Fuzzy rule
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Fuzzy rules are used within fuzzy logic systems to infer an output based on input variables. Modus ponens and modus tollens are the most important rules of inference.[1] A modus ponens rule is in the form
- Premise: x is A
- Implication: IF x is A THEN y is B
- Consequent: y is B
In crisp logic, the premise x is A can only be true or false. However, in a fuzzy rule, the premise x is A and the consequent y is B can be true to a degree, instead of entirely true or entirely false.[2] This is achieved by representing the linguistic variables A and B using fuzzy sets.[2] In a fuzzy rule, modus ponens is extended to generalised modus ponens:.[2]
- Premise: x is A*
- Implication: IF x is A THEN y is B
- Consequent: y is B*
The key difference is that the premise x is A can be only partially true. As a result, the consequent y is B is also partially true. Truth is represented as a real number between 0 and 1, where 0 is false and 1 is true.
As an example, consider a rule used to control a three-speed fan. A binary IF-THEN statement may be then
- IF temperature 30
- THEN fan speed is 3
The disadvantage of this rule is that it uses a strict temperature as a threshold, but the user may want the fan to still function at this speed when temperature = 29.9. A fuzzy IF-THEN statement may be
- IF temperature is hot
- THEN fan speed is fast
where hot and fast are described using fuzzy sets.
