Bipolar fuzzy sets and relations: a computational framework for cognitive modeling and multiagent decision analysis

A bipolar fuzzy set theory is presented for cognitive modeling and multiagent decision analysis. Firstly, notions of bipolar fuzziness are introduced. Secondly an interval-based bipolar fuzzy logic is defined which generalizes a real-valued bipolar fuzzy logic by allowing interval-based linguistic variables x and y to be substituted into (S, =, V, ⊗) or (S, =, ∪, ⊙) where S={∀(x,y)|(x,y)∈([-1,0]×[0,+1])}. Thirdly, a fuzzy number-based bipolar logic is presented which. Further generalizes the interval-based model by allowing α-level fuzzy number-based linguistic variables x and y to be substituted into (S, =, V, ⊗) or (S, =, ∪, ⊙), S={∀(x,y)|(x,y) maps ([-1,0]×[0,+1]) to [0,1]}. Bipolar fuzzy set operations of disjunction composition (V-⊗), union-composition (∪, ⊙), are proved commutative and associative; V respect to ⊗ and ∪ respect to ⊙ are proved distributive. It is shown that a interval-based bipolar variable is a nesting of a real-valued bipolar variable; a trapezoidal-fuzzy number-based bipolar variable is an 2-level nesting of an interval-based bipolar variable; and an α-level (α is an integer) fuzzy number-based bipolar variable is an α+1 level nesting of an interval-based bipolar variable. Based on the nesting features, it is proved that α-level fuzzy number-based bipolar operations can be converted to interval-based and then real-valued bipolar operations. The conversions lead to significant computational simplification on bipolar fuzzy relations. Major advantages of the bipolar fuzzy set theory include: (1) it formalizes a unified approach to polarity and fuzziness; (2) it captures die bipolar or double-sided (negative and positive, or effect and side effect) nature of human perception and cognition; and (3) it provides a basis for bipolar cognitive modeling and multiagent decision analysis