Fuzzy Implication Rules. Adnan Yazıcı Dept. of Computer Engineering, Middle East Technical University Ankara/Turkey

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1 Fuzzy Implication Rules Adnan Yazıcı Dept. of Computer Engineering, Middle East Technical University Ankara/Turkey

2 Fuzzy If-Then Rules Remember: The difference between the semantics of fuzzy mapping rules and fuzzy implication rules can be seen from the difference in their inference behavior. Even though these two types of rules behave the same when their antecedents are satisfied, they behave differently when their antecedents are not satisfied. Example: Implication rule, Mapping rule (logic representation) (procedural representation) Given:x [1,3] y [7,8], Stm: If x [1,3], Then y [7,8] Input: x=5 Variable value: x = 5 Infer: y is unkown (y [0,10]) Execution result: no action

3 Fuzzy Implication Rules Any logic system has two major components: 1. a formal language for constructing statements about the world, 2. a set of inference mechanisms for inferring additional statements about the world from those already given. Fuzzy logic is the most commonly used reasoning scheme in applications of fuzzy logic (narrow sense). The subject is complicated by the fact that there isn t a unique definition of fuzzy implications.

4 Fuzzy Implication Rules An important goal of fuzzy logic is to be able to make reasonable inference even when the condition of an implication rule is partially satisfied. This capability is sometimes referred to as approximate reasoning. This is achieved in fuzzy logic by two related techniques: 1. representing the meaning of a fuzzy implication rule using a fuzzy relation, and 2. obtaining an inferred conclusion by applying the compositional rule of inference to the fuzzy implication relation.

5 Fuzzy Implication Rules Fuzzy rule-based inference is a generalization of a logical reasoning scheme (inference) called modus ponens (MP) and modus tollens (MT). It combines the conclusion of multiple fuzzy rules in a manner similar to linear interpolation. For example: Rule: If a person s IQ is high Then the person is smart Fact: Jack s IQ is high Infer: Jack is smart. Rule: If a person s IQ is high Then the person is smart Fact: Jack is not smart Infer: Jack s IQ is not high.

6 Fuzzy Implication Rules First, these inferences insist on perfect matching. However, common sense reasoning suggest that we can infer Jack is more or less smart when the Jack s IQ is more or less high is given. Secondly, these inferences cannot handle uncertainty. For instance, if Jack told us his IQ is high but cannot provide documents supporting the claim, we may be somewhat uncertain about the claim. Under such a circumstance, however, ordinary logic cannot reason about the uncertainty.

7 Fuzzy Implication Rules These limitations motivated L.A. Zadeh to develop a reasoning scheme that generalizes classical logic so that It can conduct common-sense reasoning under partial matching, and It can reason about the certainty degree of a statement In particular, logic implications are generalized to allow partial matching. Rule: A person s IQ is high the person is smart Fact: Jack s IQ is somewhat high Infer: Jack is somewhat smart

8 Fuzzy Implication Rules The second limitation of logic (i.e., inability to deal with uncertainty) has motivated another extension to classical logic: multivalued logic. Since fuzzy logic also generalizes the truth-values in classical logic beyond true and false, it is related to multivalued logic. However, fuzzy logic differs from multivalued logic in that it also addresses the first limitation of logic (i.e., restricted to perfect matching) by using linguistic variables in its antecedent. Consequently, the statement in the antecedent describes an elastic condition that can be partially satisfied.

9 Fuzzy Implication Rules Other approaches for reasoning under uncertainty include: 1. Bayesian probabilistic inference, 2. Dempster-Shafer theory, 3. nonmonotonic logic. Fuzzy logic, among these, is unique in that it addresses both the uncertainty management problem and the partial matching issue.

10 Fuzzy Implication Rules Let us consider an implication involving fuzzy sets (i.e., fuzzy implication): (x is A) (y is B) where A and B are fuzzy subsets of U and V, respectively. This implication also specifies the possibility of various point-to-point implications. The possibilities are a matter of degree. Therefore, the meaning of the fuzzy implication can be represented by an implication relation R defined as R l (x i,y j ) = Π l ((x = x i ) (y = y j )) Where Π l denotes the possibility distribution imposed by the implication.

11 Fuzzy Implication Rules In fuzzy logic, this possibility distribution is constructed from the truth values of the instantiated implications obtained by replacing variables in the implication (i.e., x and y) with pairs of their possible values (i.e., x i and y j ): Π ((x = x i ) (y = y j )) = t ((x i is A) (y j is B)) where t denotes the truth value of a proposition. For the convenience of our discussion, we refer to the truth values as α i and β j as follows: t(x i is A) = α i t(y j is B) = β j t((x i is A) (y j is B)) = I(α i,β j ) we call the function I an implication function.

12 Fuzzy Implication Rules There is not a unique definition for implication function. Different implication functions lead to different fuzzy implication relations. Various definitions of implication functions have been developed from both the fuzzy logic and multivalued logic research communities. However, all of them at least satisfy the following rules: I(0, β j ) = 1 I(α i,1) = 1

13 Given a possibility distribution of the variable X and the implication possibility from X to Y, we infer the possibility distribution of Y. Given: X = x i is possible AND Infer: X = x i Y = y j is possible Y = y j is possible More generally, we have Given: Infer: Π(X= x i ) = a AND Π(X = x i Y = y j ) = b Π(Y = y j ) a b Where is a fuzzy conjunction operator.

14 When different values of X imply an identical value of Y say y j with potential varying possibility degrees, these inferred possibilities about Y = y j need to be combined using fuzzy disjunction. Hence, the complete formula for computing the inferred possibility distribution of Y is Π(Y = y j ) = xi (Π(X= x i ) Π((X = x i Y = y j ))) which is the compositional rule of inference.

15 Even though both fuzzy implication and fuzzy mapping rules use the compositional rule of inference to compute their inference results, their usage differ in two ways. First, the compositional rule of inference is applied to individual implication rules, while composition is applied to a set of fuzzy mapping rules that approximate a functional mapping. Second, the fuzzy relation of a fuzzy mapping rule is a Cartesian product of the rule s antecedent and its consequent part. An entry in the fuzzy implication relation, however, is the possibility that a particular input value implies a particular output value.

16 Criteria of fuzzy Implications: The criteria of desired inference involving fuzzy implication results can be grouped into six: 1. The basic criterion of modus ponens 2. The generalized criterion of modus ponens involving hedges, 3. The mismatch criterion 4. The basic criterion of modus tolens 5. The generalized criterion of modus tolens involving hedges, and 6. The chaining criterion of implications

17 1. The basic criterion of modus ponens The basic criterion of modus ponens Given: x is A y is B x is A Infer: y is B

18 2. The generalized criterion of modus ponens involving hedges, Given: x is A y is B x is very A Infer: y is very B Ex: If the color of a tomato is red, Then the tomato is ripe The color of this tomato is very red Infer: This tomato is very ripe Or Given: x is A y is B x is very A Infer: y is B Ex: If the color of a tomato is red, then the tomato is ripe The color of this tomato is more or less red Infer: This tomato is ripe

19 A more general version of criterion is to state that the inference result is desired to be the consequent whenever the given fact about x is a subset of A. Given: Infer: x is A y is B x is A A A y is B

20 3. The mismatch criterion Given: x is A y is B x is not A Infer: y is V (unkown) Where V is the universe of discourse of y.

21 4. The basic criterion of modus tolens Given: Infer: x is A y is B y is not B x is not A

22 5. The generalized criterion of modus tolens involving hedges, Given: x is A y is B y is not (very B) Infer: x is not (very A)

23 6. Given: x is A y is B y is B Infer: x is U (unkown) Where U is the universe of discourse of x.

24 7. Chaining: Given: x is A y is B y is B z is C Infer: x is A z is C

25 Fuzzy implications can be classified into three families. 1. The first family of fuzzy implication is obtained by generalizing implications in two-valued logic to fuzzy logic. A material implication p q is defined as p q. Generalizing this to fuzzy logic is defined as follows: t(p q) = t( p q). More specifically, fuzzy implications in this family can be generically defined as: t (x i is A y j is B) = t( (x i is A) (y j is B)) = ((1- μ A (x i )) μ A (y j ))

26 The second family of fuzzy implication is based on logic equivalence between implications implication p q, defined as p (p q). Fuzzy implications in this family thus have the following form: t (x i is A y j is B) = t( (x i is A) [(x i is A) (y j is B)] = (1- μ A (x i )) (μ A (x i ) μ A (y j ))

27 The third family of fuzzy implication generalizes the standard sequence of many valued logic and its variants. The implication in this logic system is defined to be true whenever the consequent is as true as or truer than the antecedent. This property is important since it allows the following tautology: a logic formula always implies itself, regardless of its truth-value. The fuzzy implication function in this family can all be described in the following form: t (x i is A y j is B) = sup {α α [0,1], α t(x i is A) t (y j is B)} = sup {α α [0,1], α μ A (x i ) μ B (y j )}

28 Major Fuzzy Implication Functions: We introduce below several major implication functions in these families. 1. Zadeh s arithmetic fuzzy implication (family 1) t (x i is A y j is B) = 1 (1- (μ A (x i )) + μ B (y j )) 2. Zadeh s maximum fuzzy implication (Family 2) t (x i is A y j is B) = (1- μ A (x i )) (μ A (x i ) μ B (y j )), using min for fuzzy conjunction and max for fuzzy disjunction.

29

30 Example: Let x and y be two variables taking values from U and V respectively, where U = {1,2,3,4} and V = {0,10,20}. A and B are fuzzy subsets of U and V, respectively, defined as follows: A = 1/ / /3 B = 0.5/10 + 1/20 Consider the following fuzzy implication rule: R1: If x is A Then y is B 1. Give the complete formula for computing the inferred possibility distribution of y (the compositional rule of inference). Then use the compositional rule of inference to compute the result of y if the value of x is VeryA (VeryA = 1/ / /3). Use the sup-min compositional operator to infer the value of y.

31 5. Using the Zadeh s maximum fuzzy implication operator, construct the fuzzy implication relation for R1. t(x i is A y j is B) = 1 ((1-μ A (x i )) μ B (y j )) R1: If x is A Then y is B A = 1/ / /3, B = 0.5/10 + 1/20 y = VeryA R = [ ] o y = [ ]

32 5. Using the Zadeh s maximum fuzzy implication operator, construct the fuzzy implication relation for R1. t(x i is A y j is B) = (1-μ A (x i )) (μ A (x i ) μ B (y j )) R1: If x is A Then y is B A = 1/ / /3 B = 0.5/10 + 1/20 y = VeryA R = [ ] y = [ ] o

33 2. Construct the fuzzy implication relation R1 using standard sequence fuzzy implication operator. Then use the compositional rule of inference to compute the result of y if the value of x is VeryA using the standard sequence fuzzy implication. 1 if (μ A (x i ) μ B (y j )), t(x i is A y j is B)= 0 if (μ A (x i ) > μ B (y j )), R1: If x is A Then y is B A = 1/ / /3 B = 0.5/10 + 1/20 y = VeryA R = [ ] y = [ ] o

34 3. Using the Godelian s fuzzy implication operator, construct the fuzzy implication relation for R1. 1 if (μ A (x i ) μ B (y j )), t(x i is A y j is B) = μ B (y i ) if (μ A (x i ) > μ B (y j )), R1: If x is A Then y is B A = 1/ / /3 B = 0.5/10 + 1/20 y = VeryA R = [ ] o y = [ ]

35 4. Using the Goguen s fuzzy implication operator, construct the fuzzy implication relation for R1. 1 (μ A (x i ) μ B (y j )), t(x i is A y j is B) = μ B (y i )/ μ A (x i ) (μ A (x i ) > μ B (y j )), R1: If x is A Then y is B A = 1/ / /3 B = 0.5/10 + 1/20 y = VeryA R = [ ] o y = [ ]

36 All the latter three fuzzy implications came from multivalued logic systems. Even though implication functions in multivalued logic systems can be used for constructing fuzzy implication relations, approximate reasoning in fuzzy logic is fundamentally different from logic inference in multi-valued logic. Approximate reasoning infers possible values of a variable, whereas multivalued logic infers the truth-values of propositions. Approximate reasoning benefits from the following important concepts and techniques in fuzzy logic: the compositional rule of inference, fuzzy relations, and possibility distributions. Without them, approximate reasoning would not have been possible. Among the five fuzzy implication functions, the standard sequence fuzzy implication satisfies most criteria.

37 Fuzzy If-Then Rules Fuzzy Implication Rules Fuzzy Mapping Rules Purpose Generalizes implications for handling imprecision Approximate functional mappings Desired Inference Generalizes modus ponens and modus tollens Forward only Application Diagnostics, high-level decision making Control, system modeling, and signal processing Related Disciplines Classical logic, multivalued logic (other extended logic systems) System ID, piecewise linear interpolation, neural networks Typical Design Approach Designed individually Designed as a rule set Suitable Problem Domains Domains with continuous and discrete variables Continuous nonlinear domains

38

39 A = 1/ / /3 B = 0.5/10 + 1/20 R1: If x is A Then y is B y = VeryA R = [ ] y = [ ] o VeryA = 1/ / /3). B= A= R = (A x B) = (0.0/(1,0) + 0.5/(1,10) + 1/(1,20) 0.0/(2,0) + 0.5/(2,10) /(2,20) 0.0/(3,0) + 0.4/(3,10) /(3,20) 0.0/(4,0) + 0.0/(4,10) /(4,20))

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