Uncertain Logic with Multiple Predicates

Uncertain Logic with Multiple Predicates Kai Yao, Zixiong Peng Uncertainty Theory Laboratory, Department of Mathematical Sciences Tsinghua University, Beijing 100084, China yaok09@mails.tsinghua.edu.cn, pengzx07@mails.tsinghua.edu.cn Abstract An uncertain proposition is a statement comprising uncertain quantifier, uncertain subject and uncertain predicate, which essentially are all uncertain sets. This paper verifies some theorems on an uncertain proposition with multiple predicates, extending the application of uncertain logic in artificial intelligence. Besides, it applies uncertain logic to some indexes of regional construction material industry, and discovers the relationship between the indexes. Keywords: logic, uncertainty theory, uncertain predicate, data mining 1 Introduction Logic plays an important role in artificial intelligence. Classical two-valued logic, assuming each proposition is either right or wrong, can only deal with precise knowledge. It was extended to multi-valued logic by Lukasiewicz in 1920 s to describe vague knowledge, which fails to be absolutely right or wrong. In 1965, Zadeh [15] founded a fuzzy set theory. After that, fuzzy logic as a form of multi-valued logic was derived to deal with approximate reasoning. The truth value for a fuzzy proposition was defined by many researchers such as Zadeh [17], Yager [13] and Bosc and Lietard [1] in different ways. However, they all failed to consider the opposite of a proposition. For example, almost all of young teachers are tall and almost none of young teachers are not tall share the same semantics, but they may have different truth value by their definitions. In 1978, Zadeh [16] introduced possibility theory as an extension of fuzzy sets theory. Based on the possibility theory, Dubios and Prade [2] proposed a possibilistic logic in 1987. Almost at the same time, Nilsson [12] proposed probabilistic logic based on probability theory. In addition, Nilsson [12] proved the consistency between probabilistic logic and classical logic. In 2007, Liu [6] founded an uncertainty theory to deal with imprecise knowledge in human language. Based on the uncertainty theory, Li and Liu [5] proposed a propositional logic and Zhang and Peng [18] proposed a predicate logic in uncertain environment. However, the proposition logic and predicate 1

logic find little use in daily life. In 2010, Liu [9] proposed an uncertain set theory in the framework of uncertain theory. After that, Liu [11] proposed uncertain logic via uncertain set theory, which can be well used in artificial intelligence. Meanwhile, Liu [11] defined a truth value for uncertain proposition. In the definition, both the proposition and its opposite were considered. Now, the uncertain logic deals with simple proposition such as most young teachers are tall very well. This paper will extend uncertain logic to the case with multiple predicates, such as most young teachers are young and tall and run quickly. It is not equivalent to most young teachers are all, and most young teachers run quickly. In this paper, we will present uncertain logic with multiple predicates to deal with the above problem, and prove some theorems on its truth value. The remainder of this paper is structured as follows. The next section is intended to introduce some concepts of uncertain set theory and uncertain logic. Section 3 gives a concept of uncertain compound predicate. After that, an uncertain proposition with multiple predicates is given in Section 4 and its truth value is also verified. Section 5 gives a numerical example on uncertain proposition with multiple predicates based on some data of regional construction material industry. Finally, some remarks are made in Section 6. 2 Preliminary Uncertainty theory was founded by Liu [6] in 2007 and refined by Liu [8] in 2010. Many researchers contributed a lot in uncertainty theory. Gao [4] proved some properties of continuous uncertain measure, and You [14] gave some convergence theorems on uncertain sequences. In 2010, Liu [10] applied uncertainty theory to risk analysis, and proposed uncertain risk analysis. Now, it has become a branch of axiomatic mathematics, and find many application in mathematical programming (Liu [7]). In 2010, Liu [9] proposed an uncertain set theory. Based on uncertain set theory, Liu [11] proposed uncertain logic to deal with uncertain knowledge. In this section, we will introduce some useful definitions about uncertain set theory and uncertain logic. Definition 1. (Liu [9]) An uncertainty 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 {ξ B} = {γ Γ ξ(γ) B} is an event. Definition 2. (Liu [9]) The uncertain sets ξ 1, ξ 2 are said to be independent if M {(ξ 1 B 1 ) (ξ 2 B 2 )} = M{ξ 1 B 1 } M{ξ 2 B 2 } M {(ξ 1 B 1 ) (ξ 2 B 2 )} = M{ξ 1 B 1 } M{ξ 2 B 2 } for any Borel sets B 1, B 2 of real numbers. 2

Definition 3. (Liu [11]) Let ξ be an uncertain set. Then its membership function is defined as µ(x) = M{x ξ} for any x R. Uncertain quantifier is an uncertain set representing the number of individuals in the text. Uncertain subject is an uncertain set containing some specified individuals in the universe. Uncertain predicate is an uncertain set representing a property that the individuals in the context have in common. They all are elements of uncertain proposition. Definition 4. (Liu [11]) Assume that Q is an uncertain quantifier, S is an uncertain subject, and ξ is an uncertain predicate. Then is called an uncertain proposition. (Q, S, ξ) = Q of S are ξ Example 1. Most young teachers are tall is an uncertain proposition in which most is an uncertain quantifier, young teachers is an uncertain subject, and tall is an uncertain predicate. Definition 5. (Liu [11]) Let (Q, S, ξ) be an uncertain proposition on the universe A. Assume that Q is a unimodal uncertain quantifier with a membership function λ and a dual membership function λ, S is an uncertain subject with a membership function ν, and ξ is an uncertain predicate with a membership function µ and a negated membership function µ(x). Then the truth value of (Q, S, ξ) with respect to the universe A is defined by T (Q, S, ξ) = ( ω inf µ(a) K K ω inf µ(a) where K ω = {K S ω λ( K ) ω}, K ω = {K S ω λ ( K ) ω}, S ω = {a A ν(a) ω}, and K represents the cardinality of K. ) 3 Uncertain Compound Predicate An uncertain compound predicate is a finite sequence of uncertain predicates and connective symbols that must make sense. It is also an uncertain predicate in essence. Usually, there are only three kinds of connective symbols, namely union, intersection and complement denoted by, and, respectively. Example 2. Let ξ be a uncertain predicate. Then τ = ξ is an uncertain compound predicate which means not ξ. Example 3. Let ξ, η be two uncertain predicates. Then τ = ξ η is an uncertain compound predicate which means ξ or η. 3

Example 4. Let ξ, η be two uncertain predicates. Then τ = ξ η is an uncertain compound predicate which means ξ and η. Example 5. Let ξ, η be two uncertain predicates. Then τ = ξ η is an uncertain compound predicate which means ξ but not η. Definition 6. Let ξ be a compound predicate of uncertain predicate ξ 1, ξ 2,, ξ n. If each ξ i appears once in the compound sequence, then ξ is called a simplex compound predicate of ξ 1, ξ 2,, ξ n denoted by ξ = f(ξ 1, ξ 2,, ξ n ). Theorem 1. Let ξ 1, ξ 2,, ξ n be some independent uncertain predicates with membership functions µ 1, µ 2,, µ n, and ξ = f(ξ 1, ξ 2,, ξ n ) be a simplex compound predicate. Then ξ has a membership function µ(x) = f(µ 1 (x 1 ), µ 2 (x 2 ),, µ n (x n )) for any vector x = (x 1, x 2,, x n ) R n. Proof: It follows from the definition of membership function and independence of uncertain sets that µ(x) = M{(x 1, x 2,, x n ) ξ} = M{(x 1, x 2,, x n ) f(ξ 1, ξ 2,, ξ n )} = f(m{x 1 ξ 1 }, M{x 2 ξ 2 },, M{x n ξ n }) = f(µ 1 (x 1 ), µ 2 (x 2 ),, µ n (x n )). The theorem is thus verified. Example 6. Let ξ be a uncertain predicate with a membership function µ. Then τ = ξ is a simplex compound predicate with a membership function λ(x) = 1 µ(x) = µ(x) for any x R. Example 7. Let ξ, η be two independent uncertain predicates with uncertain membership functions µ and ν, respectively. Then τ = ξ η is a simplex compound predicate with a membership function λ(x) = µ(x 1 ) ν(x 2 ) for any vector x = (x 1, x 2 ) R 2. Example 8. Let ξ, η be two independent uncertain predicates with uncertain membership functions µ and ν, respectively. Then τ = ξ η is a simplex compound predicate with a membership function λ(x) = µ(x 1 ) ν(x 2 ) for any vector x = (x 1, x 2 ) R 2. 4

Example 9. Let ξ, η be two independent uncertain predicates with uncertain membership functions µ and ν, respectively. Then τ = ξ η is a simplex compound predicate with a membership function λ(x) = µ(x 1 ) (1 ν(x 2 )) = µ(x 1 ) ν(x 2 ) for any vector x = (x 1, x 2 ) R 2. 4 Uncertain Proposition with Multiple Predicates In human language, propositions are sometimes made with multiple predicates, for example, most teachers are young and tall and almost all the students do well in the mathematics or English exams. This section will model such kind of propositions. Theorem 2. Let Q be a unimodal uncertain quantifier with a membership function λ and a dual membership function λ, S be an uncertain subject with a membership function ν, and ξ 1, ξ 2,, ξ n be independent uncertain predicates with membership functions µ 1, µ 2,, µ n, respectively. If ξ = f(ξ 1, ξ 2,, ξ n ) is a simplex compound quantifier, then the truth value of (Q, S, ξ) with respect to the universal A is ( ) T (Q, S, ξ) = ω inf f(µ 1(a),, µ n (a)) K K ω inf f(µ 1(a),, µ n (a)) (1) where K ω = {K S ω λ( K ) ω}, K ω = {K S ω λ ( K ) ω}, S ω = {a A ν(a) ω}. Moreover, where T (Qaξ(a)) = (ω (k ω ) (kw)) (2) k w = min{x λ(x) ω}, k w = min{x λ (x) ω}, (k ω ) = the k ω -th largest value of {f(µ 1 (a i ),, µ n (a i )) a i S ω }, (k ω) = the k ω-th largest value of { f(µ 1 (a i ),, µ n (a i )) a i S ω }. Proof: It follows from Theorem 1 that the uncertain quantifier ξ = f(ξ 1, ξ 2,, ξ n ) has a membership function µ(x) = f(µ 1 (x 1 ), µ 2 (x 2 ),, µ n (x n )) for any vector x = (x 1, x 2,, x n ). Then the equation (1) follows from the Definition 5 immediately. Since the renum is achieved at the subset with minimum cardinality, we have The equation (2) is verified. inf f(µ 1(a),, µ n (a)) = inf f(µ 1(a),, µ n (a)) = K S ω, K =k ω K S ω, K =k ω inf f(µ 1(a),, µ n (a)) = (k ω ), inf f(µ 1(a),, µ n (a)) = (kω). 5

A truth value algorithm for calculating the truth value of uncertain proposition was first proposed by Liu [11]. Here, we give a truth value algorithm for calculating the truth value of an uncertain proposition with multiple predicates based on Liu s algorithm. Truth Value Algorithm Step 0: Calculate µ(x) = f(µ 1 (x 1 ), µ 2 (x 2 ),, µ n (x n )). Step 1: Find S 1 = {a A ν(a) = 1}, k = min{x λ(x) = 1} and k = S 1 max{x λ(x) = 1}. If (k) (k ) = 1, then T = 1 and stop. Step 2: Find S 0 = {a A ν(a) > 0}, k = min{x λ(x) > 0} and k = S 0 max{x λ(x) > 0}. If (k) (k ) = 0, then T = 1 and stop. Step 3: Set b = 0 and t = 1. Step 4: Set c = (b + t)/2. Step 5: Find S c = {a A ν(a) c}k = min{x λ(x) c} and k = n max{x λ(x) c}. If (k) (k ) > c, then b = c; otherwise t = c. Step 6: If b t > ɛ (a predetermined precision), then go to Step 4; otherwise T = (b + t)/2 and stop. Corollary 1. Let Q be a unimodal uncertain quantifier with a membership function λ and a dual membership function λ, S be an uncertain subject with a membership function ν, and ξ 1 and ξ 2 be two independent uncertain predicates with membership functions µ 1 and µ 2, respectively. Then the truth value of (Q, S, ξ 1 ξ 2 ) with respect to A is ( T (Q, S, ξ 1 ξ 2 ) = ω inf (µ 1(a) µ 2 (a)) K K ω inf (1 µ 1(a) µ 2 (a)) where K ω = {K S ω λ( K ω)}, K ω = {K S ω λ ( K ω)}, S ω = {a A ν(a) ω}. ), (3) Proof: It follows from Example 6 and Example 7 that the uncertain predicate ξ = ξ 1 ξ 2 has a membership function and a negated membership function µ(x) = µ 1 (x 1 ) µ 2 (x 2 ) µ(x) = 1 µ 1 (x 1 ) µ 2 (x 2 ) for any vector x = (x 1, x 2 ) R 2. Then the equation (3) follows from the Theorem 2 immediately. Corollary 2. Let Q be a unimodal uncertain quantifier with a membership function λ and a dual membership function λ, S be an uncertain subject with a membership function ν, and ξ 1 and ξ 2 be two 6

independent uncertain predicates with membership functions µ 1 and µ 2, respectively. Then the truth value of (Q, S, ξ 1 ξ 2 ) with respect to A is ( T (Q, S, ξ 1 ξ 2 ) = ω inf (µ 1(a) µ 2 (a)) K K ω inf (1 µ 1(a) µ 2 (a)) ) (4) where K ω = {K S ω λ( K ω)}, K ω = {K S ω λ ( K ω)}, S ω = {a A ν(a) ω}. Proof: It follows from Example 6 and Example 8 that the uncertain predicate ξ = ξ 1 ξ 2 has a membership function µ(x) = µ 1 (x 1 ) µ 2 (x 2 ) and a negated membership function µ(x) = 1 µ 1 (x 1 ) µ 2 (x 2 ) for any vector x = (x 1, x 2 ) R 2. Then the equation (4) follows from the Theorem 2 immediately. Corollary 3. Let Q be a unimodal uncertain quantifier with a membership function λ and a dual membership function λ, S be an uncertain subject with a membership function ν, and ξ 1 and ξ 2 be two independent uncertain predicates with membership functions µ 1 and µ 2, respectively. Then the truth value of Q, S, ξ 1 ξ 2 ) with respect to A is ( T (Q, S, ξ 1 ξ 2 ) = ω inf (µ 1(a) (1 µ 2 (a))) K K ω inf ((1 µ 1(a)) µ 2 (a)) where K ω = {K S ω λ( K ω)}, K ω = {K S ω λ ( K ω)}, S ω = {a A ν(a) ω}. Proof: ), (5) It follows from Example 6 and Example 8 that the uncertain predicate ξ = ξ 1 ξ 2 has a membership function and a negated membership function µ(x) = µ 1 (x 1 ) (1 µ 2 (x 2 )) µ(x) = 1 µ 1 (x 1 ) µ 2 (x 2 ) for any vector x = (x 1, x 2 ) R 2. Then the equation (5) follows from the Theorem 2 immediately. 5 Numerical Examples Appendix gives some raw data on three indexes of regional construction material industry in mainland China, which are current asset turnover rate, return on sales rate, and sales growth rate. This section aims at discovering the relationship between the first one and the other two via uncertain logic. 7

It is well known that an company with a low current asset turnover rate usually has a low return on sales rate. So is an industry in a province. We now calculate the truth value of the statement via uncertain logic. The statement can be presented by an uncertain proposition (Q, S, ξ 1 ) = (most, provinces with low turnover-rate, low return on sales rate). Assume that the uncertain quantifier (percentage) Q = most has a membership function 0, if 0 x 0.7 20(x 0.7), if 0.7 < x 0.75 λ(x) = 1, if 0.75 < x 0.85 20(0.9 x), if 0.85 < x 0.9 0, if 0.9 < x 1, the uncertain subject S = provinces with low turnover-rate has a membership function 1, if 0 x 2.6 ν(x) = 2(3.1 x), if 2.6 < x 3.1 0, if x > 3.1, and the uncertain predicate ξ 1 = low return on sales rate has a membership function 1, if 0 x 10 µ 1 (x) = (12 x)/2, if 10 < x 12 0, if x > 12. By Liu s truth value algorithm, we get that the uncertain proposition has a truth value T (Q, S, ξ 1 ) = 0.9. Thus the statement that the province with a low current asset turnover rate usually has a low return on sales rate has a belief degree 0.9. Assume that the uncertain predicate ξ 2 = low sales growth rate has a membership function Then uncertain proposition µ 2 (x) = 1, if 0 x 45 (50 x)/5, if 45 < x 50 0, if x > 50. (Q, S, ξ 2 ) = (most, provinces with low turnover-rate, low sales growth rate) has a truth value T (Q, S, ξ 2 ) = 0.9. So the statement that a company with a low current asset turnover rate usually has a low sales growth rate has a belief degree 0.9, too. Now, we turn to the relationship between the three indexes. Consider an uncertain proposition (Q, S, ξ 1 ξ 2 ) = (most, provinces with low turnover-rate, 8

low return on sales rate and sales growth rate). By the truth value algorithm, we have T (Q, S, ξ 1 ξ 2 ) = 0. It means the statement is incorrect that a province with a low current asset rate usually has both a low return on sale rate and a low sales growth rate. In fact, about half of the provinces with a low current asset turnover rate have both a low return on sales rate and a low sales growth rate, and the belief degree is 0.92 by the truth value algorithm. 6 Conclusions An uncertain proposition with multiple predicates was presented in this paper, and its truth value was given via a concept of truth function. Uncertain logic with multiple predicates can be used in linguistic summarizer and extract more information. The contribution of this paper is to make uncertain logic more applicable in dealing with human language. Acknowledgements This work was ported by National Natural Science Foundation of China Grant No.60874067. References [1] Bosc P., and Lietard L., Monotonic quantified statements and fuzzy integrals, Proceeding of NAFIPS/IFIS/NASA 94, 8-12, 1994. [2] Dubois D., and Prade H., Necessity measure and resolution principle, IEEE Transactions on Man Cybenet, Vol.17, 474-478, 1987. [3] Elkan C., The paradoxical success of fuzzy logic, IEEE Expert, Vol.9, No.4, 3-8, 1994. [4] Gao X., Some properties of continuous uncertain measure, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, Vol.17, No.3, 419-426, 2009. [5] Li X. and Liu B., Hybrid logic and uncertain logic, Journal of Uncertain Systems, Vol.3, No.2, 83-94, 2009. [6] Liu B., Uncertainty Theory, 2nd ed., Springer-Verlag, Berlin, 2007. [7] Liu B., Theory and Practice of Uncertain Programming, 2nd ed., Springer-Verlag, Berlin, 2009. [8] Liu B., Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, 2010. [9] Liu B., Uncertain set theory and uncertain inference rule with application to uncertain control, Journal of Uncertain Systems, Vol.4, No.2, 83-98, 2010. [10] Liu B., Uncertain risk analysis and uncertain reliability analysis, Journal of Uncertain Systems, Vol.4, No.3, 163-170, 2010. 9

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