On Liu s Inference Rule for Uncertain Systems Xin Gao 1,, Dan A. Ralescu 2 1 School of Mathematics Physics, North China Electric Power University, Beijing 102206, P.R. China 2 Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221-0025, USA *Email: gao-xin00@mails.tsinghua.edu.cn Abstract: Liu s inference is a process of deriving consequences from uncertain knowledge or evidence via the tool of conditional uncertainty. Using identication 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 with multiple -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 rom nor fuzzy, we cannot deal with it by probability theory or credibility theory. In order to deal with subjective uncertainty, Liu [7] founded an uncertainty theory that is based on normality, monotonicity, self-duality, countable subadditivity, product measure axioms. Considerable work has been done based on uncertainty theory. Gao [3] gave some mathematical properties of uncertain measure. You [14] studied some convergence theorems of uncertain sequences. Liu [11] founded uncertain programming to solve optimization problems in uncertain environments. Liu [8] proposed uncertain calculus the concept of uncertain process. Li Liu [6] proposed uncertain logic defined the truth value as the uncertain measure that an uncertain proposition is true. Uncertainty theory has become a new tool to describe subjective uncertainty has wide applications in engineering. For a detailed exposition, the interested reader may consult the book [10]. Uncertain inference is the process of deriving consequences from uncertain knowledge or evidence. When systems exhibit fuzziness, fuzzy inference has been studied inference rules have been suggested. For example, Zadeh s compositional rule of inference, Lukasiewicz s inference rule, Mamdani s inference rule are widely used in fuzzy inference systems. Dferent from the above inference rules, Liu [9, 10] proposed uncertain inference including a new inference rule. This new inference rule uses the tool of conditional uncertain set. Proceedings of the Eighth International Conference on Information Management Sciences, Kunming, China, July 20-28, 2009, pp. 958-964. In this paper, we will investigate the expressions of Liu s inference rule for uncertain systems via identication functions. Section 2 introduces some basic concepts 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 identication functions in Section 3. We also investigate Liu s inference rule with multiple antecedents with multiple -then rules in Section 4 Section 5, respectively. Section 6 concludes this paper with a brief summary. 2 Preliminaries In this section, we will introduce some useful definitions about conditional uncertainty uncertain sets. Let Γ be a nonempty set, let L be a σ-algebra over Γ. Each element Λ L is called an event. In order to measure uncertain events, Liu [7] introduced an uncertain measure M as a set function satisfying normality, monotonicity, self-duality, 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 [7] defined a conditional uncertain measure M{A B} as follows: Definition 1 (Liu [7]) Let (Γ, L, M) be an uncertainty space, A, B L. Then the conditional uncertain measure of A given B is defined by M{A B} provided that M{B} > 0. M{A B} M{A B}, M{B} M{B} 1 M{Ac B}, M{Ac B} M{B} M{B} 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, the set {ξ B} {γ Γ ξ(γ) B} is an event in L. Dferent from the definition of uncertain variable, the concept of uncertain set was proposed by Liu [7] in 2007.
ON LIU S INFERENCE RULE FOR UNCERTAIN SYSTEMS 959 Definition 2 (Liu [7]) An uncertain set is a measurable function ξ from an uncertainty space (Γ, L, M) to a collection of subsets of real numbers, i.e., for any Borel set B, the set is an event in L. {γ Γ ξ(γ) B } Definition 3 (Liu [10]) An uncertain set ξ on (Γ, L, M) is said to have a first identication function λ(x) 0 λ(x) M{γ}, x R, γ Γ ξ(γ) {x R λ(x) M{γ}}, γ Γ. It is not true that any uncertain set has an identication function. A necessary condition is that ξ(γ 1 ) ξ(γ 2 ) whenever M{γ 1 } M{γ 2 }. In addition, a first identication function must meet the conditions 0 λ(x) 1, (λ(x) + λ(y)) 1. x y Example 1 Let ξ be an uncertain set taking values A 1, A 2,, A n with first identication coefficients λ 1, λ 2,, λ n, i.e., A 1 with first identication coefficient λ 1 A ξ 2 with first identication coefficient λ 2 A n with first identication coefficient λ n, where A i, i 1, 2,, n are sets of real numbers. Then for any collection A of sets in {A 1, A 2,, A n }, we have M{ξ A} λ i, 1 c λ i, λ i λ i 0.5. Definition 4 An uncertain variable ξ is said to have a second identication function ρ (i) ρ(x) is a nonnegative integrable function on R such that ρ(x)dx 1; R (ii) for any Borel set B of real numbers, we have ρ(x)dx, ρ(x)dx B B M{ξ B} 1 ρ(x)dx, ρ(x)dx B c B c Example 2 Let ξ be an uncertain set taking values A 1, A 2,, A n with second identication coefficients ρ 1, ρ 2,, ρ n, i.e., A 1 with second identication coefficient ρ 1 A ξ 2 with second identication coefficient ρ 2 A n with second identication coefficient ρ n, where A i, i 1, 2,, n are sets of real numbers. Then for any collection A of sets in {A 1, A 2,, A n }, we have ρ i, ρ i M{ξ A} 1 ρ i, ρ i c c Definition 5 (Liu [10]) 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 ξ B (γ) ξ(γ), γ Γ in the sense of classical set theory. Let ξ ξ be two uncertain sets. In order to define the matching degree that ξ matches ξ, we first introduce the following symbols: {ξ ξ} {γ Γ ξ (γ) ξ(γ)}, {ξ ξ c } {γ Γ ξ (γ) ξ(γ) c }, {ξ ξ c } {γ Γ ξ (γ) ξ(γ) }. Definition 6 (Liu [10]) Let ξ ξ be two uncertain sets. Then the event ξ ξ (i.e., ξ matches ξ) is defined as { {ξ ξ}, M{ξ ξ} > M{ξ ξ c } {ξ {ξ ξ c }, M{ξ ξ} M{ξ ξ c }. The matching degree of ξ ξ is defined as { M{ξ ξ}, M{ξ ξ} > M{ξ ξ c } M{ξ ξ} 3 Inference Rule M{ξ ξ c }, M{ξ ξ} M{ξ ξ c }. Liu s Inference Rule 1 (Liu [9, 10]) Let X Y be two concepts. Assume a rule of the form 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
960 XIN GAO, DAN A. RALESCU 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, ξ η be independent uncertain sets with continuous first identication functions λ, λ ν, respectively. Then Liu s Inference Rule 1 yields that η has a first identication function ν (y) λ(x) λ (x) 0.5, 1 λ(x) λ (x) {λ(x) λ(x) < λ (x)} 0.5, λ(x) λ (x) 0.5. Proof: From the continuity of λ, λ ν, we have λ(x) λ (x) ν (y) 0.5. Both the definition of conditional uncertainty that of independence of uncertain sets imply M{y η ξ M{(y η) (ξ ξ)} M{(y η) (ξ ξ)} M{ξ 1 M{(y η) (ξ ξ)} M{(y η) (ξ ξ)} M{ξ M{y η} M{ξ M{y η} M{ξ M{ξ 1 M{y η} M{ξ M{y η} M{ξ M{ξ M{y η} M{y η} M{ξ M{y η} 1 M{y η} M{ξ It follows from Liu s Inference Rule 1 that ν (y) M{y η } 0.5 M{y η ξ 0.5 M{y η} M{ξ 0.5 for any y R. By using the definition of first identication function, we have {, M{y η} 1, 0.5. Thus for any y R, we have ν (y) M{ξ 0.5. (1) Now we calculate M{ξ. We consider two cases. Case 1: If then we get λ(x) λ (x), M{ξ ξ c } λ(x) λ (x) immediately. Since {ξ ξ} {ξ ξ c } {ξ ξ c } {ξ ξ c } c, we have M{ξ ξ} M{ξ ξ c }, 0.5 < 1 M{ξ ξ c } M{ξ ξ c }. Hence the matching degree is M{ξ M{ξ ξ c } λ(x) λ (x), for this case we obtain Case 2: If ν (y) λ(x) λ (x) 0.5. λ(x) λ (x) 0.5, then M {ξ ξ c } M{ } 0. Thus M{ξ ξ} M{ξ ξ c }, the matching degree is M{ξ M{ξ ξ} 1 M{ξ ξ} 1 {λ(x) λ(x) < λ (x)}.
ON LIU S INFERENCE RULE FOR UNCERTAIN SYSTEMS 961 For this case, we obtain ν (y) 1 {λ(x) λ(x) < λ (x)} 0.5. The theorem is proved. Note that λ(x) λ (x) 0, then we set ν (y) 0.5. This means that nothing is known then anything is possible. Theorem 2 Let ξ, ξ η be independent uncertain sets with first identication functions λ, λ ν, respectively. If λ (x) λ(x) for all x, then Liu s Inference Rule 1 yields that η has a first identication function ν (y). Proof: Since λ (x) λ(x) for all x, we have Hence λ(x) λ (x) 0.5. λ(x) λ (x) λ (x) 0.5. Thus M {ξ ξ c } M{ } 0, the matching degree is M{ξ M{ξ ξ} M{Γ} 1. It follows from Liu s Inference Rule 1 the definition of conditional uncertainty that M{y η } M{y η ξ M{y η} for any y R. From the definition of first identication functions, we have ν (y). Theorem 3 Let ξ η be independent uncertain sets with continuous first identication functions λ ν, respectively. If ξ is a constant a, then Liu s Inference Rule 1 yields that η has a first identication function ν 0.5, λ(a) (y) λ(a), λ(a) 0.5. Proof: From Equation (1), we have ν (y) M{a 0.5. It follows from the definition of matching degree that { λ(a), λ(a) M{a M{a ξ} 1, λ(a) 0.5. Therefore, ν 0.5, λ(a) (y) λ(a), λ(a) 0.5. Theorem 4 Let ξ be an uncertain set taking values A 1, A 2,, A n with first identication coefficients λ 1, λ 2,, λ n, let η be an uncertain set taking values B 1, B 2,, B m with first identication coefficients ν 1, ν 2,, ν m, respectively. If ξ η are independent ξ is a constant a, then Liu s Inference Rule 1 yields that η is an uncertain set taking values B 1, B 2,, B m with first identication coefficients ν1, ν2,, νm, respectively, where ν k /T, ν k /T νk 1 (1 ν k )/T, (1 ν k )/T for k 1, 2,, m max {λi a Ai}, T 1 max {λi a Ai}, max {λi a Ai} max {λi a Ai} 0.5. Proof: It follows from Liu s Inference Rule 1 the definition of conditional uncertainty that ν k M{η B k a M{η B k } M{a, M{η B k} M{a 1 M{η B k} M{a, M{η B k} M{a From the definition of first identication coefficients, we have M{η B k } ν k, M{η B k } 1 ν k for each k. From the definition of matching degree, we obtain M{a max {λ i a A i }, 1 max {λ i a A i }, which is defined as T. The theorem is veried. max {λ i a A i } max {λ i a A i } 0.5. Theorem 5 Let ξ be an uncertain set taking values A 1, A 2,, A n with second identication coefficients ρ 1, ρ 2,, ρ n, let η be an uncertain set taking values B 1, B 2,, B m with second identication coefficients ν 1, ν 2,, ν m, respectively. If ξ η are independent ξ is a constant a, then Liu s Inference Rule 1 yields that η is an uncertain set (not necessarily having second identication coefficients) such that ν i B i B T, ν i B i B T M{η B} 1 ν i B i B T, ν i B i B T
962 XIN GAO, DAN A. RALESCU for any collection B of sets in {B 1, B 2,, B m }, where n n {ρ i a A i }, {ρ i a A i } i1 i1 T 1 n n {ρ i a A i }, {ρ i a A i } i1 i1 Proof: It follows from Liu s Inference Rule 1 the definition of conditional uncertainty that M{η B} M{η B}, M{a M{a M{η B} M{η B} M{η B} 1, M{a M{a Denoting M{a by T, we get n n {ρ i a A i }, {ρ i a A i } i1 i1 T 1 n n {ρ i a A i }, {ρ i a A i } i1 i1 From the definition of second identication coefficients, we have B i B B i B M{η B} 1 B i B B i B, B i B B i B M{η B} 1 B i B B i B We divide the proof into three cases. Case 1: If M{η B}/T, then M{η B}<0.5 M{η B} {ν i B i B}. Hence M{η B} ν i T. Case 2: If B i B M{η B}/T, then M{η B} M{η B} {ν i B i B}. Hence M{η B} 1 ν i T. Case 3: If B i B M{η B}/T 0.5, M{η B}/T 0.5, then we have M{η B} 0.5. The theorem is proved. 4 Inference Rule with Multiple Antecedents Liu s Inference Rule 2 Let X, Y Z be three concepts. Assume a rule X is an uncertain set ξ Y is an uncertain set η then Z is an uncertain set τ. From X is an uncertain set ξ 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 η η. Liu s inference rule is represented by Rule: If X is ξ Y is η then Z is τ From: X is ξ Y is η Infer: Z is τ τ (ξ ξ) (η η) Theorem 6 Let ξ, ξ, η, η τ be independent uncertain sets with continuous first identication functions functions λ, λ, ν, ν ψ respectively. Then Liu s Inference Rule 2 yields that τ has a first identication function ψ (z) 1 λ(x) λ (x) ν (y) 0.5, λ(x) λ (x) ν (y) {λ(x) λ(x)<λ (x)} { <ν (y)} 0.5, λ(x) λ (x) ν (y) 0.5. Proof: From the continuity of λ, λ, ν, ν τ, we have λ(x) λ (x) ν (y) τ(z) τ (z) 0.5. z R z R Both the definition of conditional uncertainty that of independence of uncertain sets give M{z τ (ξ ξ) (η η)} M{z τ} M{ξ M{η η}, M{z τ} M{ξ M{η η} M{z τ} 1 M{ξ M{η η}, M{z τ} M{ξ M{η η} It follows from Liu s Inference Rule 2 that ψ (z) M{z τ} M{ξ M{η η} 0.5
ON LIU S INFERENCE RULE FOR UNCERTAIN SYSTEMS 963 for all z R. By using the definition of first identication function, we have {, M{z τ} 1, 0.5. Thus for all z R, we have ψ (z) M{ξ M{ξ M{η 0.5. (2) η} It follows from the definition of matching degree that λ(x) λ (x), M{η η} 1 λ(x) λ (x) {λ(x) λ(x) < λ (x)}, λ(x) λ (x) 0.5, ν (y), 1 We divide the proof into two cases. Case 1: If then we get ν (y) { < ν (y)}, λ(x) λ (x) 0.5. λ(x) λ (x) ν (y), M{ξ ξ} M{η η} λ(x) λ (x) ν (y) ψ (z) Case 2: If then we get λ(x) λ (x) ν (y) 0.5. λ(x) λ (x) ν (y) 0.5, M{ξ M{η η} 1 {λ(x) λ(x) < λ (x)} { < ν (y)} ψ (z) 1 {λ(x) λ(x)<λ (x)} { <ν (y)} 0.5. The theorem is proved. Theorem 7 Let ξ, η, τ be independent uncertain sets with continuous first identication functions λ, ν, ψ, respectively. If ξ is a constant a η is a constant b, then Liu s Inference Rule 2 yields that τ has a first identication function ψ 0.5, λ(a) ν(b) (z) λ(a) ν(b), λ(a) ν(b) 0.5. Proof: From Equation (2), we have ψ (z) M{a M{b η} 0.5. It follows from the definition of matching degree that { λ(a), λ(a) M{a 1, λ(a) 0.5, Therefore, M{b η} M{a ξ} M{b η} ψ (z) { ν(b), ν(b) 1, ν(b) 0.5. { λ(a) ν(b), λ(a) ν(b) 1, λ(a) ν(b) 0.5. λ(a) ν(b) 0.5, λ(a) ν(b), λ(a) ν(b) 0.5. 5 Inference Rule with Multiple If-Then Rules Assume that we have n rules X is ξ 1 then Y is η 1, X is ξ 2 then Y is η 2,, X is ξ n then Y is η n. From X ξ we may determine an index k such that M{ξ ξ k } max M{ξ ξ i } which is just the index such that ξ ξ k is the most matching event (i.e., the event with the largest uncertain measure). Thus we infer that Y is a conditional uncertain set of η given the most matching event ξ ξ k. Liu s Inference Rule 3 Let X Y be two concepts. Assume n rules X is an uncertain set ξ 1 then Y is an uncertain set η 1, X is an uncertain set ξ 2 then Y is an uncertain set η 2,, X is an uncertain set ξ n then Y is an uncertain set η n. From X is an uncertain set ξ we infer that Y is an uncertain set η η k ξ ξ k (3)
964 XIN GAO, DAN A. RALESCU where ξ ξ k is the most matching event. Liu s 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 Rule n: If X is ξ n then Y is η n From: X is ξ Infer: Y is η determined by (3) Theorem 8 Let ξ i, η i, i 1, 2,, n, ξ be independent uncertain sets with continuous first identication functions λ i, ν i, i 1, 2,, n, λ, respectively. Then Liu s Inference Rule 3 yields that η has a first identication function ν ν k (y) (y) M{ξ ξ k } 0.5 where k is determined as follows: Step 1: Calculate M{ξ ξ i } for i 1, 2,, n by λ i (x) λ (x), λ i (x) λ (x) M{ξ ξ i } 1 {λ i (x) λ i (x) < λ (x)}, λ i (x) λ (x) 0.5. Step 2: Find k such that M{ξ ξ k } max M{ξ ξ i }. Proof: From the definition of matching degree λ i (x) λ (x), λ i (x) λ (x) M{ξ ξ i } 1 {λ i (x) λ i (x) < λ (x)}, λ i (x) λ (x) 0.5. Find k such that M{ξ ξ k } max M{ξ ξ i }, thus ξ ξ k is the most matching event. Following Liu s Inference Rule 3, we have ν (y) M{y η k ξ 0.5 M{y η k} M{ξ ξ k } 0.5 ν k (y) M{ξ ξ k } 0.5. 6 Conclusions We have obtained some explicit expressions for Liu s inference rule by identication functions for uncertain systems. Furthermore, we extended Liu s inference rule to uncertain inference systems with multiple antecedents with multiple -then rules. Acknowledgments This work was ported by National Natural Science Foundation of China Grant No. 60874067. Dan A. Ralescu s work was partly ported by a Taft Travel for Research Grant. References [1] Chen X, Liu B, Existence uniqueness theorem for uncertain dferential equations, Fuzzy Optimization Decision Making, to be published. [2] Chen X, Ralescu D A, A note on truth value in uncertain logic, http://orsc.edu.cn/online/090211.pdf, 2009. [3] Gao X, Some properties of continuous uncertain measure, International Journal of Uncertainty, Fuzziness Knowledge-Based Systems, Vol.17, No.3, pp.419-426, 2009. [4] Gao X, You C, Maximum entropy membership functions for discrete fuzzy variables, Information Sciences, Vol.179, No.14, pp.2353-2361, 2009. [5] Gao X, Sun S, Variance formula for trapezoidal uncertain variables, Proceedings of the Eighth International Conference on Information Management Sciences, pp.853-855, 2009. [6] Li X, Liu B, Hybrid logic uncertain logic, Journal of Uncertain Systems, Vol.3, No.2, pp.83-94, 2009. [7] Liu B, Uncertainty Theory, 2nd ed., Springer-Verlag, Berlin, 2007. [8] Liu B, Fuzzy process, hybrid process uncertain process, Journal of Uncertain Systems, Vol.2, No.1, pp.3-16, 2008. [9] Liu B, Some research problems in uncertainty theory, Journal of Uncertain Systems, Vol.3, No.1, pp.3-10, 2009. [10] Liu B, Uncertainty Theory, 3rd ed., http://orsc.edu.cn/liu/ut.pdf. [11] Liu B, Theory Practice of Uncertain Programming, 2nd ed., Springer-Verlag, Berlin, 2009. [12] Liu L, Gao X, Fuzzy weighted equilibrium assignment problem genetic algorithm, Applied Mathematical Modelling, Vol.33, No.10, 3926-3935, 2009. [13] Qin Z, Gao X, Fractional Liu process with application to finance, Proceedings of the Seventh International Conference on Information Management Sciences, pp.277-280, 2008. [14] You C, Some convergence theorems of uncertain sequences, Mathematical Computer Modelling, Vol.49, Nos.3-4, pp.482-487, 2009. [15] Zhu Y, Uncertain optimal control with application to a portfolio selection model, http://orsc.edu.cn/online/090524.pdf, 2009.