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Truth Value Solver: A Software for Calculating Truth Value with Uncertain Logic Yang Zuo Uncertainty Theory Laboratory, Department of Mathematical Sciences Tsinghua University, Beijing 100084, China [email protected] Abstract Uncertain logic based on uncertainty theory is a generalization of classical logic for dealing with uncertain knowledge. One advantage of uncertain logic is that it is consistent with the law of excluded middle and the law of contraction. The truth value is defined as the uncertain measure that a proposition is true. Truth value theorem gives a formula for computing the truth value of independent uncertain propositions. In this paper, a software based on truth value theorem is given, which is used to computing truth value. Keywords: Uncertain measure, uncertain variable, uncertain logic, truth value 1 Introduction Classical logic is used to deal with certain information, but on the other hand multi-valued logic is used to deal with uncertain information. Multi-valued logic was proposed by Lukasiewicz and extended by many researchers such as Adams [1], Shafer [9], Zadeh [10], etc. Multi-valued logic has been well developed, but the interpretation of the truth value is controversial. Although multi-valued logic theory was well developed and widely applied, there is a fatal defect that it disobeys the law of excluded middle, which is one of the most basic laws in classical logic. In 1976, Nilsson [8] considered the truth value as a probability and initiated probability logic. Recently, Li and Liu [3] introduced credibilistic logic to deal with fuzzy knowledge. Li and Liu [4] proposed uncertain logic, which can be seen as a generalization of all these multi-valued logics, in which the truth value is defined as the uncertain measure that the proposition is true. One advantage of uncertain logic is its consistency with classical logic. Chen and Ralescu [2] proved truth value theorem that provides a numerical method for calculating the truth value of uncertain formulas. 1 Although truth value theorem was given, it is still a complexity of the calculations to compute the truth value of uncertain formula. In this paper, a software to calculate truth value is introduced, which is based on truth value theorem. The rest of this paper is organized as follows: some basic concepts of uncertainty theory are recalled in Section 2. Several basic definitions of uncertain logic are contained in Section 3. A software to calculate truth value is introduced and some examples are calculated in Section 4. Finally, ta brief summary is given in Section 5. 2 Preliminaries Let Γ be a nonempty set, and L a σ-algebra over Γ. Each element Λ ∈ L is called an event. Definition 1. (Liu[5]) The set function M is called an uncertain measure if it satisfies the following four axioms: Axiom 1. (Normality) M{Γ} = 1; Axiom 2. (Monotonicity) M{Λ1 } ≤ M{Λ2 } whenever Λ1 ⊂ Λ2 ; Axiom 3. (Self-Duality) M{Λ} + M{Λc } = 1 for any event Λ; Axiom 4. (Countable Subadditivity) For every countable sequence of events {Λi }, we have (∞ ) ∞ X [ M{Λi }. M Λi ≤ i=1 i=1 . An uncertain variable is a measurable function from an uncertainty space (Γ, L, M) to the set of real numbers. Definition 2. (Liu [6]) The uncertain variables X1 , X2 , · · · , Xm are said to be independent if (m ) \ M Xi ∈ Bi = min M{Xi ∈ Bi } 1≤i≤m i=1 (1) for any Borel sets B1 , B2 , · · · , Bm of real numbers. Theorem 1. (Liu [7], Operational Law) Let ξ1 , ξ2 , · · · , ξn be independent uncertain variables, and f : <n → < a measurable function. Then ξ = f (ξ1 , ξ2 , · · · , ξn ) is an uncertain variable such that M{ξ ∈ B} = sup min Mk {Bk }, f (B1 ,B2 ,··· ,Bn )⊂B 1≤k≤n 1− sup if min Mk {Bk }, if f (B1 ,B2 ,··· ,Bn )⊂B c 1≤k≤n 0.5, sup sup min Mk {Bk } > 0.5 f (B1 ,B2 ,··· ,Bn )⊂B c 1≤k≤n otherwise 2 min Mk {Bk } > 0.5 f (B1 ,B2 ,··· ,Bn )⊂B 1≤k≤n 3 Uncertain Logic Uncertain logic was designed by Li and Liu [4] as a generalization of classical logic. Li and Liu gave the definition of uncertain proposition which is a statement with truth value in [0, 1]. In fact, the uncertain proposition X is essentially an uncertain variable taking values 0 or 1, where X = 1 means X is true and X = 0 means X is false. Uncertain propositions are called independent if they are independent uncertain variables. An uncertain formula is a finite sequence of uncertain propositions and connective symbols that are well defined. If ξ, η, τ are uncertain propositions, then X = ¬ξ,X = ξ∧η,X = (ξ∨η) → τ are all uncertain formulas. Uncertain formulas are called independent if they are independent uncertain variables. Truth value is a key concept in uncertain logic and is defined as the uncertain measure that the uncertain formula is true. Definition 3. (Li and Liu [4]) Let X be an uncertain formula. Then the truth value of X is defined as the uncertain measure that the uncertain formula X is true, i.e., T (X) = M{X = 1}. Theorem 2. (Truth Value Theorem [2]) Let X be an uncertain formula containing independent uncertain propositions ξ1 , ξ2 , · · · , ξn whose truth function is f . Then the truth value of X is sup min vi (xi ), if sup min vi (xi ) < 0.5 f (x1 ,x2 ,··· ,xn )=1 1≤i≤n f (x1 ,x2 ,··· ,xn )=1 1≤i≤n T (X) = sup min vi (xi ), if sup min vi (xi ) ≥ 0.5 1− f (x1 ,x2 ,··· ,xn )=0 1≤i≤n f (x1 ,x2 ,··· ,xn )=1 1≤i≤n where xi take values either 0 or 1, and vi are uncertainty mass functions of ξi with vi (1) = M{ξi = 1}, vi (0) = M{ξi = 0} for i = 1, 2, · · · , n, respectively. 4 Truth Value Solver Truth Value Solver is a free software for computing the truth values of uncertain formula based on truth functions. This software and its source code may be downloaded from http://orsc.edu.cn/liu/resources.htm. This software can compute truth values of uncertain formula when they are composed of no more than eight propositions. The usage of this software is very simple. All you have to do is inputting truth values of propositions and the truth function of the uncertain formula. Then it shows you the truth value of the formula in the result window. Now let us perform it via some numerical examples. 3 Example 1. Assume that ξ1 , ξ2 , ξ3 , ξ4 , ξ5 are independent uncertain propositions with truth values 0.1, 0.3, 0.5, 0.7, 0.9, respectively. Let X = only 1 propositions of ξ1 , ξ2 , ξ3 , ξ4 , ξ5 are true. (2) It is clear that the truth function is 1, if x1 + x2 + x3 + x4 + x5 = 1 f (x1 , x2 , x3 , x4 , x5 ) = 0, if x + x + x + x + x 6= 1. 1 2 3 4 5 A run of the truth value solver shows that T (X) = 0.3. Example 2. Assume that ξ1 , ξ2 , ξ3 , ξ4 , ξ5 are independent uncertain propositions with truth values 0.1, 0.3, 0.5, 0.7, 0.9, respectively. Let X = only 3 propositions of ξ1 , ξ2 , ξ3 , ξ4 , ξ5 are true. (3) It is clear that the truth function is 1, if x1 + x2 + x3 + x4 + x5 = 3 f (x1 , x2 , x3 , x4 , x5 ) = 0, if x + x + x + x + x 6= 3. 1 2 3 4 5 A run of the truth value solver shows that T (X) = 0.5. Example 3. Assume that ξ1 , ξ2 , ξ3 , ξ4 , ξ5 are independent uncertain propositions with truth values 0.1, 0.3, 0.5, 0.7, 0.9, respectively. Let X = 5 of propositions of ξ1 , ξ2 , ξ3 , ξ4 , ξ5 are true. (4) It is clear that the truth function is 1, if x1 + x2 + x3 + x4 + x5 = 5 f (x1 , x2 , x3 , x4 , x5 ) = 0, if x + x + x + x + x 6= 5. 1 2 3 4 5 A run of the truth value solver shows that T (X) = 0.1. Example 4. Assume that ξ1 , ξ2 , ξ3 , ξ4 , ξ5 are independent uncertain propositions with truth values 0.1, 0.3, 0.5, 0.7, 0.9, respectively. Let X = odd number of propositions of ξ1 , ξ2 , ξ3 , ξ4 , ξ5 are true. It is clear that the truth function is 1, if x1 + x2 + x3 + x4 + x5 ∈ {1, 3, 5} f (x1 , x2 , x3 , x4 , x5 ) = 0, if x + x + x + x + x ∈ {0, 2, 4}. 1 2 3 4 5 A run of the truth value solver shows that T (X) = 0.5. 4 (5) Example 5. Assume that ξ1 , ξ2 , ξ3 , ξ4 , ξ5 are independent uncertain propositions with truth values 0.1, 0.3, 0.5, 0.7, 0.9, respectively. Let X = even number of propositions of ξ1 , ξ2 , ξ3 , ξ4 , ξ5 are true. (6) It is clear that the truth function is 1, if x1 + x2 + x3 + x4 + x5 ∈ {0, 2, 4} f (x1 , x2 , x3 , x4 , x5 ) = 0, if x + x + x + x + x ∈ {1, 3, 5}. 1 2 3 4 5 A run of the truth value solver shows that T (X) = 0.5. Example 6. Assume that ξ1 , ξ2 , ξ3 , ξ4 , ξ5 are independent uncertain propositions with truth values 0.1, 0.3, 0.5, 0.7, 0.9, respectively. Let X = (ξ1 ∧ ξ2 ) ∨ (ξ2 ∧ ξ3 ) ∨ (ξ3 ∧ ξ4 ) ∨ (ξ4 ∧ ξ5 ). (7) It is clear that the truth function is 1, if x1 + x2 = 2 1, if x2 + x3 = 2 f (x1 , x2 , x3 , x4 , x5 ) = 1, if x3 + x4 = 2 1, if x4 + x5 = 2 0, otherwise. A run of the truth value solver shows that T (X) = 0.7. 5 Conclusion Uncertain logic was defined within the framework of uncertainty theory as a generalization of classical logic. One advantage of uncertain logic is its consistency with classical logic. We gave a software to deal with the computing of truth value of uncertain formula. Finally, some examples were given. Acknowledgments This work was supported by National Natural Science Foundation of China Grant No.60874067. References [1] Adams, E., and Levine, H., On the uncertainties transmitted from premises to conclusions in deductive inferences, Synthese, vol.30, 429-460, 1975. 5 [2] Chen XW, and Ralescu DA, A note on truth value in uncertain logic,Proceedings of the Eighth International Conference on Information and Management Sciences, Kunming, China, July 20-28, 2009, pp.739-741. [3] Li, X., Liu, B., Foundation of credibilistic logic, Fuzzy Optimization and Decision Making, vol.8, no.1, 91-102, 2009. [4] Li, X., Liu, B., Hybrid logic and uncertain logic, Journal of Uncertain Systems, vol.3, no.2, 83-94, 2009. [5] Liu, B., Uncertainty Theory, 2nd ed., Springer-Verlag, Berlin, 2007. [6] Liu, B., Uncertainty Theory, 3nd ed., http://orsc.edu.cn/liu/ut.pdf. [7] Liu, B., Some research problems in uncertainty theory, Journal of Uncertain Systems, vol.3, no.1, 3-10, 2009. [8] Nilsson, N., Probability logic, Artificial Intelligence, vol.28, 71-78, 1986. [9] Shafer, G., Mathematical Theory of Evidence, Princeton, New Jersey, 1979. [10] Zadeh, L., Fuzzy logic and approximate reasoning, Synthese, vol.30, 407-428, 1975. 6