Abstract: In this work we generalize Pawlak approximation space to bitopological approximation space. One binary relation can define two subbases of two topological spaces. Membership, equillity and inclusion relations using rough approximations are defined and studied in bitopological aapproximation space. Some new measures that measure the accuracy and the quality of approximations are defined and studied. An example to data reduction in multivalued information system are introduced.
Keywords: Topological Spaces; Rough Sets; Rough Approximations; Accuracy Measures; Data Reduction.
1. Introduction
The problem addressed in this paper is how to construct knowledge from datasets by a topological generalization of rough set theory. In the classical approach of Pawlak knowledge theory has an extended and wealthy history. For understanding, representing, and manipulating knowledge, there is a variety of opinions and approaches in this area. The process of extracting knowledge based on the ability to classify objects. Knowledge in this approach is necessarily connected with the selection of classification patterns, related to specific parts of the real or abstract world. This view for knowledge and information is more near to the theory of abstract topological spaces, which is based on a topological structure, consists of class of subsets of the universe classified according to its satisfaction to some axioms. Consequently, the abstract topological space is not metric model for most types of information representations.
There are many methods for studying data in information systems; a recent one which related to topology is rough set. This theory depends on a special class of topological spaces known by quasi-discrete topological space in which every open set is closed. The basic assumption of this theory concerns the fundamental problem of knowledge representation: in this theory one assumes that objects under study are described in terms of functions which map these objects into the corresponding, value sets.
The starting point of this paper is any general binary relation. From this relation we generate two sets one is the right set and the another is the left set. Using the collection of these sets we also generate two subbases of two topologies that used in this approach. Then we used these topologies to generalize Pawlak approximation space to bitopological approximation space. Also we defined and studied membership, equillity and inclusion relations in this bitopological space. Some measures of accuracy also defined and studied. Finally we determined the reduct of multivalued information system as an application.
This paper is organized as follows:
Section 2 introduced the previous work of this research. In Section 3 we discussed the needed fundamentals of Pawlak’s rough set theory and topological spaces. Section 4 introduced and investigated the concept of bitopological approximation space and studied their properties. Section5 is devoted to introduce the basic concepts of bitopological lower and the mixed bitopological upper approximations. An application to data reduction in multivalued information systems using these approximations is introduced in Section 6. The paper conclusion is given in Section 7.
2. Previous Works
Rough sets theory introduced by Pawlak in [22] is a mathematical tool to imperfect knowledge, decision analysis, and knowledge discovery from databases. This theory has paying attention of a lot of researchers and practitioners all over the world, who contributed fundamentally to its improvement and applications. Usually rough sets used together with other methods such as fuzzy sets [4, 5, 7, 11, 34, 39], covering and fuzzy covering [10, 19, 17], statistical methods (probabilistic space) [9], tolerance and similarity relations [20, 21, 28, 29], topological generalizations [1, 16, 25, 27, 32, 38, 40] to locate accurate approaches used in applications. Recently many generalizations of this theory have developed to help in applications such as in information retrieval [30], reduction of finite information systems [3, 42], establishment rules of interval-valued fuzzy information systems [12, 13], and foundation attributes missing values in incomplete information systems [15].
The original rough set theory depended totally on equivalence relations to approximate concepts. But these types of relations are still restrictive for many applications. Many researchers studied this issue and several interesting and meaningful generalizations to equivalence relation have been proposed. For instance, binary, tolerance, similarity, reflexive, and transitive relations are some solutions of this issue [2, 24, 41, 43, 44]. Some researchers generalized rough sets by combining fuzzy sets with rough sets [8, 18, 26, 31]. Another group has characterized a measure of uncertainty by the concept of fuzzy relations [37]. Wiweger in [32] is the first researcher that defined topological rough set that classified to be the very important topological generalizations of rough sets. Different approaches generalized approximation space, generalized rough set models and using relational interpretations for approximate operators using rough sets [23, 33, 35, 36]. In [6] the necessary and sufficient conditions for the lower and upper approximations are considered to formulate rough sets to certain families of exact sets. .
3. Basic Concepts of Rough Sets and Topological Spaces
The main benefit of rough set theory in data reduction is that it does not need any introduction or supplementary information about data.
The basic tool of Pawlak theory is the approximation space [23]. An approximation space is apair , where is a set called the universe and is an equivalence relation. The blocks made by the equivalence relation is the elementary sets of this theory that are used in approximations. Any subset of the universe can approximated using these elementary sets from lower and from upper. Classical rough lower approximation of is defined by Pawlak by and the classical upper is defined by . The lower and upper approximations are the keys to define another regions using the subset . The positive, negative and boundary regions of the subset are defined as follows:
1) The positive region of is defined by .
2) The negative region of is defined by .
3) The boundary region of is defined by
Now the set is exact with respect to , if the boundary region is empty that is . The set is rough with respect to , if the boundary region is nonempty that is .
The degree of completeness (accuracy measure) of the subset is defined by where denotes the cardinality of . Obviously . If , is exact and otherwise, if , is rough.
The pair of a non empty set and a family of subsets of is a topological space when and is closed under arbitrary union and finite intersection [14]. The equation is for the closure of . But this is for the interior.
If we considered that the class is a subbase of a topology on . Then it is easily to prove that the Pawlak lower and the upper approximations of the subset are identical with the interior and the closure operations in .
In this paper we define two subbases by a general binary relation (not equevalence) on . Subbase generates the topology and subbase generates the topology .
Rough approximations of using defined as follows:
and .
Also if we can define two other approximations as follows: and .
4. Bitopological Appriximation Space
In this section, we introduce and investigate the concept of bitopological approximation space. Also, we introduce the concepts of bitopological lower and bitopological upper approximations and study their properties.
In the topological space and for we define the set: .
Definition 4.1 In the bitopological approximation space we define: and .
Definition 4.2 In the bitopological approximation space we define:
.
By the same manner we can define the -bitopological lower and -bitopological upper approximations as follows:
Definition 4.3 Let be a bitopological approximation space, the mixed bitopological lower and mixed bitopological upper approximations of any subset defined as follows:
Definition 4.4 Let be a bitopological approximation space. Then we define the following degrees of completeness for a subset as follows:
1- Pawlak accuracy measure ,
2- Topological accuracy measure , .
3- Bitopological accuracy measure .
4- Mixed bitopological accuracy measure .
Example 4.1 Given a universe and a general binary relation defined on by Then by Definition 3.1 we can generate the following:
, , and . Also , , and . Then is the subbase of the topology and is the subbase of the topology . Then applying Definitions 3.3&3.4&3.5 of the chosen subsets of Table 1 then we have a comparision among the degree of accuracy measure , and as given in Table 1.
{c} 1/2 1/2 1 1/3 1 1
{d} 0 1/3 1/3 1/2 1/3 1
{a,b} 1/3 2/3 1/3 1/2 2/3 1
{a,d} 1/4 1/2 1/2 1/2 1/2 1
{b,c} 1/4 1/4 1 1/3 1 1
{c,d} 1/3 2/3 1/3 2/3 2/3 1
{a,b,c} 1/3 1/3 1/3 2/3 1/2 1
{a,b,d} 2/3 1/2 3/4 3/4 2/3 1
Table 1: Comparison of the three accuracy measures
Using the mixed bitopological accuracy measure the results of Table 1 has been improved. Consequently the mixed bitopological accuracy measure is more accurate than other measures.
The following subsets are all disjoint and can seperate the elements of the universe by the relation and more elements will be well defined using new types of memberships relations.
, , , , , , , , , , , , , , , , , , , , , and
5. Properties of bitopological approximations
In this section, we study the basic concepts of bitopological lower and the mixed bitopological upper approximations. We introduce six membership relations using these approximations and study their properties. Also we introduce the equality and inclusion relations with stuyding some basic properties.
Definition 5.1 In the bitopological approximation space we define for :
1) if and only if ,
2) if and only if ,
3) if and only if ,
4) if and only if ,
5) if and only if ,
6) if and only if .
Remark 5.1 According to Definition 4.1, mixed bitopological lower and mixed bitopological upper approximations of a set can be rewritten as: , .
Remark 5.2 Let be a bitopological approximation space. For any subset , and
Example 5.1 In Example 4.1, if , then and , hence , and . Also , but .
Example 5.2 In Example 4.1, if , then and . So , , but and , but .
We investigate mixed bitopological rough equality and mixed bitopological rough inclusion based on rough equality and inclusion.
Definition 5.2 In the bitopological approximation space and for we define:
(i) Roughly bottom equal by mixed bitopology if and only if
(ii) Roughly top equal by mixed bitopology if and only if
(iii) Roughly equal by mixed bitopology if and only if and
Example 5.3 According to Example 4.1 the set and are roughly bottom equal by mixed bitopology. But the set and are roughly top equal by mixed bitopology.
Definition 5.3 In the bitopological approximation space and for we define:
(i) is roughly bottom included by mixed bitopology in if .
(ii) is roughly top included by mixed bitopology in if
(iii) is roughly included by mixed bitopology in if and .
Example 5.4 In Example 4.1, we have , , and , then is mixed bitopological roughly bottom included in and is mixed bitopological roughly top included in .
Definition 5.4 Let be a bitopological approximation space. The subset is called:
(i) Mixed bitopological definable , if .
(ii) Mixed bitopological rough, if .
(iii) Roughly mixed bitopological definable, if and
(iv) Internally mixed bitopological undefinable, if and
(v) Externally mixed bitopological undefinable, if and
(vi) Totally mixed bitopological undefinable, if and
Proposition 5.1 Let be a bitopological approximation space. Then we have:
(i) Every exact set in is mixed bitopological exact.
(ii) Every exact set in is mixed bitopological exact.
(iii) Every mixed bitopological rough set in is rough.
(iv) Every mixed bitopological rough set in is rough.
Proof. Obvious.
The converse of Proposition 5.1, may not be true in general and we declare this by the following example.
Example 5.5 In Example 4.1, the sets , , , and are mixed bitopological exact, but neither exact nor exact.
Remark 5.2 Let be a bitopological approximation space. Then we have:
(i) The intersection of two mixed bitopological exact sets need not be mixed bitopological exact set.
(ii) The union of two mixed bitopological exact sets need not be mixed bitopological exact set.
The following example illustrates the above remark.
Example 5.6 According to Example 3.1, suppose , , and , are mixed bitopological exact. Then and are not mixed bitopological exact.
5. An Example to Data Reduction in Multivalued Information systems
The structure is attribute system, here is the system universe, is a set of attributes and is set of values of the attribute . is the information function such that .
For any subset we define the relation : , for we define the class as follows: , where
The structure is a decision table, where is the set of decisions.
We define the relation of the decision attribute by:
The class of this relation is The set of all classes is .
The collection is the subbase of the first topology and the class is the subbase of the second topology , by the same manner construct the topology .
We consider the set is a reduct of , if and is a minimal ,where, iff s.t.
D C B A U
{d3} {c1} {b1,b2,b4} {a2} p1
{d3} {c1,c3} {b1,b2} {a1,a2} p2
{d3} {c1} {b1,b3} {a3} p3
{d1} {c4} {b1,b2,b4} {a1} p4
{d2} {c1,c2} {b5} {a1} p5
{d3} {c1} {b1,b2} {a1} p6
{d3} {c1,c3} {b1,b3,b4} {a1} p7
Table 2: Multivalued Information System
To obtained the subbases of the two topologies applying the relation to Table 2.
Then we have:
, where and where . From Table 2, we have the following couples of topologies:
,
,
,
,
,
,
,
,
,
The decision attribute with the relation: gives the subbase , and the decision topology is ,
The complement decision topology is
Then we have , and , this leads to is the reduct of Table 2.
7. Conclusion
The objectives of this work are to study new alternative method of data reduction. This method is about using generalizations of rough sets by two topological spaces. The advantage of this generalization is to use a general binary relation that defines two subbases of the two topologies used in our generalizations.
The bitopological lower and bitopological upper approximations that induced by to topologies are new tools that give the higher accuracy measure of data. Also, we defined the concept of a rough bitopological membership function as alternative approach for measuring data accuracy. The bitopological approach is used to reduct multivalued information system.
The future work of this work is to suggest simplifications on these generalizations by make new algorithms, which simplify the calculations on it.