Prn:6/04/2018; 13:56] P.6 (1-10)
Y. Du, P. Zhu / International Journal of Approximate Reasoning ••• (••••) •••–•••
6
1 2 3
(1) ψ ◦ ϕi ≤ ϕi ◦ ψ , for every ϕi ; (2) ψ ◦ ϕi ◦ ψ = ϕi ◦ ψ , for every ϕi ; (3) If ϕi is a k-ary fuzzy relation, we suppose that
4
1 2 3 4
ψ i (a, b) =
5
∧
5
6
a2 ,a3 ,··· ,ak ∈U
6
7
(ϕi ◦ ψ)(a, a2 , · · · , ak−1 , ak ) ↔ (ϕi ◦ ψ)(b, a2 , · · · , ak−1 , ak ).
7
8 9 10
15 16 17 18 19
10 11
i∈I
12 14
9
ψ(a, b) ≤ ∧ ψ i (a, b).
11 13
8
Then for all a, b ∈ U we have
12
Proof. (1) ⇔ (2). If ψ ◦ ϕi ◦ψ = ϕi ◦ψ for every ϕi , then ψ ◦ ϕi = ψ ◦ ϕi ◦ I U ≤ ψ ◦ ϕi ◦ψ = ϕi ◦ψ . Conversely, if ψ ◦ ϕi ≤ ϕi ◦ψ then ψ ◦ ϕi ◦ ψ ≤ ϕi ◦ ψ ◦ ψ = ϕi ◦ ψ , and ϕi ◦ ψ = I U ◦ ϕi ◦ ψ ≤ ψ ◦ ϕi ◦ ψ , we conclude that ψ ◦ ϕi ◦ ψ = ϕi ◦ ψ . (2) ⇒ (3). Suppose that ψ ◦ ϕi ◦ ψ = ϕi ◦ ψ for every ϕi . Then by the Equation (9), for all b, a, a2 , · · · , ak−1 , ak ∈ U we have that
ψ(a, b) ⊗ (ϕi ◦ ψ)(b, a2 , · · · , ak−1 , ak )
13 14 15 16 17 18 19
20
≤ (ψ ◦ (ϕi ◦ ψ))(a, a2 , · · · , ak−1 , ak )
20
21
= (ψ ◦ ϕi ◦ ψ)(a, a2 , · · · , ak−1 , ak )
21
= (ϕi ◦ ψ)(a, a2 , · · · , ak−1 , ak ),
23
22 23 24 25 26 27 28 29 30 31 32
24
and by the adjunction property we obtain that
ψ(a, b) ≤ (ϕi ◦ ψ)(b, a2 , · · · , ak−1 , ak ) → (ϕi ◦ ψ)(a, a2 , · · · , ak−1 , ak ).
35 36 37
ψ(a, b) = ψ(b, a) ≤ (ϕi ◦ ψ)(a, a2 , · · · , ak−1 , ak ) → (ϕi ◦ ψ)(b, a2 , · · · , ak−1 , ak ),
ψ(a, b) ≤ (ϕi ◦ ψ)(a, a2 , · · · , ak−1 , ak ) ↔ (ϕi ◦ ψ)(b, a2 , · · · , ak−1 , ak ). Since it is satisfied for all a2 , a3 , · · · , ak ∈ U and ϕi (i ∈ I ), we conclude that the condition (3) holds. (3) ⇒ (2). If the condition (3) holds, then for all b, a, a2 , · · · , ak−1 , ak ∈ U and ϕi (i ∈ I ) we have that
46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61
31
34 35 36 37 38
ψ(a, b) ≤ (ϕi ◦ ψ)(a, a2 , · · · , ak−1 , ak ) ↔ (ϕi ◦ ψ)(b, a2 , · · · , ak−1 , ak ) ≤ (ϕi ◦ ψ)(b, a2 , · · · , ak−1 , ak ) → (ϕi ◦ ψ)(a, a2 , · · · , ak−1 , ak ), and by the adjunction property we have that
43 45
30
33
41
44
27
32
and hence,
40 42
26
29
38 39
25
28
By symmetry,
33 34
22
39 40 41 42 43
ψ(a, b) ⊗ (ϕi ◦ ψ)(b, a2 , · · · , ak−1 , ak ) ≤ (ϕi ◦ ψ)(a, a2 , · · · , ak−1 , ak ). Note that
44 45 46
(ψ ◦ ϕi ◦ ψ)(a, a2 , · · · , ak−1 , ak ) = (ψ ◦ (ϕi ◦ ψ))(a, a2 , · · · , ak−1 , ak ) = ∨ ψ(a, b) ⊗ (ϕi ◦ ψ)(b, a2 , · · · , ak−1 , ak ) b ∈U
48 49 50 51 52
≤ (ϕi ◦ ψ)(a, a2 , · · · , ak−1 , ak ), and hence ψ ◦ ϕi ◦ ψ ≤ ϕi ◦ ψ . Because
47
53
ϕi ◦ ψ ≤ ψ ◦ ϕi ◦ ψ , we conclude that ψ ◦ ϕi ◦ ψ = ϕi ◦ ψ for every ϕi . 2
Lemma 4.1. Let A = (U , (ϕi )i ∈ I ) be a fuzzy relational structure and ψ a fuzzy equivalence relation on U . Suppose that
ψ R (a, b) = ∧ ψ i (a, b), i∈I
for all a, b ∈ U . Then we have that ψ ≤ ψ R and ψ i (i ∈ I ) and ψ R are fuzzy equivalence relations.
54 55 56 57 58 59 60 61
JID:IJA AID:8202 /FLA
[m3G; v1.235; Prn:6/04/2018; 13:56] P.7 (1-10)
Y. Du, P. Zhu / International Journal of Approximate Reasoning ••• (••••) •••–•••
1 2 3
7
Proof. From the assertion (3) of Theorem 4.1, we get that ψ ≤ ψ R . Next, since ↔ is reflexive and symmetric, it is easily verified that ψ i (i ∈ I ) is reflexive and symmetric. Note that for all a, b, c , a2 , a3 , · · · , ak ∈ U
4
∧
a2 ,a3 ,··· ,ak ∈U
7
(ϕi ◦ ψ)(a, a2 , · · · , ak −1 , ak ) ↔ (ϕi ◦ ψ)(b, a2 , · · · , ak −1 , ak )
8
13
i
33 34
21 22 23 24 25
It follows that for all a, b, c ∈ U
26 27
i
ψ (a, b) ⊗ ψ (b, c )
30 32
20
≤ (ϕi ◦ ψ)(a, a2 , · · · , ak−1 , ak ) ↔ (ϕi ◦ ψ)(c , a2 , · · · , ak−1 , ak ).
≤
∧
a2 ,a3 ,··· ,ak ∈U
28
(ϕi ◦ ψ)(a, a2 , · · · , ak−1 , ak ) ↔ (ϕi ◦ ψ)(c , a2 , · · · , ak−1 , ak )
37 38 39
31
Thus ψ (i ∈ I ) is a fuzzy equivalence relation. Analogously, we can conclude that ψ R is a fuzzy equivalence relation.
32
i
2
42
45 46 47 48 49 50 51
Lemma 4.2. Let A = (U , (ϕi )i ∈ I ) be a fuzzy relational structure, and let ψ1 and ψ2 be fuzzy equivalence relations on U . If ψ1 ≤ ψ2 , then ψ1R ≤ ψ2R .
54
≤ (ϕi ◦ ψ1 )(a, a2 , · · · , ak−1 , ak ) ⊗ ψ2 (ak , c ) ↔ (ϕi ◦ ψ1 )(b, a2 , · · · , ak−1 , ak ) ⊗ ψ2 (ak , c ). Then, by (4) we have that for all c ∈ U ,
ψ1i (a, b)
= =
≤
59 60 61
39 41 42
≤
44 45 46 47 48 49 50
∧
a2 ,a3 ,··· ,ak ∈U
51
(ϕi ◦ ψ1 )(a, a2 , · · · , ak−1 , ak ) ↔ (ϕi ◦ ψ1 )(b, a2 , · · · , ak−1 , ak )
52
∧
{ ∧
a2 ,a3 ,··· ,ak−1 ∈U ak ∈U
(ϕi ◦ ψ1 )(a, a2 , · · · , ak−1 , ak ) ↔ (ϕi ◦ ψ1 )(b, a2 , · · · , ak−1 , ak )}
56 58
38
43
(ϕi ◦ ψ1 )(a, a2 , · · · , ak−1 , ak ) ↔ (ϕi ◦ ψ1 )(b, a2 , · · · , ak−1 , ak )
55 57
37
40
Proof. Let ϕi be a k-ary fuzzy relation and a, b ∈ U . From ψ1 ≤ ψ2 , it follows ψ1 ◦ ψ2 = ψ2 , and by (3), for all c , a2 , a3 , · · · , ak ∈ U we get that
52 53
34 36
43 44
33 35
We also have the following.
40 41
29 30
= ψ i (a, c ).
35 36
16
19
(ϕi ◦ ψ)(b, a2 , · · · , ak−1 , ak ) ↔ (ϕi ◦ ψ)(c , a2 , · · · , ak−1 , ak )
28
15
18
i
i
14
17
≤ (ϕi ◦ ψ)(a, a2 , · · · , ak−1 , ak ) ↔ (ϕi ◦ ψ)(b, a2 , · · · , ak−1 , ak ) ⊗
27
31
◦ ψ)(c , a2 , · · ·
, ak−1 , ak )
ψ (a, b) ⊗ ψ (b, c )
23
29
Then for all a, b, c , a2 , a3 , · · · , ak ∈ U
22
26
13
, ak−1 , ak ) ↔ (ϕi
≤ (ϕi ◦ ψ)(b, a2 , · · · , ak−1 , ak ) ↔ (ϕi ◦ ψ)(c , a2 , · · · , ak−1 , ak ).
20
25
9
12
a2 ,a3 ,··· ,ak ∈U
(ϕi ◦ ψ)(b, a2 , · · ·
19
24
∧
16
21
8
11
ψ i (b, c ) =
15
18
7
10
and
14
17
6
≤ (ϕi ◦ ψ)(a, a2 , · · · , ak−1 , ak ) ↔ (ϕi ◦ ψ)(b, a2 , · · · , ak−1 , ak ),
9
12
3 5
6
11
2 4
ψ i (a, b) =
5
10
1
∧
{ ∧ [(ϕi ◦ ψ1 )(a, a2 , · · · , ak−1 , ak ) ⊗ ψ2 (ak , c )
53 54 55 56 57
a2 ,a3 ,··· ,ak−1 ∈U ak ∈U
58
↔ (ϕi ◦ ψ1 )(b, a2 , · · · , ak−1 , ak ) ⊗ ψ2 (ak , c )]}
59
∧
a2 ,a3 ,··· ,ak−1 ∈U
{[ ∨ (ϕi ◦ ψ1 )(a, a2 , · · · , ak−1 , ak ) ⊗ ψ2 (ak , c )] ak ∈U
60 61
JID:IJA AID:8202 /FLA
[m3G; v1.235; Prn:6/04/2018; 13:56] P.8 (1-10)
Y. Du, P. Zhu / International Journal of Approximate Reasoning ••• (••••) •••–•••
8
↔ [ ∨ (ϕi ◦ ψ1 )(b, a2 , · · · , ak−1 , ak ) ⊗ ψ2 (ak , c )]}
1
=
3
2
∧
{((ϕi ◦ ψ1 ) ◦ ψ2 )(a, a2 , · · · , ak−1 , c )
4
a2 ,a3 ,··· ,ak−1 ∈U
5
↔ ((ϕi ◦ ψ1 ) ◦ ψ2 )(b, a2 , · · · , ak−1 , c )}
6
=
7 8
∧
a2 ,a3 ,··· ,ak−1 ∈U
=
10
4 5 6
{(ϕi ◦ (ψ1 ◦ ψ2 ))(a, a2 , · · · , ak−1 , c )
∧
7 8 9
{(ϕi ◦ ψ2 )(a, a2 , · · · , ak−1 , c )
11
a2 ,a3 ,··· ,ak−1 ∈U
12
↔ (ϕi ◦ ψ2 )(b, a2 , · · · , ak−1 , c )}.
14
3
↔ (ϕi ◦ (ψ1 ◦ ψ2 ))(b, a2 , · · · , ak−1 , c )}
9
13
1
ak ∈U
2
10 11 12
Namely, ψ1i (a, b) ≤ ψ2i (a, b) for all a, b ∈ U . Thus ψ1R ≤ ψ2R .
13
2
14
15 16 17
15
Theorem 4.2. Let A = (U , (ϕi )i ∈ I ) be a fuzzy relational structure. The set E b (U ) of all bisimulation fuzzy equivalence relations on U forms a complete lattice. This lattice is a complete join-subsemilattice of the lattice E (U ) of all fuzzy equivalence relations on U .
18 19 20 21 22 23 24
27 28
Proof. Because the equality relation I U is a bisimulation on U , it can be proved that E (U ) is a complete join-subsemilattice of E (U ). Let {ψ j } j ∈ J be a family of bisimulation fuzzy equivalence relations on U , and let ψ be its join in E (U ). Then for any j ∈ J , according to ψ j ≤ ψ and Lemma 4.2 we obtain that ψ j ≤ ψ jR ≤ ψ R . Thus ψ ≤ ψ R . Hence, by the assertion (3) of Theorem 4.1 we obtain that ψ is a bisimulation fuzzy equivalence relation. 2
31 32
ψ1 = ψ,
35
ψk+1 =
ψk ∧ ψkR ,
f or each k ∈ N .
40
43 44 45 46 47 48 49 50 51 52 53
58 59 60 61
26 27 28 30 31 32
37
(1) ψ b ≤ · · · ≤ ψk+1 ≤ ψk ≤ · · · ≤ ψ1 = ψ ; (2) If ψk = ψk+m , for some k, m ∈ N, then ψk = ψk+1 = ψ b ; (3) If A is finite and L is locally finite, then ψk = ψ b for some k ∈ N.
38 39 40 41
Proof. (1) It is easy to see that ψk+1 ≤ ψk for each k ∈ N and ψ b ≤ ψ1 . Let ψ b ≤ ψk for some k ∈ N. Then ψ b ≤ (ψ b ) R ≤ ψkR , so ψ b ≤ ψk ∧ ψkR . Thus, by induction we get ψ b ≤ ψk for each k ∈ N. (2) Suppose that ψk = ψk+m , for some k, m ∈ N. Then ψk = ψk+m ≤ ψk+1 = ψk ∧ ψkR ≤ ψkR , what means that ψk is a bisimulation fuzzy equivalence relation. Since ψ b is the largest bisimulation fuzzy equivalence relation contained in ψ , we conclude that ψk = ψk+1 = ψ b . (3) Because A is finite and L is locally finite, there exist k, m ∈ N such that ψk = ψk+m , and by (2) we conclude that ψk = ψ b . 2
Remark 4.1. If A is finite and L is locally finite, Theorem 4.3 gives a method of computing the largest bisimulation fuzzy equivalence relation contained in a given fuzzy equivalence relation ψ . But it does not necessarily work if L is not locally finite.
42 43 44 45 46 47 48 49 50 51 52 53 54 55
5. Computational examples and fuzzy rough set analysis
56 57
24
36
54 55
23
35
Then
41 42
22
34
37 39
21
33
34
38
20
29
Theorem 4.3. Let A = (U , (ϕi )i ∈ I ) be a fuzzy relational structure, let ψ be a fuzzy equivalence relation on U and let ψ b be the largest bisimulation fuzzy equivalence relation contained in ψ . Define inductively a sequence {ψk }k∈ N of fuzzy equivalence relations on U as follows:
33
36
19
25
From Theorem 4.2 we have that for any fuzzy equivalence relation ψ on U there exists the largest bisimulation fuzzy equivalence relation contained in ψ . It will be denoted by ψ b . In the next theorem we consider the problem how to construct it.
29 30
17 18
b
25 26
16
56
Example 5.1. Let L be the product structure, A = (U , (ϕ1 , ϕ2 )) a fuzzy relational structure over L with U = {a, b}, and the binary fuzzy relation ϕ1 , the ternary fuzzy relation ϕ2 given by
ϕ1 =
1
(a, b)
+
1 /2
(b, a)
,
ϕ2 =
1
(a, a, b)
+
1
(a, b, b)
+
1 /2
(b, a, a)
+
1 /2
(b, b, a)
57 58 59 60
,
61
JID:IJA AID:8202 /FLA
[m3G; v1.235; Prn:6/04/2018; 13:56] P.9 (1-10)
Y. Du, P. Zhu / International Journal of Approximate Reasoning ••• (••••) •••–•••
1 2 3
respectively. Let ψ be the universal relation on U , i.e., for every a, b ∈ U we have that ψ(a, b) = 1. Applying to ψ the method from Theorem 4.3, we get a sequence {ψk }k∈ N of fuzzy equivalence relations given by
4
1
ϕk =
5
8 9
+
(a, a)
6 7
9
1/2k−1
(a, b)
+
1/2k−1
(b, a)
+
1
(b, b)
,
1
ψb = IU =
11
(a, b)
+
1
(b, a)
6
16
19 20
13 14
We can see that the method of computing the largest bisimulation fuzzy equivalence relation does not work since L is not locally finite. Example 5.2. We continue with Example 3.1. Because L is locally finite, the method of computing the largest bisimulation fuzzy equivalence relation works. Then, the largest bisimulation fuzzy equivalence relation ψ b is given by
ψb =
23
1
(a, a)
0.5
+
(a, b)
+
0.5
(b, a)
+
1
28 29 30 31 32 33 34 35 36 37 38
(b, b)
22 23
Let L = [0, 1]. For any fuzzy set A ∈ [0, 1]U , the lower and upper approximations of A, ψ b ( A ) and ψ b ( A ), with respect
36
⊥
to the fuzzy approximation space (U , ψ b ) are fuzzy sets of U whose membership functions, for each a ∈ U , are
49
54
ψb
57 58 59
( A )(a) = ∨ { A (b)ψ (a, b)},
(13)
b ∈U
31 32 33 34 35 37
40
A=
0.8 a
+
0.4 b
44 45
where a binary composition on [0, 1] is a t-norm and a binary composition ⊥ on [0, 1] is a t-conorm. We illustrate fuzzy approximations based on ψ b with Example 5.2. Suppose that ⊥ = ∨ and = ∧. Let us approximate the concept
46 47 48 49 50
.
51 52
It follows from Equations (12) and (13) that their membership functions ψ
b
∧
∨
( A ) and ψ b ( A ) of A are
53 54 55
ψ b ∨( A) = ∧
ψ b ( A) =
60 61
30
43
b
55 56
29
42
52 53
28
41
and
50 51
27
38
(12)
b ∈U
45
48
26
39
ψ b ⊥ ( A )(a) = ∧ { A (b)⊥(1 − ψ b (a, b))},
43
47
20
25
41
46
19
24
40
44
18
Now we discuss the fuzzy rough set analysis of fuzzy relational structures by bisimulations. In essence, fuzzy relations of (ϕi )i ∈ I with regard to fuzzy relational structures can be understood as the knowledge that has been acquired. The notion of largest bisimulations collects the available knowledge represented by (ϕi )i ∈ I to the greatest extent. It is known that equivalence relations are used to model indiscernibility in the original model of rough sets. In this section, largest bisimulations are used to model indiscernibility of fuzzy relational structures. Based on largest bisimulations, the lower and upper fuzzy approximations are given, which can be used to deduce the unavailable knowledge represented by fuzzy sets with respect to fuzzy relational structures. Formally, for the largest bisimulation ψ b , (U , ψ b ) is a fuzzy approximation space, which is induced by (U , (ϕi )i ∈ I ). Based on ψ b , the lower and upper fuzzy approximations of a target fuzzy set over (U , ψ b ) can be achieved. In the literature, there are several kinds of fuzzy approximation operators (see, for example, [1,28–30,38,43–47,12,13,20,35]). This paper considers the lower and upper fuzzy approximation operators proposed by Hu et al. [20]. The details are as follows.
39
42
16
21
.
24
27
15 17
22
26
9
12
since the equality relation is the only bisimulation fuzzy equivalence relation on U .
21
25
8
11
17 18
7
10
,
14 15
3 5
12 13
2 4
k∈N
whose all members are different. We have that the largest bisimulation fuzzy equivalence relation ψ b is the equality relation on U , i.e.,
10
1
respectively.
0.5 a 0.8 a
+ +
0.4 b 0.5 b
,
56 57 58
,
59 60 61
JID:IJA AID:8202 /FLA
1
[m3G; v1.235; Prn:6/04/2018; 13:56] P.10 (1-10)
Y. Du, P. Zhu / International Journal of Approximate Reasoning ••• (••••) •••–•••
10
6. Conclusion
2 3 4 5 6 7 8
2
In this research, we focus on fuzzy relational structures which are the generalizations of relational structures in fuzzy environment. This paper considers the concept of bisimulations as indiscernibility with respect to fuzzy relational structures. Using bisimulations, we collect the knowledge represented by fuzzy relations in fuzzy relational structures. Based on bisimulations, the lower and upper approximations are given to deduce knowledge hidden in fuzzy relational structures. Moreover, we have given an effective method of computing the largest bisimulation, which is applicable to fuzzy relational structures over an arbitrary complete residuated lattice.
9 10
13
18 19 20 21 22 23 24 25
28 29
[12] [13] [14]
30 31 32 33 34 35
[15] [16] [17] [18] [19] [20]
36 37 38
[21] [22] [23]
39 40 41 42 43 44 45 46 47 48
[24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34]
49 50 51 52 53 54 55 56 57 58 59 60 61
6 7 8 10 12 13 15 16
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
26 27
5
14
References
16 17
4
11
The authors would like to thank the anonymous referees for their valuable comments and helpful suggestions. The work is supported by the National Natural Science Foundation of China under Grants 61672107 and 61772035.
14 15
3
9
Acknowledgements
11 12
1
[35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48]
D. Boixader, J. Jacas, J. Recasens, Upper and lower approximations of fuzzy sets, Int. J. Gen. Syst. 29 (1999) 555–568. P. Buchholz, Bisimulation relations forweighted automata, Theor. Comput. Sci. 393 (2008) 109–123. R. Bˇelohlávek, Fuzzy Relational Systems: Foundations and Principles, Kluwer, New York, 2002. R. Bˇelohlávek, V. Vychodil, Fuzzy Equational Logic, Studies in Fuzziness and Soft Computing, Springer, Berlin, Heidelberg, 2005. Y. Cao, G. Chen, E. Kerre, Bisimulations for fuzzy-transition systems, IEEE Trans. Fuzzy Syst. 19 (3) (2011) 540–552. Y. Cao, S. Sun, H. Wang, G. Chen, A behavioral distance for fuzzy-transition systems, IEEE Trans. Fuzzy Syst. 21 (4) (2013) 735–747. T. Chen, T. Han, Y. Cao, Polynomial-time algorithms for computing distances of fuzzy transition systems, arXiv preprint, arXiv:1701.06644. ´ c, ´ A. Stamenkovic, ´ J. Ignjatovic, ´ T. Petkovic, ´ Factorization of fuzzy automata, in: Proc. Fundament. Comput. Theor., vol. 4639, 2007, pp. 213–225. M. Ciri ´ c, ´ A. Stamenkovic, ´ J. Ignjatovic, ´ T. Petkovic, ´ Fuzzy relation equations and reduction of fuzzy automata, J. Comput. Syst. Sci. 76 (7) (2010) 609–633. M. Ciri ´ c, ´ J. Ignjatovic, ´ N. Damljanovic, ´ M. Bašic, ´ Bisimulations for fuzzy automata, Fuzzy Sets Syst. 186 (2012) 100–139. M. Ciri ´ c, ´ J. Ignjatovic, ´ I. Janˇcic, ´ N. Damljanovic, ´ Computation of the greatest simulations and bisimulations between fuzzy automata, Fuzzy Sets Syst. M. Ciri 208 (2012) 22–42. D. Dubois, H. Prade, Rough fuzzy sets and fuzzy rough sets, Int. J. Gen. Syst. 17 (1990) 191–209. L. D’eer, N. Verbiest, C. Cornelis, L. Godo, A comprehensive study of implicator-conjunctor-based and noise-tolerant fuzzy rough sets: definitions, properties and robustness analysis, Fuzzy Sets Syst. 275 (2015) 1–38. Y. Deng, H. Wu, Modal characterisations of probabilistic and fuzzy bisimulations, in: International Conference on Formal Engineering Methods, vol. 8829, Springer, 2014. Y. Deng, H. Wu, Logical characterizations of simulation and bisimulation for fuzzy transition systems, Fuzzy Sets Syst. 301 (6) (2016) 19–36. A. Dovier, C. Piazza, A. Policriti, An efficient algorithm for computing bisimulation equivalence, Theor. Comput. Sci. 311 (2004) 221–256. T.F. Fan, Rough set analysis of relational structures, Inf. Sci. 221 (2013) 230–244. T.F. Fan, Fuzzy bisimulation for Gödel modal logic, IEEE Trans. Fuzzy Syst. 23 (6) (2015) 2387–2396. R. Gentilini, C. Piazza, A. Policriti, From bisimulation to simulation: coarsest partition problems, J. Autom. Reason. 31 (2003) 73–103. B.Q. Hu, Z.H. Huang, (⊥, )-generalized fuzzy rough sets based on fuzzy composition operations, in: B.-y. Cao, C.-y. Zhang, T.-f. Li (Eds.), Fuzzy Information and Engineering, in: Adv. Soft Comput., vol. 54, Springer-Verlag, Berlin, Heidelberg, 2009, pp. 647–659. ´ c, ´ S. Bogdanovic, ´ M. Ciri ´ Fuzzy homomorphisms of algebras, Fuzzy Sets Syst. 160 (16) (2009) 2345–2365. J. Ignjatovic, G.J. Klir, B. Yuan, Fuzzy Sets and Fuzzy, Logic Theory and Application, Prentice-Hall, Englewood Cliffs, NJ, 1995. C. Liau, T. Lin, Reasoning about relational granulation in modal logics, in: Proceedings of the First IEEE International Conference on Granular Computing, IEEE Press, 2005, pp. 534–538. N. Lynch, F. Vaandrager, Forward and backward simulations: part I. Untimed systems, Inf. Comput. 121 (1995) 214–233. R. Milner, A calculus of communicating systems, in: G. Goos, J. Hartmanis (Eds.), Lect. Notes Comput. Sci., vol. 92, Springer, 1980. R. Milner, Communication and Concurrency, Prentice-Hall International, 1989. R. Milner, Communicating and Mobile Systems: The π -Calculus, Cambridge University Press, Cambridge, 1999. J.S. Mi, Y. Leung, H.Y. Zhao, T. Feng, Generalized fuzzy rough sets determined by a triangular norm, Inf. Sci. 178 (2008) 3203–3213. J.S. Mi, W.X. Zhang, An axiomatic characterization of a fuzzy generalization of rough sets, Inf. Sci. 160 (2004) 235–249. N.N. Morsi, M.M. Yakout, Axiomatics for fuzzy rough sets, Fuzzy Sets Syst. 100 (1998) 327–342. Z. Pawlak, Rough sets, Int. J. Comput. Inf. Sci. 11 (5) (1982) 341–356. Z. Pawlak, Rough Set: Theoretical Aspects of Reasoning About Data, Kluwer, Norwell, MA, 1991. Z. Pawlak, A. Skowron, Rudiments of rough sets, Inf. Sci. 177 (2007) 3–27. D. Park, Concurrency and automata on infinite sequences, in: P. Deussen (Ed.), Proc. 5th GI Conf., Karlsruhe, Germany, in: Lect. Notes Comput. Sci., vol. 104, Springer-Verlag, 1981, pp. 167–183. D.W. Pei, A generalized model of fuzzy rough sets, Int. J. Gen. Syst. 34 (2005) 603–613. ´ Congruences and homomorphisms of fuzzy automata, Fuzzy Sets Syst. 157 (2006) 444–458. T. Petkovic, M. Roggenbach, M. Majster-Cederbaum, Towards a unified view of bisimulation: a comparative study, Theor. Comput. Sci. 238 (2000) 81–130. A.M. Radzikowska, E.E. Kerre, A comparative study of fuzzy rough sets, Fuzzy Sets Syst. 126 (2002) 137–155. ´ c, ´ J. Ignjatovic, ´ M. Ciri ´ Reduction of fuzzy automata by means of fuzzy quasi-orders, Inf. Sci. 275 (2014) 168–198. A. Stamenkovic, J. Scott, Social Network Analysis: A Handbook, second ed., SAGE Publications, 2000. D. Sangiorgi, On the origins of bisimulation and coinduction, ACM Trans. Program. Lang. Syst. 31 (4) (2009) 111–151. D.D. Sun, Y.M. Li, W.W. Yang, Bisimulation relations for fuzzy finite automata, Fuzzy Syst. Math. 23 (2009) 92–100. W.Z. Wu, J.S. Mi, W.X. Zhang, Generalized fuzzy rough sets, Inf. Sci. 151 (2003) 263–282. W.Z. Wu, W.X. Zhang, Constructive and axiomatic approaches of fuzzy approximation operators, Inf. Sci. 159 (2004) 233–254. W.Z. Wu, Y. Leung, J.S. Mi, On characterizations of (I,T)-fuzzy rough approximation operators, Fuzzy Sets Syst. 154 (2005) 76–102. Y.Y. Yao, A comparative study of fuzzy sets and rough sets, Inf. Sci. 109 (1998) 227–242. W. Zhu, F.Y. Wang, Reduction and axiomization of covering generalized rough sets, Inf. Sci. 152 (2003) 217–230. P. Zhu, H.Y. Xie, Q.Y. Wen, Rough approximations based on bisimulations, Int. J. Approx. Reason. 81 (2017) 49–62.
17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61
[m3G; v1.235; Prn:6/04/2018; 13:56] P.1 (1-10)
International Journal of Approximate Reasoning ••• (••••) •••–•••
1
Contents lists available at ScienceDirect
2
1 2
3
International Journal of Approximate Reasoning
4 5
3 4 5
6
6
www.elsevier.com/locate/ijar
7
7
8
8
9
9
10
10
11 12 13 14 15
Fuzzy approximations of fuzzy relational structures a b
17
c
14
Automation School, Beijing University of Posts and Telecommunications, 100876 Beijing, China University of Foreign Languages, 471003 Luoyang, Henan, China School of Science, Beijing University of Posts and Telecommunications, 100876 Beijing, China
18
23 24 25
i n f o
28 29 30 31 32
17
a b s t r a c t
20 21
Article history: Received 6 January 2017 Received in revised form 31 January 2018 Accepted 4 April 2018 Available online xxxx
26 27
16
19
a r t i c l e
21 22
15
18
19 20
12 13
Yibin Du a,b,∗ , Ping Zhu c
16
11
Keywords: Fuzzy relational structure Bisimulation Approximation method Rough set Fuzzy set Complete residuated lattice
In social network analysis, relational structures play a crucial role in the processing of complex data. This paper proposes a fuzzy relational structure which consists of a non-empty universal set and a set of fuzzy relations of finite arity. In order to deduce knowledge hidden in a fuzzy relational structure we present the concept of bisimulations as indiscernibility with respect to the fuzzy relational structure. Furthermore, we give a method of computing the largest bisimulations over fuzzy relational structures and present computational examples of the method. Finally, the fuzzy rough set analysis of fuzzy relational structures is discussed. © 2018 Elsevier Inc. All rights reserved.
22 23 24 25 26 27 28 29 30 31 32
33
33
34
34
35 36
1. Introduction
37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54
59 60 61
38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55
56 58
36 37
The theory of rough sets was firstly proposed by Pawlak in 1982 [31–33]. It is a useful mathematical tool for dealing with attribute data. In this theory, attribute data can be represented by a data table which comprises a set of objects and a set of attributes. The objects of a given data table can be identified within the limits of imprecise data by an indiscernibility relation induced by different values of attributes characterizing these objects. In other words, the indiscernibility relation induced by different values of attributes enables us to characterize a set of objects by a pair of sets, called the lower and upper approximation of the set of objects. Although databases only contain attribute information about objects in many practical cases, data about the relationships between objects has become increasingly important in decision analysis in recent years [17]. A remarkable example can be found in social network analysis, where the principal types of data are attribute data and relational data [40]. Liau et al. [23] studied granulation based on relational information between objects from the viewpoint of modal logic. More recently, Fan [17] further investigated relational information systems and extended rough set analysis from attribute information systems to relational structures which consists of a non-empty universal set and a set of relations. In this paper we pay attention to the generalization of relational structures in fuzzy environment and present bisimulations as indiscernibility with respect to fuzzy relational structures. Milner [25] and Park [34] introduced bisimulation as a concept that provides an effective method for reducing the complexity of concurrent processes and studying the equivalence of automata. In particular, bisimulation is exploited to reduce the state space of a system by combining bisimilar states. In the last thirty years, it has been applied to computer science, modal logic and set theory [16,19,24,26,27,37,41,14,18]. Recently, bisimulations have been introduced to fuzzy systems by
55 57
35
56
*
Corresponding author at: Automation School, Beijing University of Posts and Telecommunications, 100876 Beijing, China. E-mail addresses: [email protected] (Y. Du), [email protected] (P. Zhu).
https://doi.org/10.1016/j.ijar.2018.04.003 0888-613X/© 2018 Elsevier Inc. All rights reserved.
57 58 59 60 61
JID:IJA AID:8202 /FLA
[m3G; v1.235; Prn:6/04/2018; 13:56] P.2 (1-10)
Y. Du, P. Zhu / International Journal of Approximate Reasoning ••• (••••) •••–•••
2
1 2 3 4 5 6 7 8 9 10 11
two general approaches. One is based on a binary relation on the state space of a fuzzy system such that related states have exactly the same possibility degree of making a transition into every class of related states [36,2,42,5,6,48,7,15]. The other ´ c´ and his colleagues [8,21,9,10,39,11] proposed two types of simulations is based on a fuzzy relation on the state space. Ciri and four types of bisimulations for fuzzy automata. The purpose of the paper is to discuss the fuzzy rough set analysis of fuzzy relational structures by bisimulations which are similar to forward bisimulations considered in [10]. The rest of this paper is structured as follows. In Section 2 we introduce necessary terminology and notions of residuated lattices, fuzzy sets, and n-ary fuzzy relations and extend the concept of composition from binary fuzzy relations to n-ary fuzzy relations. Sections 3 presents a fuzzy relational structure which consists of a non-empty universal set and a set of fuzzy relations. In Section 4 we give main results on the computing of the largest bisimulations over fuzzy relational structures. Examples presented in Section 5 demonstrate the application of our method. Section 6 concludes this work.
12 13
16
19 20
25 26 27 28 29
5 6 7 8 9 10 11
15 16 17
(1) ( L , ∧, ∨, 0, 1) is a lattice with the least element 0 and the greatest element 1, (2) ( L , ⊗, 1) is a commutative monoid with the unit 1, (3) ⊗ and → form an adjoint pair, i.e., they satisfy the adjunction property: for all a, b, c ∈ L,
18 19 20 21
a ⊗ b ≤ c ⇔ a ≤ b → c.
22 24
4
14
For the terminology and basic notions in this section we refer to [10,3,4,22]. A residuated lattice is an algebra L = ( L , ∧, ∨, ⊗, →, 0, 1) such that
21 23
3
13
17 18
2
12
2. Preliminaries
14 15
1
22
In addition, a residuated lattice is called complete if ( L , ∧, ∨, 0, 1) is a complete lattice. If any finitely generated subalgebra of residuated lattice L is finite, then L is called locally finite. For any family ai , i ∈ I , of elements of L, we write ∨ ai for the supremum of {ai }i ∈ I , and ∧ ai for the infimum. An operai∈ I
i∈ I
tion ↔ is defined by a ↔ b = (a → b) ∧ (b → a). It can be verified that ⊗ is isotonic in both arguments and for all a, b, c ∈ L and {ai }i ∈ I , {b i }i ∈ I ⊆ L the following hold:
23 24 25 26 27 28 29
( ∨ ai ) ⊗ a = ∨ (ai ⊗ a),
(1)
32
(a ↔ a) = 1, (a ↔ b) = (b ↔ a), (a ↔ b) ⊗ (b ↔ c ) ≤ (a ↔ c ),
(2)
32
33
a ↔ b ≤ a ⊗ c ↔ b ⊗ c,
(3)
33
∧ (ai ↔ bi ) ≤ ( ∨ ai ) ↔ ( ∨ bi ).
(4)
30 31
34 35 36 37 38 39 40 41 42 43
i∈I
i∈I
46 47 48 49 50 51 52 53 54 55 56 57
i∈I
A=
A (a1 ) a1
+
A (a2 ) a2
i∈ I
of { A i }i ∈ I is defined as ( ∧ A i )(a) = ∧ A i (a), for all a ∈ U , the join ∨ A i of { A i }i ∈ I is defined as ( ∨ A i )(a) = ∨ A i (a), for all i∈ I
i∈ I
i∈ I
a ∈ U , and the product A ⊗ B is a fuzzy set defined by ( A ⊗ B )(a) = A (a) ⊗ B (a), for all a ∈ U . A fuzzy set in U 1 × · · · × U n is called an n-ary fuzzy relation between U 1 , U 2 , · · · , and U n . For simplicity, we call an n-ary fuzzy relation as a fuzzy relation. If U 1 = · · · = U n = U , we speak of n-ary fuzzy relation on a set U . The set of all n-ary n fuzzy relations on U will be denoted by L U . 2 2 For ϕ , ψ ∈ L U , the composition ϕ ◦ ψ is a fuzzy relation from L U defined by
38 39 40 41 42 43
47 48 49 50 51 52 53 54 55 56 57 58
(ϕ ◦ ψ)(a, c ) = ∨ ϕ (a, b) ⊗ ψ(b, c ), b ∈U
60 61
37
46
A (a )
i∈ I
36
45
an
where the term a i and i = 1, · · · , n, signifies that A (ai ) is the degree of membership of ai in A and the plus sign i represents the union. Let A, B and A i ∈ L U (i ∈ I ). A and B are said to be equal, which is denoted by A = B, if A (a) = B (a) for all a ∈ U . The inclusion of A and B is defined by A ≤ B if A (a) ≤ B (a) for all a ∈ U . Further, given this partial order the meet ∧ A i i∈ I
35
44
A (an )
58 59
34
i∈I
+ ··· +
30 31
Let L be the real unit interval [0, 1], a ∧ b = min{a, b} and a ∨ b = max{a, b}. Suppose that a ⊗ b = min{a, b} and a → b = 1 if a ≤ b, and = b otherwise. Then the structure of a residuated lattice with assumptions above is called Gödel structure. The structure where a ⊗ b = a · b and a → b = 1 if a ≤ b, and = b/a otherwise is the Goguen (product) structure. From now on, L = ( L , ∧, ∨, ⊗, →, 0, 1) will be a complete residuated lattice. Let U be a non-empty set. A fuzzy set A in U is a mapping A : U → L. The set U is called a universe. A (a) is called the degree of membership of a in A. The set of all fuzzy sets in U will be denoted by L U . The support of a fuzzy set A is a set defined as supp ( A ) = {a ∈ U : A (a) > 0}. If supp ( A ) = {a1 , · · · , an } is a finite set, then we often use Zadeh’s notation
44 45
i∈I
for all (a, c ) ∈ U 2 .
59 60 61
JID:IJA AID:8202 /FLA
[m3G; v1.235; Prn:6/04/2018; 13:56] P.3 (1-10)
Y. Du, P. Zhu / International Journal of Approximate Reasoning ••• (••••) •••–•••
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
2
For ϕ ∈ L U , the inverse of ϕ is a fuzzy relation 2 for all ϕ1 , ϕ2 , and ϕi ∈ L U (i ∈ I ), we have that
25 26 27 28
ϕ −1 ∈ L U defined by ϕ −1 (b, a) = ϕ (a, b), for all a, b ∈ U . Furthermore, 2
3
(5)
( ∨ ϕi )−1 = ∨ ϕi−1 .
(6)
i∈I
i∈I
A binary fuzzy relation
31 32 33
10 11 12
k1 +k
a (k1 + k)-ary fuzzy relation where ϕ ∈ L U , and let ψ be a (k + k2 )-ary fuzzy relation where ψ ∈ L U k-composition of ϕ and ψ is a (k1 + k2 )-ary fuzzy relation defined by
∨
b=(b1 ,b2 ,··· ,bk )∈U k
ϕ (a, b) ⊗ ψ(b, c),
k+k2
. Then, the
38 39 40 41
46 47 48 49
52 53 54 55 56 57 58 59 60 61
16 17 18 19 21
(7)
(8)
The details are as follows. By the definition of compositions, for all (a, d) ∈ U k1 +k4 where a = (a1 , a2 , · · · , ak1 ) and d = (d1 , d2 , · · · , dk4 ) we have
22 24 25 26 27 28
= = = = =
∨
c=(c 1 ,c 2 ,··· ,ck3 )∈U k3
∨
c=(c 1 ,c 2 ,··· ,ck3 )∈U
∨
k3
31 32 33 35
(ϕ1 ◦k2 ϕ2 )(a, c) ⊗ ϕ3 (c, d)
36
{
38
∨
b=(b1 ,b2 ,··· ,bk2 )∈U
{
∨
k2
c=(c 1 ,c 2 ,··· ,ck3 )∈U k3 b=(b1 ,b2 ,··· ,bk2 )∈U k2
ϕ1 (a, b) ⊗ ϕ2 (b, c)} ⊗ ϕ3 (c, d)
40 41 42
ϕ1 (a, b) ⊗ {
∨
ϕ1 (a, b) ⊗ (ϕ2 ◦k3 ϕ3 )(b, d)
b=(b1 ,b2 ,··· ,bk2 )∈U k2
39
ϕ1 (a, b) ⊗ ϕ2 (b, c) ⊗ ϕ3 (c, d)}
∨
b=(b1 ,b2 ,··· ,bk2 )∈U k2
37
∨
c=(c 1 ,c 2 ,··· ,ck3 )∈U k3
ϕ2 (b, c) ⊗ ϕ3 (c, d)}
43 44 45 46 47
= (ϕ1 ◦k2 (ϕ2 ◦k3 ϕ3 ))(a, d). Next, let
30
34
((ϕ1 ◦k2 ϕ2 ) ◦k3 ϕ3 )(a, d)
48
ϕ be a binary fuzzy relation, and let ψ be a k-ary fuzzy relation. Then we have
49
50 51
15
29
44 45
14
23
(ϕ1 ◦k2 ϕ2 ) ◦k3 ϕ3 = ϕ1 ◦k2 (ϕ2 ◦k3 ϕ3 ).
42 43
13
20
for all (a, c) ∈ U k1 +k2 where a = (a1 , a2 , · · · , ak1 ) and c = (c 1 , c 2 , · · · , ck2 ). When k = 1, omit the subscript and write the composition as ϕ ◦ ψ . For the concept of k-composition, we have the following properties. Let ϕ1 be a (k1 + k2 )-ary fuzzy relation, let ϕ2 be a (k2 + k3 )-ary fuzzy relation, and let ϕ3 be a (k3 + k4 )-ary fuzzy relation. Then, by the definition of k-composition we have
35 37
6
9
If ϕ is reflexive and transitive, then ϕ ◦ ϕ = ϕ . A binary fuzzy relation on U which is reflexive, symmetric and transitive is called a fuzzy equivalence relation. The set of all fuzzy equivalence relations on U is denoted by E (U ). Given the ordering of binary fuzzy relations, E (U ) is a complete lattice, in which the meet coincide with the ordinary meet of binary fuzzy relations. For later need, we extend the concept of composition from binary fuzzy relations to n-ary fuzzy relations. Let ϕ be
34 36
5
8
29 30
4
7
ϕ on U is
(1) reflexive if ϕ (a, a) = 1, for every a ∈ U ; (2) symmetric if ϕ (a, b) = ϕ (b, a), for all a, b ∈ U ; (3) transitive if ϕ (a, b) ⊗ ϕ (b, c ) ≤ ϕ (a, c ), for all a, b, c ∈ U .
(ϕ ◦k ψ)(a, c) =
1 2
(ϕ1 ◦ ϕ2 )−1 = ϕ2−1 ◦ ϕ1−1 ,
23 24
3
50
(ϕ ◦ ψ) ◦ ϕ = ϕ ◦ (ψ ◦ ϕ ).
(9)
In fact, Equation (9) is a special case of Equation (8). Further, let ϕ0 be a (k1 + k2 )-ary fuzzy relation, let k3 )-ary fuzzy relations, and let ϕ3 be a (k3 + k4 )-ary fuzzy relation. Then we have that
ϕ1 ≤ ϕ2 implies ϕ0 ◦k2 ϕ1 ≤ ϕ0 ◦k2 ϕ2 and ϕ1 ◦k3 ϕ3 ≤ ϕ2 ◦k3 ϕ3 .
ϕ1 , ϕ2 be (k2 +
i∈I
i∈I
i∈I
i∈I
52 53 54
(10)
Finally, let ϕ , ϕi (i ∈ I ) be (k1 + k2 )-ary fuzzy relations, and let ψ , ψi (i ∈ I ) be (k2 + k3 )-ary fuzzy relations. Then we can easily verify that
ϕ ◦k2 ( ∨ ψi ) = ∨ (ϕ ◦k2 ψi ), ( ∨ ϕi ) ◦k2 ψ = ∨ (ϕi ◦k2 ψ).
51
(11)
55 56 57 58 59 60 61
JID:IJA AID:8202 /FLA
[m3G; v1.235; Prn:6/04/2018; 13:56] P.4 (1-10)
Y. Du, P. Zhu / International Journal of Approximate Reasoning ••• (••••) •••–•••
4
1
3. Fuzzy relational structures
1
2 3 4 5
2
In order to deal with relational data, Fan [17] considers the rough set analysis of relational structures. In this section, we attempt to conduct a further study along this line. We present fuzzy relational structures as basic models for some fuzzy approximation spaces and state the key definition of bisimulations for fuzzy relational structures.
6 7 8 9
12 13 14
Definition 3.1. A fuzzy relational structure is a tuple A = (U , (ϕi )i ∈ I ), where U is the universe of the structure; I is an index set; and for each i ∈ I , ϕi is a fuzzy relation on U .
17 18
k
ϕi for each i ∈ I are finite. For each fuzzy relation ϕi there exists a natural
number k in N such that ϕi ∈ L U . A fuzzy relational structure whose universe and set of fuzzy relations (ϕi )i ∈ I are finite is called a fuzzy finite relational structure. Example 3.1. Let L be the Gödel structure, A = (U , (ϕ1 , ϕ2 )) a fuzzy relational structure over L with U = {a, b}, and the binary fuzzy relation ϕ1 , the ternary fuzzy relation ϕ2 given by
ϕ1 =
20 21
0.5
(a, a)
+
0.5
(b, a)
,
ϕ2 =
0.8
(a, a, b)
+
0.6
(a, b, b)
+
0.5
(b, a, a)
+
0.5
25 26 27 28 29 30 31 32
(b, b, a)
Definition 3.2. Let A = (U , (ϕi )i ∈ I ) be a fuzzy relational structure. A binary fuzzy relation ψ is called a bisimulation on U if for every ϕi , the following hold:
43 44
47 48
51
18
24 25 26 27 28 29
33
I U (a, b) =
34 35
1 if a = b 0 other wise .
36 37 38
Thus, the set of all bisimulations on U is non-empty. The concept of bisimulations is similar to the concept of forward bisimulations which is considered in [10]. For any bisimulation ψ , (U , ψ) is a fuzzy approximation space. Then based on bisimulation the lower and upper fuzzy approximations of a target fuzzy set over (U , ψ) are given, which can be used to deduce the unavailable knowledge represented by fuzzy sets with respect to fuzzy relational structures. Now we are ready to state and prove some properties of bisimulations.
39 40 41 42 43 44 45
Proposition 3.1. Let A = (U , (ϕi )i ∈ I ) be a fuzzy relational structure. Suppose that ψ1 , ψ2 ∈ L
U2
2
are bisimulations on U and
{ψ j } j ∈ J ⊆ L U is a family of bisimulations on U . Then the fuzzy relations ψ1 ◦ ψ2 and ∨ ψ j are bisimulations on U . j∈ J
Proof. According to Definition 3.2 and Equations (5), (8) and (10), for each i ∈ I we have that
52
(ψ1 ◦ ψ2 ) ◦ ϕi = ψ1 ◦ (ψ2 ◦ ϕi )
53 54 55 56 57 58 59 60 61
17
32
It is easy to see that ψ −1 is also a bisimulation on U . A simple example of a bisimulation is the equality relation on U
49 50
16
31
45 46
14
30
(1) ψ −1 ◦ ϕi ≤ ϕi ◦ ψ −1 ; (2) ψ ◦ ϕi ≤ ϕi ◦ ψ .
38
42
13
22
The fuzzy approximation space is the basic construction of fuzzy rough set theory. Given a fuzzy relational structure A = (U , (ϕi )i ∈ I ), we can derive its fuzzy approximation spaces based on bisimulations over A .
37
41
12
23
36
40
11
21
respectively.
35
39
9
20
33 34
8
19
,
23 24
7
15
19
22
5
10
This paper assumes that the index set I and
15 16
4 6
10 11
3
and
46 47 48 49 50 51 52 53
≤ ψ1 ◦ (ϕi ◦ ψ2 )
54
= (ψ1 ◦ ϕi ) ◦ ψ2
56
55
≤ (ϕi ◦ ψ1 ) ◦ ψ2
57
= ϕi ◦ (ψ1 ◦ ψ2 ),
59
58 60 61
JID:IJA AID:8202 /FLA
[m3G; v1.235; Prn:6/04/2018; 13:56] P.5 (1-10)
Y. Du, P. Zhu / International Journal of Approximate Reasoning ••• (••••) •••–•••
1 2
(ψ1 ◦ ψ2 )−1 ◦ ϕi = (ψ2−1 ◦ ψ1−1 ) ◦ ϕi
1 2
= ψ2−1 ◦ (ψ1−1 ◦ ϕi )
3
3
≤ ψ2−1 ◦ (ϕi ◦ ψ1−1 )
4 5
4 5
= (ψ2−1 ◦ ϕi ) ◦ ψ1−1
6
6
≤ (ϕi ◦ ψ2−1 ) ◦ ψ1−1
7 8
7 8
= ϕi ◦ (ψ2−1 ◦ ψ1−1 )
9
−1
= ϕi ◦ (ψ1 ◦ ψ2 )
10 11
5
9 10
.
12
Thus ψ1 ◦ ψ2 is a bisimulation on U . Analogously, by Definition 3.2 and Equations (6) and (11) it holds that ∨ ψ j is a
13
bisimulation on U .
2
14 15 16 17 18 19 20 21 22 23 24 25
j∈ J
28
Since the set of all bisimulations on U is non-empty, we have that the union of all bisimulations gives rise to the largest bisimulation. Furthermore, it can be proved that the largest bisimulation is a fuzzy equivalence relation. Proposition 3.2. Let A = (U , (ϕi )i ∈ I ) be a fuzzy relational structure. Then the largest bisimulation on U is a fuzzy equivalence relation. 2
Proof. First, let {ψ j } j ∈ J ⊆ L U be the set of all bisimulations on U . Then we have that the largest bisimulation is ∨ ψ j . j∈ J
1 −1 Third, by ψ − j ∈ {ψ j } j ∈ J and Proposition 3.1, we derive ∨ ψ j = ∨ (ψ j ∨ ψ j ). Then for all a, b ∈ U , j∈ J
33
41 42 43 44 45 46 47 48 49
52 53
35
∨ ψ j ))(b, a)
36 37 38 j∈ J
j∈ J
j∈ J
j∈ J
j∈ J
( ∨ ψ j ) ◦ ( ∨ ψ j )(a, c ) = ∨ (( ∨ ψ j )(a, b) ⊗ ( ∨ ψ j )(b, c )) ≤ ( ∨ ψ j )(a, c ) j∈ J
j∈ J
b ∈U
j∈ J
j∈ J
58
j∈ J
j∈ J
61
41 42
45 46 47
2
48 49 50
Remark 3.1. Because of symmetry, we have that a fuzzy equivalence relation ψ on U is a bisimulation on U if and only if ψ ◦ ϕi ≤ ϕi ◦ ψ for every ϕi . It is worthy to note that the largest bisimulation as an indiscernibility relation reflects the classification ability of the fuzzy relational structure to the greatest extent.
51 52 53 54
4. Computation of the largest bisimulation
55 56
We will provide a method for computing the largest bisimulations over fuzzy relational structures adapting the method developed in [9]. Now preparing for the method, we are ready to prove the following results.
59 60
40
44
( ∨ ψ j )(a, b) ⊗ ( ∨ ψ j )(b, c ) ≤ ( ∨ ψ j )(a, c ) for all a, b, c ∈ U . Thus we obtain Proposition 3.2.
39
43
for all a, c ∈ U . Then we have
56 57
25
34
54 55
24
33
50 51
23
32
Last, according to Proposition 3.1, we have ( ∨ ψ j ) ◦ ( ∨ ψ j ) ∈ {ψ j } j ∈ J . Then ( ∨ ψ j ) ◦ ( ∨ ψ j ) ≤ ∨ ψ j . It means that
j∈ J
22
1 (b, a) ∨ ψ j (b, a)) = ∨ (ψ − j
j∈ J
j∈ J
21
30
= ( ∨ ψ j )(b, a).
38
20
31
j∈ J
37
19
= ∨ (ψ j (a, b) ∨ ψ j (b, a))
= ( ∨ (ψ j
36
18
29
−1
35
17
= ∨ (ψ j (a, b) ∨ ψ j (a, b))
j∈ J
34
16
28
j∈ J
32
15
27
j∈ J
j∈ J
31
13
26
−1
30
40
j∈ J
1 ( ∨ ψ j )(a, b) = ( ∨ (ψ j ∨ ψ − j ))(a, b )
29
39
j∈ J
Second, from I U ∈ {ψ j } j ∈ J , we can easily conclude that ( ∨ ψ j )(a, a) = 1 for every a ∈ U .
j∈ J
12 14
26 27
11
57 58 59
Theorem 4.1. Let A = (U , (ϕi )i ∈ I ) be a fuzzy relational structure and ψ a fuzzy equivalence relation on U . Then the following conditions are equivalent:
60 61
JID:IJA AID:8202 /FLA
[m3G; v1.235; Prn:6/04/2018; 13:56] P.6 (1-10)
Y. Du, P. Zhu / International Journal of Approximate Reasoning ••• (••••) •••–•••
6
1 2 3
(1) ψ ◦ ϕi ≤ ϕi ◦ ψ , for every ϕi ; (2) ψ ◦ ϕi ◦ ψ = ϕi ◦ ψ , for every ϕi ; (3) If ϕi is a k-ary fuzzy relation, we suppose that
4
1 2 3 4
ψ i (a, b) =
5
∧
5
6
a2 ,a3 ,··· ,ak ∈U
6
7
(ϕi ◦ ψ)(a, a2 , · · · , ak−1 , ak ) ↔ (ϕi ◦ ψ)(b, a2 , · · · , ak−1 , ak ).
7
8 9 10
15 16 17 18 19
10 11
i∈I
12 14
9
ψ(a, b) ≤ ∧ ψ i (a, b).
11 13
8
Then for all a, b ∈ U we have
12
Proof. (1) ⇔ (2). If ψ ◦ ϕi ◦ψ = ϕi ◦ψ for every ϕi , then ψ ◦ ϕi = ψ ◦ ϕi ◦ I U ≤ ψ ◦ ϕi ◦ψ = ϕi ◦ψ . Conversely, if ψ ◦ ϕi ≤ ϕi ◦ψ then ψ ◦ ϕi ◦ ψ ≤ ϕi ◦ ψ ◦ ψ = ϕi ◦ ψ , and ϕi ◦ ψ = I U ◦ ϕi ◦ ψ ≤ ψ ◦ ϕi ◦ ψ , we conclude that ψ ◦ ϕi ◦ ψ = ϕi ◦ ψ . (2) ⇒ (3). Suppose that ψ ◦ ϕi ◦ ψ = ϕi ◦ ψ for every ϕi . Then by the Equation (9), for all b, a, a2 , · · · , ak−1 , ak ∈ U we have that
ψ(a, b) ⊗ (ϕi ◦ ψ)(b, a2 , · · · , ak−1 , ak )
13 14 15 16 17 18 19
20
≤ (ψ ◦ (ϕi ◦ ψ))(a, a2 , · · · , ak−1 , ak )
20
21
= (ψ ◦ ϕi ◦ ψ)(a, a2 , · · · , ak−1 , ak )
21
= (ϕi ◦ ψ)(a, a2 , · · · , ak−1 , ak ),
23
22 23 24 25 26 27 28 29 30 31 32
24
and by the adjunction property we obtain that
ψ(a, b) ≤ (ϕi ◦ ψ)(b, a2 , · · · , ak−1 , ak ) → (ϕi ◦ ψ)(a, a2 , · · · , ak−1 , ak ).
35 36 37
ψ(a, b) = ψ(b, a) ≤ (ϕi ◦ ψ)(a, a2 , · · · , ak−1 , ak ) → (ϕi ◦ ψ)(b, a2 , · · · , ak−1 , ak ),
ψ(a, b) ≤ (ϕi ◦ ψ)(a, a2 , · · · , ak−1 , ak ) ↔ (ϕi ◦ ψ)(b, a2 , · · · , ak−1 , ak ). Since it is satisfied for all a2 , a3 , · · · , ak ∈ U and ϕi (i ∈ I ), we conclude that the condition (3) holds. (3) ⇒ (2). If the condition (3) holds, then for all b, a, a2 , · · · , ak−1 , ak ∈ U and ϕi (i ∈ I ) we have that
46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61
31
34 35 36 37 38
ψ(a, b) ≤ (ϕi ◦ ψ)(a, a2 , · · · , ak−1 , ak ) ↔ (ϕi ◦ ψ)(b, a2 , · · · , ak−1 , ak ) ≤ (ϕi ◦ ψ)(b, a2 , · · · , ak−1 , ak ) → (ϕi ◦ ψ)(a, a2 , · · · , ak−1 , ak ), and by the adjunction property we have that
43 45
30
33
41
44
27
32
and hence,
40 42
26
29
38 39
25
28
By symmetry,
33 34
22
39 40 41 42 43
ψ(a, b) ⊗ (ϕi ◦ ψ)(b, a2 , · · · , ak−1 , ak ) ≤ (ϕi ◦ ψ)(a, a2 , · · · , ak−1 , ak ). Note that
44 45 46
(ψ ◦ ϕi ◦ ψ)(a, a2 , · · · , ak−1 , ak ) = (ψ ◦ (ϕi ◦ ψ))(a, a2 , · · · , ak−1 , ak ) = ∨ ψ(a, b) ⊗ (ϕi ◦ ψ)(b, a2 , · · · , ak−1 , ak ) b ∈U
48 49 50 51 52
≤ (ϕi ◦ ψ)(a, a2 , · · · , ak−1 , ak ), and hence ψ ◦ ϕi ◦ ψ ≤ ϕi ◦ ψ . Because
47
53
ϕi ◦ ψ ≤ ψ ◦ ϕi ◦ ψ , we conclude that ψ ◦ ϕi ◦ ψ = ϕi ◦ ψ for every ϕi . 2
Lemma 4.1. Let A = (U , (ϕi )i ∈ I ) be a fuzzy relational structure and ψ a fuzzy equivalence relation on U . Suppose that
ψ R (a, b) = ∧ ψ i (a, b), i∈I
for all a, b ∈ U . Then we have that ψ ≤ ψ R and ψ i (i ∈ I ) and ψ R are fuzzy equivalence relations.
54 55 56 57 58 59 60 61
JID:IJA AID:8202 /FLA
[m3G; v1.235; Prn:6/04/2018; 13:56] P.7 (1-10)
Y. Du, P. Zhu / International Journal of Approximate Reasoning ••• (••••) •••–•••
1 2 3
7
Proof. From the assertion (3) of Theorem 4.1, we get that ψ ≤ ψ R . Next, since ↔ is reflexive and symmetric, it is easily verified that ψ i (i ∈ I ) is reflexive and symmetric. Note that for all a, b, c , a2 , a3 , · · · , ak ∈ U
4
∧
a2 ,a3 ,··· ,ak ∈U
7
(ϕi ◦ ψ)(a, a2 , · · · , ak −1 , ak ) ↔ (ϕi ◦ ψ)(b, a2 , · · · , ak −1 , ak )
8
13
i
33 34
21 22 23 24 25
It follows that for all a, b, c ∈ U
26 27
i
ψ (a, b) ⊗ ψ (b, c )
30 32
20
≤ (ϕi ◦ ψ)(a, a2 , · · · , ak−1 , ak ) ↔ (ϕi ◦ ψ)(c , a2 , · · · , ak−1 , ak ).
≤
∧
a2 ,a3 ,··· ,ak ∈U
28
(ϕi ◦ ψ)(a, a2 , · · · , ak−1 , ak ) ↔ (ϕi ◦ ψ)(c , a2 , · · · , ak−1 , ak )
37 38 39
31
Thus ψ (i ∈ I ) is a fuzzy equivalence relation. Analogously, we can conclude that ψ R is a fuzzy equivalence relation.
32
i
2
42
45 46 47 48 49 50 51
Lemma 4.2. Let A = (U , (ϕi )i ∈ I ) be a fuzzy relational structure, and let ψ1 and ψ2 be fuzzy equivalence relations on U . If ψ1 ≤ ψ2 , then ψ1R ≤ ψ2R .
54
≤ (ϕi ◦ ψ1 )(a, a2 , · · · , ak−1 , ak ) ⊗ ψ2 (ak , c ) ↔ (ϕi ◦ ψ1 )(b, a2 , · · · , ak−1 , ak ) ⊗ ψ2 (ak , c ). Then, by (4) we have that for all c ∈ U ,
ψ1i (a, b)
= =
≤
59 60 61
39 41 42
≤
44 45 46 47 48 49 50
∧
a2 ,a3 ,··· ,ak ∈U
51
(ϕi ◦ ψ1 )(a, a2 , · · · , ak−1 , ak ) ↔ (ϕi ◦ ψ1 )(b, a2 , · · · , ak−1 , ak )
52
∧
{ ∧
a2 ,a3 ,··· ,ak−1 ∈U ak ∈U
(ϕi ◦ ψ1 )(a, a2 , · · · , ak−1 , ak ) ↔ (ϕi ◦ ψ1 )(b, a2 , · · · , ak−1 , ak )}
56 58
38
43
(ϕi ◦ ψ1 )(a, a2 , · · · , ak−1 , ak ) ↔ (ϕi ◦ ψ1 )(b, a2 , · · · , ak−1 , ak )
55 57
37
40
Proof. Let ϕi be a k-ary fuzzy relation and a, b ∈ U . From ψ1 ≤ ψ2 , it follows ψ1 ◦ ψ2 = ψ2 , and by (3), for all c , a2 , a3 , · · · , ak ∈ U we get that
52 53
34 36
43 44
33 35
We also have the following.
40 41
29 30
= ψ i (a, c ).
35 36
16
19
(ϕi ◦ ψ)(b, a2 , · · · , ak−1 , ak ) ↔ (ϕi ◦ ψ)(c , a2 , · · · , ak−1 , ak )
28
15
18
i
i
14
17
≤ (ϕi ◦ ψ)(a, a2 , · · · , ak−1 , ak ) ↔ (ϕi ◦ ψ)(b, a2 , · · · , ak−1 , ak ) ⊗
27
31
◦ ψ)(c , a2 , · · ·
, ak−1 , ak )
ψ (a, b) ⊗ ψ (b, c )
23
29
Then for all a, b, c , a2 , a3 , · · · , ak ∈ U
22
26
13
, ak−1 , ak ) ↔ (ϕi
≤ (ϕi ◦ ψ)(b, a2 , · · · , ak−1 , ak ) ↔ (ϕi ◦ ψ)(c , a2 , · · · , ak−1 , ak ).
20
25
9
12
a2 ,a3 ,··· ,ak ∈U
(ϕi ◦ ψ)(b, a2 , · · ·
19
24
∧
16
21
8
11
ψ i (b, c ) =
15
18
7
10
and
14
17
6
≤ (ϕi ◦ ψ)(a, a2 , · · · , ak−1 , ak ) ↔ (ϕi ◦ ψ)(b, a2 , · · · , ak−1 , ak ),
9
12
3 5
6
11
2 4
ψ i (a, b) =
5
10
1
∧
{ ∧ [(ϕi ◦ ψ1 )(a, a2 , · · · , ak−1 , ak ) ⊗ ψ2 (ak , c )
53 54 55 56 57
a2 ,a3 ,··· ,ak−1 ∈U ak ∈U
58
↔ (ϕi ◦ ψ1 )(b, a2 , · · · , ak−1 , ak ) ⊗ ψ2 (ak , c )]}
59
∧
a2 ,a3 ,··· ,ak−1 ∈U
{[ ∨ (ϕi ◦ ψ1 )(a, a2 , · · · , ak−1 , ak ) ⊗ ψ2 (ak , c )] ak ∈U
60 61
JID:IJA AID:8202 /FLA
[m3G; v1.235; Prn:6/04/2018; 13:56] P.8 (1-10)
Y. Du, P. Zhu / International Journal of Approximate Reasoning ••• (••••) •••–•••
8
↔ [ ∨ (ϕi ◦ ψ1 )(b, a2 , · · · , ak−1 , ak ) ⊗ ψ2 (ak , c )]}
1
=
3
2
∧
{((ϕi ◦ ψ1 ) ◦ ψ2 )(a, a2 , · · · , ak−1 , c )
4
a2 ,a3 ,··· ,ak−1 ∈U
5
↔ ((ϕi ◦ ψ1 ) ◦ ψ2 )(b, a2 , · · · , ak−1 , c )}
6
=
7 8
∧
a2 ,a3 ,··· ,ak−1 ∈U
=
10
4 5 6
{(ϕi ◦ (ψ1 ◦ ψ2 ))(a, a2 , · · · , ak−1 , c )
∧
7 8 9
{(ϕi ◦ ψ2 )(a, a2 , · · · , ak−1 , c )
11
a2 ,a3 ,··· ,ak−1 ∈U
12
↔ (ϕi ◦ ψ2 )(b, a2 , · · · , ak−1 , c )}.
14
3
↔ (ϕi ◦ (ψ1 ◦ ψ2 ))(b, a2 , · · · , ak−1 , c )}
9
13
1
ak ∈U
2
10 11 12
Namely, ψ1i (a, b) ≤ ψ2i (a, b) for all a, b ∈ U . Thus ψ1R ≤ ψ2R .
13
2
14
15 16 17
15
Theorem 4.2. Let A = (U , (ϕi )i ∈ I ) be a fuzzy relational structure. The set E b (U ) of all bisimulation fuzzy equivalence relations on U forms a complete lattice. This lattice is a complete join-subsemilattice of the lattice E (U ) of all fuzzy equivalence relations on U .
18 19 20 21 22 23 24
27 28
Proof. Because the equality relation I U is a bisimulation on U , it can be proved that E (U ) is a complete join-subsemilattice of E (U ). Let {ψ j } j ∈ J be a family of bisimulation fuzzy equivalence relations on U , and let ψ be its join in E (U ). Then for any j ∈ J , according to ψ j ≤ ψ and Lemma 4.2 we obtain that ψ j ≤ ψ jR ≤ ψ R . Thus ψ ≤ ψ R . Hence, by the assertion (3) of Theorem 4.1 we obtain that ψ is a bisimulation fuzzy equivalence relation. 2
31 32
ψ1 = ψ,
35
ψk+1 =
ψk ∧ ψkR ,
f or each k ∈ N .
40
43 44 45 46 47 48 49 50 51 52 53
58 59 60 61
26 27 28 30 31 32
37
(1) ψ b ≤ · · · ≤ ψk+1 ≤ ψk ≤ · · · ≤ ψ1 = ψ ; (2) If ψk = ψk+m , for some k, m ∈ N, then ψk = ψk+1 = ψ b ; (3) If A is finite and L is locally finite, then ψk = ψ b for some k ∈ N.
38 39 40 41
Proof. (1) It is easy to see that ψk+1 ≤ ψk for each k ∈ N and ψ b ≤ ψ1 . Let ψ b ≤ ψk for some k ∈ N. Then ψ b ≤ (ψ b ) R ≤ ψkR , so ψ b ≤ ψk ∧ ψkR . Thus, by induction we get ψ b ≤ ψk for each k ∈ N. (2) Suppose that ψk = ψk+m , for some k, m ∈ N. Then ψk = ψk+m ≤ ψk+1 = ψk ∧ ψkR ≤ ψkR , what means that ψk is a bisimulation fuzzy equivalence relation. Since ψ b is the largest bisimulation fuzzy equivalence relation contained in ψ , we conclude that ψk = ψk+1 = ψ b . (3) Because A is finite and L is locally finite, there exist k, m ∈ N such that ψk = ψk+m , and by (2) we conclude that ψk = ψ b . 2
Remark 4.1. If A is finite and L is locally finite, Theorem 4.3 gives a method of computing the largest bisimulation fuzzy equivalence relation contained in a given fuzzy equivalence relation ψ . But it does not necessarily work if L is not locally finite.
42 43 44 45 46 47 48 49 50 51 52 53 54 55
5. Computational examples and fuzzy rough set analysis
56 57
24
36
54 55
23
35
Then
41 42
22
34
37 39
21
33
34
38
20
29
Theorem 4.3. Let A = (U , (ϕi )i ∈ I ) be a fuzzy relational structure, let ψ be a fuzzy equivalence relation on U and let ψ b be the largest bisimulation fuzzy equivalence relation contained in ψ . Define inductively a sequence {ψk }k∈ N of fuzzy equivalence relations on U as follows:
33
36
19
25
From Theorem 4.2 we have that for any fuzzy equivalence relation ψ on U there exists the largest bisimulation fuzzy equivalence relation contained in ψ . It will be denoted by ψ b . In the next theorem we consider the problem how to construct it.
29 30
17 18
b
25 26
16
56
Example 5.1. Let L be the product structure, A = (U , (ϕ1 , ϕ2 )) a fuzzy relational structure over L with U = {a, b}, and the binary fuzzy relation ϕ1 , the ternary fuzzy relation ϕ2 given by
ϕ1 =
1
(a, b)
+
1 /2
(b, a)
,
ϕ2 =
1
(a, a, b)
+
1
(a, b, b)
+
1 /2
(b, a, a)
+
1 /2
(b, b, a)
57 58 59 60
,
61
JID:IJA AID:8202 /FLA
[m3G; v1.235; Prn:6/04/2018; 13:56] P.9 (1-10)
Y. Du, P. Zhu / International Journal of Approximate Reasoning ••• (••••) •••–•••
1 2 3
respectively. Let ψ be the universal relation on U , i.e., for every a, b ∈ U we have that ψ(a, b) = 1. Applying to ψ the method from Theorem 4.3, we get a sequence {ψk }k∈ N of fuzzy equivalence relations given by
4
1
ϕk =
5
8 9
+
(a, a)
6 7
9
1/2k−1
(a, b)
+
1/2k−1
(b, a)
+
1
(b, b)
,
1
ψb = IU =
11
(a, b)
+
1
(b, a)
6
16
19 20
13 14
We can see that the method of computing the largest bisimulation fuzzy equivalence relation does not work since L is not locally finite. Example 5.2. We continue with Example 3.1. Because L is locally finite, the method of computing the largest bisimulation fuzzy equivalence relation works. Then, the largest bisimulation fuzzy equivalence relation ψ b is given by
ψb =
23
1
(a, a)
0.5
+
(a, b)
+
0.5
(b, a)
+
1
28 29 30 31 32 33 34 35 36 37 38
(b, b)
22 23
Let L = [0, 1]. For any fuzzy set A ∈ [0, 1]U , the lower and upper approximations of A, ψ b ( A ) and ψ b ( A ), with respect
36
⊥
to the fuzzy approximation space (U , ψ b ) are fuzzy sets of U whose membership functions, for each a ∈ U , are
49
54
ψb
57 58 59
( A )(a) = ∨ { A (b)ψ (a, b)},
(13)
b ∈U
31 32 33 34 35 37
40
A=
0.8 a
+
0.4 b
44 45
where a binary composition on [0, 1] is a t-norm and a binary composition ⊥ on [0, 1] is a t-conorm. We illustrate fuzzy approximations based on ψ b with Example 5.2. Suppose that ⊥ = ∨ and = ∧. Let us approximate the concept
46 47 48 49 50
.
51 52
It follows from Equations (12) and (13) that their membership functions ψ
b
∧
∨
( A ) and ψ b ( A ) of A are
53 54 55
ψ b ∨( A) = ∧
ψ b ( A) =
60 61
30
43
b
55 56
29
42
52 53
28
41
and
50 51
27
38
(12)
b ∈U
45
48
26
39
ψ b ⊥ ( A )(a) = ∧ { A (b)⊥(1 − ψ b (a, b))},
43
47
20
25
41
46
19
24
40
44
18
Now we discuss the fuzzy rough set analysis of fuzzy relational structures by bisimulations. In essence, fuzzy relations of (ϕi )i ∈ I with regard to fuzzy relational structures can be understood as the knowledge that has been acquired. The notion of largest bisimulations collects the available knowledge represented by (ϕi )i ∈ I to the greatest extent. It is known that equivalence relations are used to model indiscernibility in the original model of rough sets. In this section, largest bisimulations are used to model indiscernibility of fuzzy relational structures. Based on largest bisimulations, the lower and upper fuzzy approximations are given, which can be used to deduce the unavailable knowledge represented by fuzzy sets with respect to fuzzy relational structures. Formally, for the largest bisimulation ψ b , (U , ψ b ) is a fuzzy approximation space, which is induced by (U , (ϕi )i ∈ I ). Based on ψ b , the lower and upper fuzzy approximations of a target fuzzy set over (U , ψ b ) can be achieved. In the literature, there are several kinds of fuzzy approximation operators (see, for example, [1,28–30,38,43–47,12,13,20,35]). This paper considers the lower and upper fuzzy approximation operators proposed by Hu et al. [20]. The details are as follows.
39
42
16
21
.
24
27
15 17
22
26
9
12
since the equality relation is the only bisimulation fuzzy equivalence relation on U .
21
25
8
11
17 18
7
10
,
14 15
3 5
12 13
2 4
k∈N
whose all members are different. We have that the largest bisimulation fuzzy equivalence relation ψ b is the equality relation on U , i.e.,
10
1
respectively.
0.5 a 0.8 a
+ +
0.4 b 0.5 b
,
56 57 58
,
59 60 61
JID:IJA AID:8202 /FLA
1
[m3G; v1.235; Prn:6/04/2018; 13:56] P.10 (1-10)
Y. Du, P. Zhu / International Journal of Approximate Reasoning ••• (••••) •••–•••
10
6. Conclusion
2 3 4 5 6 7 8
2
In this research, we focus on fuzzy relational structures which are the generalizations of relational structures in fuzzy environment. This paper considers the concept of bisimulations as indiscernibility with respect to fuzzy relational structures. Using bisimulations, we collect the knowledge represented by fuzzy relations in fuzzy relational structures. Based on bisimulations, the lower and upper approximations are given to deduce knowledge hidden in fuzzy relational structures. Moreover, we have given an effective method of computing the largest bisimulation, which is applicable to fuzzy relational structures over an arbitrary complete residuated lattice.
9 10
13
18 19 20 21 22 23 24 25
28 29
[12] [13] [14]
30 31 32 33 34 35
[15] [16] [17] [18] [19] [20]
36 37 38
[21] [22] [23]
39 40 41 42 43 44 45 46 47 48
[24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34]
49 50 51 52 53 54 55 56 57 58 59 60 61
6 7 8 10 12 13 15 16
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
26 27
5
14
References
16 17
4
11
The authors would like to thank the anonymous referees for their valuable comments and helpful suggestions. The work is supported by the National Natural Science Foundation of China under Grants 61672107 and 61772035.
14 15
3
9
Acknowledgements
11 12
1
[35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48]
D. Boixader, J. Jacas, J. Recasens, Upper and lower approximations of fuzzy sets, Int. J. Gen. Syst. 29 (1999) 555–568. P. Buchholz, Bisimulation relations forweighted automata, Theor. Comput. Sci. 393 (2008) 109–123. R. Bˇelohlávek, Fuzzy Relational Systems: Foundations and Principles, Kluwer, New York, 2002. R. Bˇelohlávek, V. Vychodil, Fuzzy Equational Logic, Studies in Fuzziness and Soft Computing, Springer, Berlin, Heidelberg, 2005. Y. Cao, G. Chen, E. Kerre, Bisimulations for fuzzy-transition systems, IEEE Trans. Fuzzy Syst. 19 (3) (2011) 540–552. Y. Cao, S. Sun, H. Wang, G. Chen, A behavioral distance for fuzzy-transition systems, IEEE Trans. Fuzzy Syst. 21 (4) (2013) 735–747. T. Chen, T. Han, Y. Cao, Polynomial-time algorithms for computing distances of fuzzy transition systems, arXiv preprint, arXiv:1701.06644. ´ c, ´ A. Stamenkovic, ´ J. Ignjatovic, ´ T. Petkovic, ´ Factorization of fuzzy automata, in: Proc. Fundament. Comput. Theor., vol. 4639, 2007, pp. 213–225. M. Ciri ´ c, ´ A. Stamenkovic, ´ J. Ignjatovic, ´ T. Petkovic, ´ Fuzzy relation equations and reduction of fuzzy automata, J. Comput. Syst. Sci. 76 (7) (2010) 609–633. M. Ciri ´ c, ´ J. Ignjatovic, ´ N. Damljanovic, ´ M. Bašic, ´ Bisimulations for fuzzy automata, Fuzzy Sets Syst. 186 (2012) 100–139. M. Ciri ´ c, ´ J. Ignjatovic, ´ I. Janˇcic, ´ N. Damljanovic, ´ Computation of the greatest simulations and bisimulations between fuzzy automata, Fuzzy Sets Syst. M. Ciri 208 (2012) 22–42. D. Dubois, H. Prade, Rough fuzzy sets and fuzzy rough sets, Int. J. Gen. Syst. 17 (1990) 191–209. L. D’eer, N. Verbiest, C. Cornelis, L. Godo, A comprehensive study of implicator-conjunctor-based and noise-tolerant fuzzy rough sets: definitions, properties and robustness analysis, Fuzzy Sets Syst. 275 (2015) 1–38. Y. Deng, H. Wu, Modal characterisations of probabilistic and fuzzy bisimulations, in: International Conference on Formal Engineering Methods, vol. 8829, Springer, 2014. Y. Deng, H. Wu, Logical characterizations of simulation and bisimulation for fuzzy transition systems, Fuzzy Sets Syst. 301 (6) (2016) 19–36. A. Dovier, C. Piazza, A. Policriti, An efficient algorithm for computing bisimulation equivalence, Theor. Comput. Sci. 311 (2004) 221–256. T.F. Fan, Rough set analysis of relational structures, Inf. Sci. 221 (2013) 230–244. T.F. Fan, Fuzzy bisimulation for Gödel modal logic, IEEE Trans. Fuzzy Syst. 23 (6) (2015) 2387–2396. R. Gentilini, C. Piazza, A. Policriti, From bisimulation to simulation: coarsest partition problems, J. Autom. Reason. 31 (2003) 73–103. B.Q. Hu, Z.H. Huang, (⊥, )-generalized fuzzy rough sets based on fuzzy composition operations, in: B.-y. Cao, C.-y. Zhang, T.-f. Li (Eds.), Fuzzy Information and Engineering, in: Adv. Soft Comput., vol. 54, Springer-Verlag, Berlin, Heidelberg, 2009, pp. 647–659. ´ c, ´ S. Bogdanovic, ´ M. Ciri ´ Fuzzy homomorphisms of algebras, Fuzzy Sets Syst. 160 (16) (2009) 2345–2365. J. Ignjatovic, G.J. Klir, B. Yuan, Fuzzy Sets and Fuzzy, Logic Theory and Application, Prentice-Hall, Englewood Cliffs, NJ, 1995. C. Liau, T. Lin, Reasoning about relational granulation in modal logics, in: Proceedings of the First IEEE International Conference on Granular Computing, IEEE Press, 2005, pp. 534–538. N. Lynch, F. Vaandrager, Forward and backward simulations: part I. Untimed systems, Inf. Comput. 121 (1995) 214–233. R. Milner, A calculus of communicating systems, in: G. Goos, J. Hartmanis (Eds.), Lect. Notes Comput. Sci., vol. 92, Springer, 1980. R. Milner, Communication and Concurrency, Prentice-Hall International, 1989. R. Milner, Communicating and Mobile Systems: The π -Calculus, Cambridge University Press, Cambridge, 1999. J.S. Mi, Y. Leung, H.Y. Zhao, T. Feng, Generalized fuzzy rough sets determined by a triangular norm, Inf. Sci. 178 (2008) 3203–3213. J.S. Mi, W.X. Zhang, An axiomatic characterization of a fuzzy generalization of rough sets, Inf. Sci. 160 (2004) 235–249. N.N. Morsi, M.M. Yakout, Axiomatics for fuzzy rough sets, Fuzzy Sets Syst. 100 (1998) 327–342. Z. Pawlak, Rough sets, Int. J. Comput. Inf. Sci. 11 (5) (1982) 341–356. Z. Pawlak, Rough Set: Theoretical Aspects of Reasoning About Data, Kluwer, Norwell, MA, 1991. Z. Pawlak, A. Skowron, Rudiments of rough sets, Inf. Sci. 177 (2007) 3–27. D. Park, Concurrency and automata on infinite sequences, in: P. Deussen (Ed.), Proc. 5th GI Conf., Karlsruhe, Germany, in: Lect. Notes Comput. Sci., vol. 104, Springer-Verlag, 1981, pp. 167–183. D.W. Pei, A generalized model of fuzzy rough sets, Int. J. Gen. Syst. 34 (2005) 603–613. ´ Congruences and homomorphisms of fuzzy automata, Fuzzy Sets Syst. 157 (2006) 444–458. T. Petkovic, M. Roggenbach, M. Majster-Cederbaum, Towards a unified view of bisimulation: a comparative study, Theor. Comput. Sci. 238 (2000) 81–130. A.M. Radzikowska, E.E. Kerre, A comparative study of fuzzy rough sets, Fuzzy Sets Syst. 126 (2002) 137–155. ´ c, ´ J. Ignjatovic, ´ M. Ciri ´ Reduction of fuzzy automata by means of fuzzy quasi-orders, Inf. Sci. 275 (2014) 168–198. A. Stamenkovic, J. Scott, Social Network Analysis: A Handbook, second ed., SAGE Publications, 2000. D. Sangiorgi, On the origins of bisimulation and coinduction, ACM Trans. Program. Lang. Syst. 31 (4) (2009) 111–151. D.D. Sun, Y.M. Li, W.W. Yang, Bisimulation relations for fuzzy finite automata, Fuzzy Syst. Math. 23 (2009) 92–100. W.Z. Wu, J.S. Mi, W.X. Zhang, Generalized fuzzy rough sets, Inf. Sci. 151 (2003) 263–282. W.Z. Wu, W.X. Zhang, Constructive and axiomatic approaches of fuzzy approximation operators, Inf. Sci. 159 (2004) 233–254. W.Z. Wu, Y. Leung, J.S. Mi, On characterizations of (I,T)-fuzzy rough approximation operators, Fuzzy Sets Syst. 154 (2005) 76–102. Y.Y. Yao, A comparative study of fuzzy sets and rough sets, Inf. Sci. 109 (1998) 227–242. W. Zhu, F.Y. Wang, Reduction and axiomization of covering generalized rough sets, Inf. Sci. 152 (2003) 217–230. P. Zhu, H.Y. Xie, Q.Y. Wen, Rough approximations based on bisimulations, Int. J. Approx. Reason. 81 (2017) 49–62.
17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61