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Semantic of intuitionistic fuzzy rough logic and its reasoning
SHEN Liping , WANG Yanping
School of Science, Liaoning University of Technology, Jinzhou 121001, China
Abstract: The theory of the intuitionistic fuzzy rough set is introduced into the logic reasoning. By the combination of such basic theories as rough sets, intuitionistic fuzzy sets and mathematical logic, the semantic and reasoning methods of intuitionistic fuzzy rough logic are given. Initially, five logic values for the intuitionistic fuzzy proposition logic are given, i.e. intuitionistic fuzzy true, intuitionistic fuzzy false, intuitionistic fuzzy rough true, intuitionistic fuzzy rough false and intuitionistic fuzzy rough incompatible, and on the basis of this, the intuitionistic fuzzy rough logic operations are given; then the semantic of intuitionistic fuzzy rough proposition formulas in approximate space is discussed; finally, the semantic reasoning methods are proposed as for the intuitionistic fuzzy rough propositional formula containing different logical conjunctions.
Key words: rough set     rough logic     fuzzy rough logic     intuitionistic fuzzy rough logic

1 预备知识

1)AB当且仅当μA(x)≤μB(x)且νA(x)≥νB(x)，∀xU

2)AB当且仅当μA(x)＝μB(x)且νA(x)＝νB(x)，∀xU

3)AB＝{〈x, min{μA(x), μB(x)}, max{νA(x), νB(x)}〉|xU}；

4)AB＝{〈x, max{μA(x), μB(x)}, min{νA(x), νB(x)}〉|xU}；

5)AC＝{〈x, νA(x), μA(x)〉|xU}．

2 直觉模糊粗糙逻辑的概念

1)若|ϕ|＝U，则称公式ϕM中真，记作|＝Mϕ;

2)若|ϕ|≠U，则称公式ϕM中假，记作|≠Mϕ;

3)若R|ϕ|＝U，则称公式ϕ在中M粗糙真，记作|＝Rϕ;

4)若R|ϕ|＝∅，则称公式ϕM中粗糙假，记作|≠Rϕ;

5)若R|ϕ|＝UR|ϕ|＝∅，则称公式ϕM中粗糙不相容。

1)若μA(x)≥νA(x) < ，则称ϕM中直觉模糊真，记作|＝IFMϕ

2)若μA(x) < νA(x)≥，则称ϕM中直觉模糊假，记作|≠IFMϕ

3)若μA(x)≥νA(x) < ，则称ϕM中直觉模糊粗糙真，记作|＝IFRϕ

4)若μA(x) < νA(x)≥，则称ϕM中直觉模糊粗糙假，记作|≠IFRϕ

5)若μA(x)≥νA(x) < μ A(x) < νA(x)≥，则称ϕM中直觉模糊粗糙不相容。

3 直觉模糊粗糙集的逻辑运算

4 直觉模糊粗糙逻辑语义推理 4.1 含有“→”公式

ϕφ是近似空间M＝(U, R)上的直觉模糊公式，则有下面的定理1~6成立。

1)若|＝IFMφ, 则|＝IFMϕφ，即若φ直觉模糊真，则ϕφ直觉模糊真；

2)若|≠IFMϕ, 则|＝IFMϕφ，即若ϕ直觉模糊假，则ϕφ直觉模糊真；

3)若|≠IFMϕφ, 则|＝IFMϕ且|≠IFMφ，即若ϕφ直觉模糊假，则ϕ直觉模糊真，φ直觉模糊假。

1)因为|＝IFMφ，则μB(x)≥νB(x) < ，所以对于T(|ϕφ|)(x)＝〈νA(x)∨μB(x), μA(x)∧νB(x)〉, 显然有νA(x)∨μB(x)≥μA(x)∧νB(x) < ，因此可得结论。

2)因为|≠IFMϕ，则μA(x) < νA(x)≥，所以对于T(|ϕφ|)(x)＝〈νA(x)∨μB(x), μA(x)∧νB(x)〉, 即有νA(x)∨μB(x)≥μA(x)∧νB(x) < ，因此可得结论。

3)因为|≠IFMϕφ，则νA(x)∨μB(x) < μA(x)∧νB(x)≥，所以νA(x) < μB(x) < μA(x)≥νB(x)≥，故|＝IFMϕ，|≠IFMφ，因此可得结论。

1)若|＝IFRφ, 则|＝IFRϕφ，即若φ直觉模糊粗糙真，则ϕφ直觉模糊粗糙真；

2)若|≠IFRφ, 则|＝IFRϕφ，即若ϕ直觉模糊粗糙假，则ϕφ直觉模糊粗糙真；

3)若|≠IFRϕφ, 则|＝IFRϕ且|≠IFRφ，即若ϕφ直觉模糊粗糙假，则ϕ直觉模糊粗糙真，φ直觉模糊粗糙假。

1)若|＝IFMφ, 则|＝IFRϕφ，即若φ直觉模糊真，则ϕφ直觉模糊粗糙真；

2)若|≠IFMϕ, 则|＝IFRϕφ，即若ϕ直觉模糊假，则ϕφ直觉模糊粗糙真。

1)因为|＝IFMφ, 则μB(x)≥νB(x) < ，又由于|φ|⊆R|φ|，所以μB(x)≥νB(x) < ，因此对于T(R|ϕφ|)(x)＝〈νA(x)∨μB(x), μA(x)∧νB(x)〉，总有νA(x)∨μB(x)≥μA(x)∧νB(x) < ，因此可得结论。

2)同理可证。

1)若|＝IFM(ϕφ)∧ϕ，则|＝IFMφ

2)若|＝IFR(ϕφ)∧ϕ，则|＝IFRφ.

1) T(|(ϕφ)∧ϕ|)(x)＝T(|(┐ϕφ)∧ϕ|)(x)＝T(|φϕ|)(x)＝T(|ϕ|)(x)∧T(|φ|)(x)＝〈μA(x)∧μB(x), νA(x)∨νB(x)〉。

2)T(R-|(ϕφ)∧ϕ|)(x)＝T(R|(┐ϕφ)∧ϕ|)(x)＝T(R|φϕ|)(x)＝T(R(|ϕ|∩|φ|))(x)＝〈a, b

1)如果|＝IFMϕφ且对于T(|ϕ|∪|φ|)(x)＝〈μA(x)∨μB(x), νA(x)∧νB(x)〉, 有μA(x)∨μB(x)≥, νA(x)∧νB(x) < ，则|＝IFMφ且|＝IFRφ.

2)如果|＝IFMφ且对于T(|ϕ|∩|φ|)(x)＝μA(x)∧μB(x), νA(x)∨νB(x)〉, 有μA(x)∧μB(x) < , νA(x)∨νB(x)≥，则|≠IFRϕφ且|≠IFMϕφ.

1)因为|＝IFMϕφ，所以对T(|ϕφ|)(x)＝〈νA(x)∨μB(x), μA(x)∧νB(x)〉, 有νA(x)∨μB(x)≥, μA(x)∧νB(x) < ，又由μA(x)∨μB(x)≥, νA(x)∧νB(x) < ，则必有μB(x)≥νB(x) < ，又因为|φ|⊆R|φ|，所以μB(x)≤μB(x), νB(x)≤νB(x) < ，因此可得结论。

2)因为|＝IFMϕ，则有μA(x)≥, νA(x) < ，又因为μA(x)∧μB(x) < , νA(x)∨νB(x)≥，所以有μB(x) < , νB(x)≥。而T(|ϕφ|)(x)＝T(|┐ϕφ|)(x)＝〈νA(x)∨μB(x), μA(x)∧νB(x)〉，因此有νA(x)∨μB(x) < , μA(x)∧νB(x)≥。又因为|ϕφ|⊇R|ϕφ|，所以对T(R|ϕφ|)(x)＝〈νA(x)∨μB(x), μA(x)∧νB(x)〉, 有νA(x)∨μB(x)≤νA(x)∨μB(x) < , μA(x)∧νB(x)≥μA(x)∧νB(x)≥，因此可证得结论。

4.2 含有“┐”公式

4.3 含有“∧, ∨”公式

5 结束语

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DOI: 10.3969/j.issn.1673-4785.201209063

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#### 文章信息

SHEN Liping, WANG Yanping

Semantic of intuitionistic fuzzy rough logic and its reasoning

CAAI Transactions on Intelligent Systems, 2014, 9(1): 83-87
http://dx.doi.org/10.3969/j.issn.1673-4785.201209063