ON LIU S INFERENCE RULE FOR UNCERTAIN SYSTEMS

Transcription

1 International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems Vol. 18, No. 1 ( c World Scientific Publishing Company DOI: /S ON LIU S INFERENCE RULE FOR UNCERTAIN SYSTEMS XIN GAO School of Mathematics and Physics, North China Electric Power University, Beijing, , China gao-xin00@mails.tsinghua.edu.cn YUAN GAO Department of Mathematical Sciences, Tsinghua University Beijing, , China g-y08@mails.tsinghua.edu.cn DAN A. RALESCU Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH , USA ralescd@uc.edu Received 7 August 2009 Revised 13 January 2010 Liu s inference is a process of deriving consequences from uncertain knowledge or evidence via the tool of conditional uncertainty. Using membership functions, this paper derives some expressions of Liu s inference rule for uncertain systems. This paper also discusses Liu s inference rule with multiple antecedents and with multiple if-then rules. Keywords: Uncertain inference; inference rule; uncertainty theory. 1. Introduction The practical systems are usually in states of uncertainty. When the uncertainty is neither random nor fuzzy, we cannot deal with it by probability theory or credibility theory. In order to deal with subjective uncertainty, Liu 6 founded an uncertainty theory that is based on normality, monotonicity, self-duality, countable subadditivity, and product measure axioms. Considerable work has been done based on uncertainty theory. Gao 3 gave some mathematical properties of uncertain measure. You 12 studiedsomeconvergencetheoremsofuncertainsequences.liu 10 foundeduncertain programming to model optimization problems in uncertain environments. Liu proposed uncertain calculus 8 and the concept of uncertain process. 7 Li and Liu 5 proposed uncertain logic and defined the truth value as the uncertain measure that an uncertain proposition is true. Uncertainty theory has become a new tool 1

2 2 X. Gao, Y. Gao & D. A. Ralescu to describe subjective uncertainty and has wide applications in engineering. For a detailed exposition, the interested reader may consult the book. 9 Uncertain inference is the process of deriving consequences from uncertain knowledge or evidence. When systems exhibit fuzziness, fuzzy inference has been studied and inference rules have been suggested. For example, Zadeh s compositional rule of inference, Lukasiewicz s inference rule, and Mamdani s inference rule are widely used in fuzzy inference systems. Different from the above inference rules, Liu 11 proposed uncertain inference including a new inference rule. This new inference rule uses the tool of conditional uncertain set. In this paper, we will investigate the expressions of Liu s inference rule for uncertain systems via membership functions. Section 2 introduces some basic concepts and results on uncertain sets as a preliminary. By embedding conditional uncertainty into the framework of Liu s inference rule, we obtain some expressions of Liu s inference rule by membership functions in Sec. 3. We also investigate Liu s inference rule with multiple antecedents and with multiple if-then rules in Sec. 4 and Sec. 5, respectively. In Sec. 6, the general inference rule is given. Section 7 concludes this paper with a brief summary. 2. Preliminaries In this section, we will introduce some useful definitions about conditional uncertainty and uncertain sets. Let Γ be a nonempty set, and let L be a σ-algebra overγ. Each element Λ L is called an event. In order to measure uncertain events, Liu 6 introduced an uncertain measure M as a set function satisfying normality, monotonicity, self-duality, and countable subadditivity axioms. Then the triplet (Γ, L, M is called an uncertainty space. In order to define the uncertain measure of an event A after it has been learned that some other event B has occurred, Liu 6 defined a conditional uncertain measure M{A B} as follows: Definition 1. (Liu 6 Let (Γ,L,M be an uncertainty space, and A,B L. Then the conditional uncertain measure of A given B is defined by M{A B} = M{A B}, if M{A B} < 0.5 M{B} M{B} 1 M{Ac B}, if M{Ac B} < 0.5 M{B} M{B} 0.5, otherwise provided that M{B} > 0. An uncertain variable is defined as a measurable function ξ from an uncertainty space (Γ,L,M to the set of real numbers, i.e., for any Borel set B of real set R,

3 On Liu s Inference Rule for Uncertain Systems 3 the set {ξ B} = {γ Γ ξ(γ B} is an event in L. Different from the definition of uncertain variable, the concept of uncertain set was proposed by Liu 11 in Definition 2. (Liu 11 An uncertain set is a measurable function ξ from an uncertainty space (Γ,L,M to a collection of sets of real numbers, i.e., for any Borel set B of real numbers, the set is an event. {ξ B} = {γ Γ ξ(γ B} A real-valued function µ is called a membership function if 0 µ(x 1, x R. Let µ be a membership function. Then for any number α [0,1], the set µ α = {x R µ(x α} is called the α-cut of µ. Let µ be a membership function. Then for any number α [0,1], the set W α = {µ β β α} is called the α-class of µ. Especially, the 1-class is called the total class of µ. Definition 3. (Liu 11 An uncertain set ξ is said to have a membership function µ if the range of ξ is just the total class of µ, and where W α is the α-class of µ. M{ξ W α } = α, α [0,1] (1 Definition 4. (Liu 11 Let ξ be an uncertain set on (Γ,L,M. A conditional uncertain set of ξ given B is a measurable function ξ B from the conditional uncertainty space (Γ,L,M{ B} to a collection of sets of real numbers such that in the sense of classical set theory. ξ B (γ ξ(γ, γ Γ Let ξ and η be two nonempty uncertain sets. What is the appropriate event that η is included in ξ? There exist two different concepts on this question. One concept uses the event {η ξ} that η is strongly included in ξ; the other one uses the event {η ξ c } that η is weakly included in ξ. It is easy to verify that {η ξ} {η ξ c }. Intuitively, it is too conservative to take the strong inclusion {η ξ}, and too adventurous to take the weak inclusion η ξ c. Thus Liu 11 introduced a new symbol to represent this inclusion relationship called imaginary inclusion. That is, {η ξ} represents the event that η is imaginarily included in ξ. Liu 11 also defined the uncertain measure that η is imaginarily included in ξ:

4 4 X. Gao, Y. Gao & D. A. Ralescu Definition 5. (Liu 11 Let ξ and η be two nonempty uncertain sets. Then the membership degree of η to ξ is defined as the averageof the uncertain measure that strongly included and weakly included, i.e., M{η ξ} = 1 2 (M{η ξ}+m{η ξc }. (2 For any uncertain sets ξ and η, the membership degree M{η ξ} reflects the truth degree that η is a subset of ξ. If M{η ξ} = 1, then η is completely included in ξ. If M{η ξ} = 0, then η and ξ have no intersection at all. 3. Inference Rule Liu s Inference Rule 1. (Liu 11 Let X and Y be two concepts. Assume a rule of the form if X is an uncertain set ξ then Y is an uncertain set η. From X is an uncertain set ξ we infer that Y is an uncertain set η = η ξ ξ which is the conditional uncertain set of η given ξ ξ. Liu s inference rule is represented by Rule: If X is ξ then Y is η From: X is ξ Infer: Y is η = η ξ ξ Theorem 1. Let ξ and η be independent uncertain sets with membership functions µ and ν, respectively. If ξ is a set A of real numbers, then Liu s Inference Rule 1 yields that η has a membership function ( ν (y = 1+ sup 2ν(y µ(x+ inf sup 2ν(y 2 µ(x+ inf µ(x, if ν(y < ( if ν(y > 1 µ(x, 0.5, otherwise. supµ(x+ inf µ(x /4 supµ(x+ inf µ(x /4 Proof. Let ν be the membership function of η. Then we have {y η } = {η W ν (y}. Thus M{y η } = M{η W ν (y} = ν (y. It follows from Liu s Inference Rule 1 that ν (y = M{y η } = M{y η A ξ}. (3

5 On Liu s Inference Rule for Uncertain Systems 5 Both the definition of conditional uncertainty and that of independence of uncertain sets imply M{(y η (A ξ} M{(y η (A ξ}, if < 0.5 M{A ξ} M{A ξ} M{y η A ξ} = M{(y η (A ξ} M{(y η (A ξ} 1, if < 0.5 M{A ξ} M{A ξ} 0.5, otherwise M{y η} M{A ξ} M{y η} M{A ξ}, if < 0.5 M{A ξ} M{A ξ} = M{y η} M{A ξ} M{y η} M{A ξ} 1, if < 0.5 M{A ξ} M{A ξ} 0.5, otherwise M{y η} M{y η}, if M{A ξ} M{A ξ} < 0.5 = M{y η} M{y η} 1, if M{A ξ} M{A ξ} < , otherwise. Since µ is the membership function of ξ, we immediately have Thus we get In addition, we have Thus {A ξ} = {ξ W α }, with α = inf µ(x. M{A ξ} = M{ξ W α } = α = inf µ(x. {A ξ c } = {ξ W α }, with α = supµ(x. M{A ξ c } = M{ξ W α } = α = supµ(x. It follows from the definition of membership degree that M{A ξ} = 1 2 ( inf µ(x+sup µ(x. (4 The equation (3 follows from M{y η} = ν(y, M{y η} = 1 ν(y and Equation (4 immediately.

6 6 X. Gao, Y. Gao & D. A. Ralescu Theorem 2. Let ξ and η be independent uncertain sets with membership functions µ and ν, respectively. If ξ is a constant a, then Liu s Inference Rule 1 yields that η has a membership function ν (y = ν(y µ(a, if ν(y < µ(a/2 ν(y+µ(a 1, if ν(y > 1 µ(a/2 µ(a 0.5, otherwise. Proof. The constant a can be considered as an uncertain singleton set {a}. It follows from Theorem 1 that and hence the equation (5 is proved. M{a ξ} = 1 (µ(a+µ(a = µ(a 2 (5 4. Inference Rule with Multiple Antecedents Liu s Inference Rule 2. Let X, Y and Z be three concepts. Assume a rule if X is an uncertain set ξ and Y is an uncertain set η then Z is an uncertain set τ. From X is an uncertain set ξ and Y is an uncertain set η we infer that Z is an uncertain set τ = τ (ξ ξ (η η which is the conditional uncertain set of τ given ξ ξ and η η. Liu s inference rule is represented by Rule: If X is ξ and Y is η then Z is τ From: X is ξ and Y is η Infer: Z is τ = τ (ξ ξ (η η Theorem 3. Let ξ, η, τ be independent uncertain sets with membership functions µ, ν, λ, respectively. If ξ is a set A of real numbers and η is a set B of real numbers, then Liu s Inference Rule 2 yields that τ has a membership function where µ = 1 2 λ (z = λ(z µ ν, µ ν if λ(z < 2 λ(z+ µ ν 1 µ ν, if λ(z > 1 µ ν 2 0.5, otherwise, µ(x+sup µ(x, and ν = 1 ( inf 2 y B ( inf ν(y+ supν(y y B.

7 On Liu s Inference Rule for Uncertain Systems 7 Proof. It follows from Liu s Inference Rule 2 that τ has a membership function λ (z = M{z τ (A ξ (B η} Both the definition of conditional uncertainty and that of independence of uncertain sets give M{z τ (A ξ (B η} = 1 M{z τ} M{A ξ} M{B η}, if M{z τ} M{A ξ} M{B η} < 0.5 M{z τ} M{A ξ} M{B η}, if M{z τ} M{A ξ} M{B η} < , otherwise. The theorem follows from M{z τ} = λ(z, M{z τ} = 1 λ(z, Equation (4 and M{(A ξ (B η} = M{A ξ} M{B η}. Theorem 4. Let ξ, η, τ be independent uncertain sets with membership functions µ,ν,λ, respectively. If ξ is a constant a and η is a constant b, then Liu s Inference Rule 2 yields that τ has a membership function λ (z = λ(z µ(a ν(b, µ(a ν(b if λ(z < 2 λ(z+µ(a ν(b 1, if λ(z > 1 µ(a ν(b µ(a ν(b 2 0.5, otherwise. Proof. µ = µ(a and ν = ν(b can be proved by a similar process following the proof of Theorem 2. The theorem follows from Theorem 3 immediately. 5. Inference Rule with Multiple If-Then Rules Liu s Inference Rule 3. Let X and Y be two concepts. Assume two rules if X is an uncertain set ξ 1 then Y is an uncertain set η 1 and if X is an uncertain set ξ 2 then Y is an uncertain set η 2. From X is an uncertain set ξ we infer that Y is an uncertain set η = M{ξ ξ 1 } η 1 ξ ξ 1 M{ξ ξ 1 }+M{ξ ξ 2 } + M{ξ ξ 2 } η 2 ξ ξ 2 M{ξ ξ 1 }+M{ξ ξ 2 }. (6 The inference rule is represented by Rule 1: If X is ξ 1 then Y is η 1 Rule 2: If X is ξ 2 then Y is η 2 From: X is ξ Infer: Y is η determined by (6

8 8 X. Gao, Y. Gao & D. A. Ralescu Theorem 5. Let ξ 1,ξ 2,η 1,η 2 be independent uncertain sets with membership functions µ 1,µ 2,ν 1,ν 2, respectively. If ξ is a set A of real numbers, then Liu s Inference Rule 3 yields η = µ 1 µ 1 + µ 2 η 1 + µ 2 µ 1 + µ 2 η 2 where η1 and η 2 are uncertain sets whose membership functions are respectively given by ν 1 (y, if ν 1 (y < µ 1 /2 µ 1 ν1(y = ν 1 (y+ µ 1 1, if ν 1 (y > 1 µ 1 /2 µ 1 0.5, otherwise, with µ 1 = 1 2 ( inf ν2 (y = µ 1(x+ sup ν 2 (y µ 2, if ν 2 (y < µ 2 /2 ν 2 (y+ µ 2 1 µ 2, if ν 2 (y > 1 µ 2 /2 0.5, otherwise, µ 1 (x, and µ 2 = 1 ( inf 2 µ 2(x+ sup µ 2 (x. Proof. The theorem follows from Liu s Inference Rule 3 and Theorem 1 immediately. Theorem 6. Let ξ 1,ξ 2,η 1,η 2 be independent uncertain sets with continuous membership functions µ 1,µ 2,ν 1,ν 2, respectively. If ξ is a constant a, then Liu s Inference Rule 3 yields η = µ 1 (a µ 1 (a+µ 2 (a η 1 + µ 2 (a µ 1 (a+µ 2 (a η 2 where η1 and η 2 are uncertain sets whose membership functions are respectively given by ν 1 (y µ 1 (a, if ν 1(y < µ 1 (a/2 ν1(y = ν 1 (y+µ 1 (a 1, if ν 1 (y > 1 µ 1 (a/2 µ 1 (a 0.5, otherwise,

9 ν2 (y = On Liu s Inference Rule for Uncertain Systems 9 ν 2 (y µ 2 (a, if ν 2(y < µ 2 (a/2 ν 2 (y+µ 2 (a 1, if ν 2 (y > 1 µ 2 (a/2 µ 2 (a 0.5, otherwise. Proof. µ 1 = µ 1 (a and µ 2 = µ 2 (a can be proved by a similar process following the proof of Theorem 2. The theorem follows from Theorem 5 immediately. 6. General Inference Rule Now it is easy to extend Liu s Inference Rule to the general case. Liu s Inference Rule 4. Let X 1, X 2,...,X m be concepts. Assume rules if X 1 is ξ i1 and and ξ im then Y is η i for i = 1,2,...,k. From X 1 is ξ1 and and ξm we infer that Y is an uncertain set η = k i=1 where the coefficients are determined by c i η i (ξ 1 ξ i1 (ξ 2 ξi2 (ξ m ξim c 1 +c 2 + +c k (7 c i = M{(ξ 1 ξ i1 (ξ 2 ξ i2 (ξ m ξ im } for i = 1,2,...,k. The inference rule is represented by Rule 1: If X 1 is ξ 11 and and X m is ξ 1m then Y is η 1 Rule 2: If X 1 is ξ 21 and and X m is ξ 2m then Y is η 2 Rule k: If X 1 is ξ k1 and and X m is ξ km then Y is η k From: X 1 is ξ 1 and and X m is ξ m Infer: Y is η determined by (7. Theorem 7. Assume ξ i1,ξ i2,...,ξ im,η i are independent uncertain sets with membership functions µ i1,µ i2,...,µ im,ν i, i = 1,2,...,k, respectively. If ξ 1,ξ 2,...,ξ m are sets A 1,A 2,...,A m, respectively, then Liu s Inference Rule 4 yields η = k i=1 c i η i c 1 +c 2 + +c k where ηi are uncertain sets whose membership functions are given by ν i (y, if ν i (y < c i /2 c i νi(y = ν i (y+c i 1, if ν i (y > 1 c i /2 c i 0.5, otherwise,

10 10 X. Gao, Y. Gao & D. A. Ralescu and c i are constants determined by c i = min 1 l m µ il with µ il = 1 ( inf µ il (x+ sup µ il (x for i = 1,2,...,k, respectively. 2 l l Proof. For each i, since A 1 ξ i1, A 2 ξ i2,...,a m ξ im are independent events, we immediately have { m } M (A l ξ il = min M{A l ξ il } = min µ il 1 l m 1 l m l=1 for i = 1,2,...,k. From those equations, we can prove the theorem by Liu s Inference Rule 4 immediately. Theorem 8. Assume ξ i1,ξ i2,...,ξ im,η i are independent uncertain sets with membership functions µ i1,µ i2,...,µ im,ν i, i = 1,2,...,k, respectively. If ξ 1,ξ 2,...,ξ m are constants a 1,a 2,...,a m, respectively, then Liu s Inference Rule 4 yields η = k i=1 c i η i c 1 +c 2 + +c k where ηi are uncertain sets whose membership functions are given by ν i (y, if ν i (y < c i /2 c i νi (y = ν i (y+c i 1, if ν i (y > 1 c i /2 c i 0.5, otherwise, and c i are constants determined by for i = 1,2,...,k, respectively. c i = min 1 l m µ il(a l Proof. The equation µ il = µ il (a l can be proved by a similar process following the proof of Theorem 2. The theorem follows from Theorem 7 immediately. 7. Conclusions We have obtained some explicit expressions for Liu s inference rule by membership functions for uncertain systems. Furthermore, we extended Liu s inference rule to uncertain inference systems with multiple antecedents and with multiple if-then rules, and finally obtained the general inference rule.

11 On Liu s Inference Rule for Uncertain Systems 11 Acknowledgments This work was supported by Chinese Universities Scientific Fund (No. 09QL47. Dan A. Ralescu s work was partly supported by a Taft Travel for Research Grant. References 1. X. Chen and B. Liu, Existence and uniqueness theorem for uncertain differential equations, Fuzzy Optimization and Decision Making, to be published. 2. X. Chen and D. A. Ralescu, A note on truth value in uncertain logic, X. Gao, Some properties of continuous uncertain measure, Int. J. Uncertainty, Fuzziness and Knowledge-Based Systems 17(3 ( X. Gao and S. Sun, Variance formula for trapezoidal uncertain variables, in Proc. Eighth Int. Conf. Information and Management Sciences, 2009, pp X. Li and B. Liu, Hybrid logic and uncertain logic, J. Uncertain Systems 3(2 ( B. Liu, Uncertainty Theory, 2nd edn. (Springer-Verlag, Berlin, B. Liu, Fuzzy process, hybrid process and uncertain process, J. Uncertain Systems 2(1 ( B. Liu, Some research problems in uncertainty theory, J. Uncertain Systems 3(1 ( B. Liu, Uncertainty Theory, 3rd edn., B. Liu, Theory and Practice of Uncertain Programming, 2nd edn. (Springer-Verlag, Berlin, B. Liu, Uncertain set theory and uncertain inference rule with application to uncertain control, J. Uncertain Systems 4(2 ( C. You, Some convergence theorems of uncertain sequences, Mathematical and Computer Modelling 49(3 4 ( Y. Zhu, Uncertain optimal control with application to a portfolio selection model,

12 Copyright of International Journal of Uncertainty, Fuzziness & Knowledge-Based Systems is the property of World Scientific Publishing Company and its content may not be copied or ed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or articles for individual use.