PART 8 Approximate Reasoning 1. Fuzzy expert systems 2. Fuzzy implications 3. Selecting fuzzy implications 4. Multiconditional reasoning 5. Fuzzy relation equations 6. Interval-valued reasoning FUZZY SETS AND FUZZY LOGIC Theory and Applications

2 Fuzzy expert systems

Fuzzy implications Extensions of classical implications: 3

Fuzzy implications S-implications 1.Kleene-Dienes implication 2.Reichenbach implication 3.Lukasiewicz implication 4

Fuzzy implications S-implications 4.Largest S-implication 5

Fuzzy implications Theorem 8.1 6

Fuzzy implications R-implications 1.Gödel implication 2.Goguen implication 7

Fuzzy implications R-implications 3.Lukasiewicz implication 4.the limit of all R-implications 8

Fuzzy implications Theorem 8.2 9

Fuzzy implications QL-implications 1.Zadeh implication 2. When i is the algebraic product and u is the algebraic sum. 10

Fuzzy implications QL-implications 3. When i is the bounded difference and u is the bounded sum, we obtain the Kleene-Dienes implication. 4. When i = i min and u = u max 11

Fuzzy implications Combined ones 12

Fuzzy implications Axioms of fuzzy implications 13

Fuzzy implications Axioms of fuzzy implications 14

Fuzzy implications Axioms of fuzzy implications 15

Fuzzy implications Theorem

Selecting fuzzy implications Generalized modus ponens any fuzzy implication suitable for approximate reasoning based on the generalized modus ponens should satisfy (8.13) for arbitrary fuzzy sets A and B. 17

Selecting fuzzy implications Theorem

Selecting fuzzy implications Theorem

Selecting fuzzy implications Generalized modus tollens Generalized hypothetical syllogism 20

Multiconditional reasoning general schema of multiconditional approximate reasoning The method of interpolation is most common way to determine B‘. It consists of the following two steps: 21

Multiconditional reasoning 22

Multiconditional reasoning 23

Multiconditional reasoning four possible ways of calculating the conclusion B': Theorem

Fuzzy relation equations Suppose now that both modus ponens and modus tollens are required. The problem of determining R becomes the problem of solving the following system of fuzzy relation equation: 25

Fuzzy relation equations Theorem 8.7

Fuzzy relation equations If then is also the greatest approximate solution to the system (8.30). 27

Fuzzy relation equations Theorem 8.8

Interval-valued reasoning Let A denote an interval-valued fuzzy set. L A,U A are fuzzy sets called the lower bound and the upper bound of A. A shorthand notation of A( x ) When L A = U A, A becomes an ordinary fuzzy set. 29

Interval-valued reasoning given a conditional fuzzy proposition (if - then rule) where A, B are interval-valued fuzzy sets defined on the universal sets X and Y. given a fact how can we derive a conclusion in the form 30

Interval-valued reasoning view this conditional proposition as an interval- valued fuzzy relation R = [L R,U R ], where It is easy to prove that L R (x, y) ≦ U R (x, y) and, hence, R is well defined. 31

Interval-valued reasoning Once relation R is determined, it facilitates the reasoning process. Given A’ = [L A’,U A’ ], we derive a conclusion B’ = [L B’,U B’ ] by the compositional rule of inference where i is a t-norm and 32

Interval-valued reasoning Example let a proposition be given, where Assuming that the Lukasiewicz implication 33

Interval-valued reasoning

Exercise