Journal of Applied Mathematics

Volume 2014 (2014), Article ID 597650, 8 pages

http://dx.doi.org/10.1155/2014/597650

Research Article

Rough Operations and Uncertainty Measures on MV-Algebras

Maosen Xie

Institute of Mathematics and Finance, Sichuan University of Arts and Science, Dazhou, Sichuan 635000, China

Received 10 February 2014; Revised 13 May 2014; Accepted 13 May 2014; Published 11 June 2014

Academic Editor: Morteza Rafei

Copyright © 2014 Maosen Xie. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We define a lower approximate operation and an upper approximate operation based on a partition on MV-algebras and discuss their properties. We then introduce a belief measure and a plausibility measure on MV-algebras and investigate the relationship between rough operations and uncertainty measures.

1. Introduction

The rough set theory, introduced by Pawlak, has been conceived as a tool to conceptualize, organize, and analyze various types of data, in particular, to deal with inexact, uncertain, or vague knowledge in applications related to artificial intelligence. Since then, the subject has been investigated in many studies and various rough set models have been used in machine learning, knowledge discovery, decision support systems, and pattern recognition. In Pawlak’s rough set model, a key concept is an equivalence relation, and given an equivalence relation on a universe, we can define a pair of rough approximations which provide a lower bound and an upper bound for each subset of the universe. Rough approximations can also be defined equivalently by a partition of the universe which is corresponding to the equivalence relation [1–6].

Dempster-Shafer theory of evidence is a method developed to model and manipulate uncertain, imprecise, incomplete, and even vague information. It was originated by Dempster’s concept of lower and upper probabilities and extended by Shafer as a theory. The basic representational structure in this theory is a belief structure, which consists of a family of subsets, called focal elements, with associated individual positive weights summing to 1. The fundamental numeric measures derived from the belief structure are a dual pair of belief and plausibility functions [7]. Combining the Dempster-Shafer theory and fuzzy set theory has been suggested to be a way to deal with different kinds of uncertain information in intelligent systems in a number of studies [8–10]. In [11–14], by introducing a pair of dual rough operations on Boolean algebras and using them to interpret some uncertainty measures on Boolean algebras, Bayesian theory and Dempster-Shafer theory are extended to be constructed on Boolean algebras. This provides a more general framework to deal with uncertainty reasoning and a better understanding of both rough operations and uncertainty measures on Boolean algebras.

The present paper extends the rough operations and Dempster-Shafer theory with respect to Boolean algebras to MV-algebra. The paper is arranged as follows. In Section 2, some results of MV-algebra which will be used in this paper are recollected. In Section 3, we introduce a pair of approximate operations based on a partition on MV-algebra, which is the generation of rough operations on Boolean algebras, and discuss their properties. In Section 4, we define a belief measure and a plausibility measure on MV-algebras and investigate the relationship between rough operations and uncertainty measures. In Section 5 concludes.

2. MV-Algebra and Its Partitions

In this section, we recall firstly the basic notions on MV-algebras. See [15–18] for further results on MV-algebras. An MV-algebra is an algebra where is an associative and commutative binary operation on having as the neutral element, a unary operation is involutive with for all , and moreover the identity is satisfied for all .

A partial order is defined on by if and only if . An additional constant 1 and two binary operations , , and are defined as follows:

If is also totally ordered then is called a totally ordered MV-algebra. An MV-algebra is called -complete if every nonempty countable subset of has a supremum in .

An MV-subalgebra of is a subset of containing the neutral element of , closed under the operations of and endowed with the restriction of these operations to .

The (finite) Cartesian product of MV-algebras , endowed with the partial order and the MV-algebra operations defined pointwise, really is also an MV-algebra and will be called product, for short.

An element satisfying is called an idempotent element (Boolean element); the set of all idempotent elements of endowed with the natural restriction of operations inherited from is a Boolean algebra.

Example 1. The real unit interval with Łukasiewicz operation , and , where , is a -complete MV-algebra called the MV-unit interval.

Example 2. Let , , where is the MV-unit interval. The order and the operation on are defined in pointwise: for

Then is an MV-algebra called the MV-cube.

Example 3. Let . The order and the operations on are defined as Figure 1, and the operations on are defined as Tables 1 and 2, respectively. Then is an MV-algebra.

Table 1

Table 2

Figure 1:

Proposition 4 (see [15–18]). Let be an MV-algebra. For any ,

(P1);(P2), ;(P3);(P4);(P5);(P6);(P7) is a commutative semigroup with unit element 0;(P8), if the supremum exists in equality;(P9), if the infimum exists in equality.

Two elements and in an MV-algebra are called orthogonal (denoted as ) if . Obviously, if and only if . A finite subset of elements of an MV-algebra is said to be orthogonal if , , .

Definition 5. A finite collection of nonneutral elements of an MV-algebra is said to be a partition of if and only if

(i) is orthogonal;(ii).

Lemma 6. Every element in partition is idempotent element; that is, , , .

Proof. , we have , and it also means that . It follows that . By we know . Obviously, . Hence, .

Theorem 7. Let be a partition of . Suppose that

where . Then is a Boolean algebra.

Proof. At first, we prove that for , where . From we know . On other hand, from we know for any . Hence, . This proves that . Hence, for every .

Secondly, we prove that is closed under and . Let , . Note that for and . If then or else . Hence, for all . By the duality of and we know for all .

Finally, from the above proof we easily obtain that and . Therefore, is a Boolean algebra.

Example 8. Let be the MV-cube in Example 2 and be a partition of in the usual sense. Denote

Then is a partition of .

Example 9. Let be the MV-algebra in Example 3. Then is a partition of .

Example 10. Let be the MV-algebra in Example 3 and the classic Boolean algebra. Then is an MV-algebra, and is a partition of .

3. Approximate Operations on MV-Algebras

In Pawlak’s rough set theory, a rough set is induced by a partition of the universe. In this section, we will extend Pawlak’s rough set theory by defining a pair of approximate operations induced by a partition of the unity of an MV-algebra.

Definition 11. Let be an MV-algebra and a partition of . Then a pair of operations and , such that

are called a lower approximation and an upper approximation based on partition , respectively. If then is called a definable element with respect to else is called a rough element. If no confusion arisen then the operations and can be abbreviated as and , respectively.

Remark 12. Let be a Boolean algebra of some subsets of a nonempty set and let   be a partition of in the usual sense. In this case, the conditions and in Definition 11 identity with that and do not hold in the usual set meaning, respectively. Hence,

This means that the operations and introduced for MV-algebras in Definition 11 are extensions of the operations and in the typical set notations, respectively.

Theorem 13. If we denote , then

Proof. Consider the following:

Theorem 14. Let be a partition of MV-algebra and and the lower and upper approximations induced by , respectively. Then

(1), ;(2), , ;(3)if then , ;(4);(5);(6), ;(7), ;(8), .

Proof. The proof of (1), (2), and (3) is obvious as follows.

(4) .

(5) .

By the duality of and , we have .

(6) By (3) we have . For any , if then it follows from that or . Hence,

Thus,

This shows that . Therefore, .

By the duality of and we know that another equation holds.

(7) By (3) and (5) we have . For any ,

if and only if there exists and such that . Hence, . It follows that

This shows that .

(8) By (3) we have . If , then . It follows from that . Since is a partition we have . Hence, . This means that . This shows that . Therefore, .

Example 15. Let the MV-algebra and the partition of be as defined in Example 9. Then , ; , .

Example 16. The MV-algebra and the partition of as defined in Example 10. Then ;   .

By Definition 11 we know that every element in is definable and the definable elements of can also be obtained by the following theorem.

Theorem 17. Let be an MV-algebra and a partition of . The definable elements of with respect to are

and forms a Boolean algebra.

Proof. At first, we prove that the definable elements set with respect to are . Suppose that is a definable element. Then . , if then . This means that . Conversely, let . Then implies . It follows that . By Theorem 14(4) we have .

Since is a definable element we also know . Hence, for any , if then . Let and . Then . Hence, ; that is . This means that . Conversely, let . Then implies . If and , then . Hence, does not hold. By and the definition of we know ; hence, . This shows that . By Theorem 14(4) we have .

This shows that the definable elements set with respect to are .

Next we prove that forms a Boolean algebra. Let . For any , if , then and . By the definition of we know and ; that is . Hence, . This shows that is closed under operation . If , then or . By the definition of we know or ; that is . Hence, . This shows that is closed under operation . This shows that forms a sublattice of .

By Theorem 14 we know . Let . Then , or . This means that , or . It follows from , or that . Since , or , we have , . Hence, . Analogously, .

This shows that is a Boolean algebra.

4. Approximate Operations and Uncertainty Measures

In this section, we will discuss the relationship between rough operations and uncertainty measures. Given an MV-algebra, we may only know the measures of some elements when information is absent in an MV-algebra. For those elements that we do not know the measure, what we can do is to define belief measure and plausibility measure on them.

Definition 18. The function such that

(1),(2),(3) is finitely additive, that is, if is orthogonal then

is called a finitely additive measure on MV-algebra .

For finitely additive measure, we have the following conclusion.

Theorem 19 (see [19, 20]). If is a finitely additive measure on the MV-algebra then

(1)if then , where ;(2)if then ;(3);(4).

Example 20. Let be an MV-algebra and the MV-unit interval. If a mapping is a homomorphism of type , that is, , , for any , then is called a Łukasiewicz-valuation (see [21]). Then it is easy to prove that the Łukasiewicz-valuation is a finitely additive measure.

Example 21. Let be an MV-algebra, let be a partition of , let be the smallest subalgebra of containing , and let be the set of all homomorphisms of type from to the MV-unit interval . An element in can be viewed as a function from to ; that is, for any , a function , can be defined. Suppose that is a probability measure space, which satisfies the fact that is a -measurable function on ; that is, for every . In this case, the element can also be viewed as a random variable from to (see [21]). Hence, we can define for every . It is easy to prove that is a finitely additive measure.

Example 22. Let be an MV-algebra. A state on a -complete MV-algebra is a mapping such that for all :

(1),(2)if then ,(3)if then ,

where stands for is a nondecreasing sequence and (see [20, 21]). Then a state is a finitely additive measure by the following Lemma 23.

Lemma 23. For any -orthogonal subset and any state of , it holds that

Proof. Since is -orthogonal we know . From the definition of a state, we have inductively

Proposition 24. Let be a partition of MV-algebra and let be the Boolean algebra defined in Theorem 7. If function satisfies then we can extend to by defining , , for every , and is a finitely additive measure on .

Proof. Obviously and . For then there are such that and . If then we assert . In fact, if then there is such that . Hence, . This shows that is a finitely additive measure on .

Definition 25. (1) A function is called a belief measure if , , and

for every positive integer and for every tuple , of subsets of .

(2) A function is called a plausibility measure if , , and

for every positive integer and for every tuple , of subsets of .

Theorem 26. Suppose that is a partition of MV-algebra and function satisfies . By using function we define a function on as follows:

Then is a plausibility measure on .

Proof. It is easy to check , , and . For any , it is easy to check

By the well-known inclusion-exclusion formulas in probability theory we have

This shows that is a plausibility measure on .

Theorem 27. Suppose that is a partition of MV-algebra and function satisfies . By using function we define a function on as follows:

Then Bel is a belief measure on .

Proof. It is easy to check , , and . For any , it is easy to check

By the well-known inclusion-exclusion formulas in probability theory we have

This shows that Bel is a belief measure on .

Let be an MV-algebra and a partition of . Suppose that function satisfies . Then we can extend to by defining

where , , and are lower and upper approximations of , respectively. The values and of an element of can be viewed as our best estimate of the measures of , given our lack of knowledge. Moreover, we can get the following theorem.

Theorem 28. and defined above are belief measure and plausibility measure on , respectively.

Proof. (1) We prove . Note that . In the following we prove that if and only if . In fact, . This shows that if and only if . Hence, . It follows from Theorem 26 that is a plausibility measure on .

(2) We prove . Note that . In the following we prove that if and only if . In fact, . This shows that if and only if . Hence, . It follows from Theorem 27 that is a belief measure on .

Example 29. The MV-algebra and the partition of as defined in Example 10. Suppose that function , , , . The uncertainty measures on with respect to are shown in Table 3.

Table 3

5. Conclusion

In this paper, a pair of dual rough operations on MV-algebras is introduced. The properties of rough operations and the relationship between rough operations and uncertainty measures are discussed. If information is absent in an MV-algebra and we may only know the measures of some elements, then what we can do is to define belief measure and plausibility measure on MV-algebra, which is used to interpret some uncertainty measures on MV-algebras.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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