Need for Expert . . . Need to Describe . . . Rules: General Case Step-by-Step . . . Introduction to Fuzzy Logic First Step: . . . Need for Interpolation Vladik Kreinovich Need to Combine . . . Defuzzification Department of Computer Science University of Texas at El Paso Fuzzy Computations: . . . 500 W. University Home Page El Paso, Texas 79968, USA Title Page vladik@utep.edu http://www.cs.utep.edu/vladik ◭◭ ◮◮ ◭ ◮ Page 1 of 31 Go Back Full Screen Close Quit
Need for Expert . . . Need to Describe . . . 1. Need for Expert Knowledge Rules: General Case • In some cases, we have a precise knowledge – e.g., an Step-by-Step . . . autopilot perfectly pilots a plane. First Step: . . . Need for Interpolation • In many other cases, we have to rely on expert knowl- Need to Combine . . . edge. Defuzzification • So far, computer-based systems have not (yet) replaced Fuzzy Computations: . . . skilled medical doctors or even skilled drivers. Home Page • In the ideal world, everyone should go to the best doc- Title Page tor. ◭◭ ◮◮ • However, in real life, this is not possible. ◭ ◮ • It is therefore desirable to have a computer-based tool Page 2 of 31 that contains the knowledge of the best doctors. Go Back • Such tool will help all other doctors make good deci- Full Screen sions. Close Quit
Need for Expert . . . Need to Describe . . . 2. Need to Describe Imprecise (Fuzzy) Knowledge Rules: General Case • Many doctors (and experts in general) are absolutely Step-by-Step . . . willing to share their knowledge. First Step: . . . Need for Interpolation • Challenge: this knowledge is often described in terms Need to Combine . . . of imprecise (“fuzzy”) words from natural language. Defuzzification • A medical doctor can say “if a patient has high fever” Fuzzy Computations: . . . or “if a skin mole has an irregular shape”. Home Page • A driver cannot say with what force to hit the brakes if Title Page the car 10 m in front slows down from 100 to 90 km/h. ◭◭ ◮◮ • He/she will say “brake a little bit”. ◭ ◮ • We thus need to translate these fuzzy words into computer- Page 3 of 31 understandable language. about the object we can use. Go Back • This is what Zadeh’s fuzzy logic is about. Full Screen Close Quit
Need for Expert . . . Need to Describe . . . 3. Toy Example: a Thermostat Rules: General Case • To illustrate the main idea of fuzzy logic, let us consider Step-by-Step . . . a simplified thermostat with a dial. First Step: . . . Need for Interpolation • Turning the dial to the left makes it cooler . Need to Combine . . . • Turning it to the right makes it warmer . Defuzzification • We want to reach a comfort temperature T 0 . Fuzzy Computations: . . . Home Page • In other words, we want the difference x = T − T 0 to Title Page be 0. ◭◭ ◮◮ • We need to describe, for each x , to which angle u we turn the dial: u = f ( x ). ◭ ◮ Page 4 of 31 Go Back Full Screen Close Quit
Need for Expert . . . Need to Describe . . . 4. Thermostat: Rules Rules: General Case • For such an easy system, we do not need any expert to Step-by-Step . . . formulate reasonable rules. First Step: . . . Need for Interpolation • We can immediately describe several reasonable con- Need to Combine . . . trol rules. Defuzzification • If the room is comfortable, no control is needed. Fuzzy Computations: . . . • So, if the difference x = T − T 0 is negligible, then the Home Page control u should also be negligible. Title Page • If the room is slightly overheated, cool it a little bit. ◭◭ ◮◮ • So, if x is positive and small, u must be negative and ◭ ◮ small. Page 5 of 31 • If the temperature is a little lower than we would like Go Back it to be, then we need to heat the room a little bit. Full Screen • In other terms, if x is small negative, then u must be small positive. Close Quit
Need for Expert . . . Need to Describe . . . 5. Thermostat: Rules (cont-d) Rules: General Case • We can formulate many similar natural rules. Step-by-Step . . . First Step: . . . • For simplicity, we will restrict ourselves to the above Need for Interpolation three: Need to Combine . . . – if x is negligible, then u must be negligible; Defuzzification – if x is small positive, then u must be small negative; Fuzzy Computations: . . . Home Page – if x is small negative, then u must be small positive. Title Page ◭◭ ◮◮ ◭ ◮ Page 6 of 31 Go Back Full Screen Close Quit
Need for Expert . . . Need to Describe . . . 6. Rules: General Case Rules: General Case • Let us denote “negligible” by N , “small positive” by Step-by-Step . . . SP, and “small negative” by SN. First Step: . . . Need for Interpolation • Then, the rules take the following form: Need to Combine . . . N ( x ) ⇒ N ( u ); SP ( x ) ⇒ SN ( u ); SN ( x ) ⇒ SP ( u ) . Defuzzification Fuzzy Computations: . . . • In general, the expert’s knowledge about the depen- Home Page dence of y on x 1 , . . . , x n can be expressed by rules: Title Page If x 1 is A r 1 , . . . , and x n is A rn , then y is B r . ◭◭ ◮◮ • Here, A ri and B r are words from natural language like ◭ ◮ “small”, “medium”, “large”, “approximately 1”. Page 7 of 31 • These rules have the form Go Back A r 1 ( x 1 ) & . . . & A rn ( x n ) ⇒ B r ( y ) . Full Screen Close Quit
Need for Expert . . . Need to Describe . . . 7. Step-by-Step Translation of These Rules Rules: General Case • Our goal is to represent rule bases in precise terms. Step-by-Step . . . First Step: . . . • A rule base has a clear structure. Need for Interpolation • A rule base consists of rules. Need to Combine . . . • Each rule, in its turn, is obtained: Defuzzification Fuzzy Computations: . . . – from properties (expressed by words from natural Home Page language) Title Page – by using logical connectives. ◭◭ ◮◮ • In view of this structure, it is reasonable to represent ◭ ◮ the rule base: Page 8 of 31 – by first representing the basic elements of the rule base, and then Go Back – by extending this representation to the rule base as Full Screen a whole. Close Quit
Need for Expert . . . Need to Describe . . . 8. Step-by-Step Translation (cont-d) Rules: General Case • So, first, we represent the properties A ri ( x i ) and B r ( y ). Step-by-Step . . . First Step: . . . • Second, we represent the logical connectives. Need for Interpolation • Third, we use logical connectives to represent each rule. Need to Combine . . . Defuzzification • Fourth, we combine the representations of different rules into a representation of a rule base. Fuzzy Computations: . . . Home Page • As a result of these four steps, we get an advising (ex- Title Page pert) system. ◭◭ ◮◮ • For example, if we apply these four steps to the medical knowledge, we ideally, get a system that ◭ ◮ Page 9 of 31 – given the patient’s symptoms, – provides the diagnostic and medical advice. Go Back Full Screen Close Quit
Need for Expert . . . Need to Describe . . . 9. Step-by-Step Translation (cont-d) Rules: General Case • For example, it can say that most probably, the patient Step-by-Step . . . has a flu, but it is also possible that he has bronchitis. First Step: . . . Need for Interpolation • Such an advice, coming from an expert system, is usu- Need to Combine . . . ally used by a specialist to make a decision. Defuzzification • However, there are situations like automatic control Fuzzy Computations: . . . where there is no time to involve a human operator. Home Page • For such control situations, we need an additional , fifth Title Page follow-up step: making a decision. ◭◭ ◮◮ • Let us describe all five steps. ◭ ◮ Page 10 of 31 Go Back Full Screen Close Quit
Need for Expert . . . Need to Describe . . . 10. First Step: Representing Natural-language Prop- Rules: General Case erties Step-by-Step . . . • For properties A like “small”, for some values x , we are First Step: . . . not 100% sure whether this value is small or not. Need for Interpolation Need to Combine . . . • A natural idea is to ask the expert to mark, on a scale Defuzzification from 0 to 1, to what extend the given value x is small. Fuzzy Computations: . . . • We can use another scale – e.g., 0 to 10 – and then Home Page divide by 10. Title Page • As a result, for several values x i , we get a degree A ( x i ) ◭◭ ◮◮ to which x i satisfies the property A . ◭ ◮ • Some experts are not comfortable marking this value. Page 11 of 31 • Then, we poll the experts and take A ( x i ) = m/n if m Go Back out of n consider x i to be, e.g., small. Full Screen Close Quit
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