Fuzzy rule

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.

Comparison between Boolean and fuzzy logic rules

As an example, consider a rule used to control a three-speed fan. A binary IF-THEN statement may be then

IF temperature

\geq

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.

Fuzzy rule connectors

Rules can connect multiple variables through fuzzy set operations using t-norms and t-conorms.

T-norms are used as an AND connector.^[3]^[4]^[5] For example,

IF temperature is hot AND humidity is high

THEN fan speed is fast

The degree of truth assigned to temperature is hot and to humidity is high. The result of a t-norm operation on these two degrees is used as the degree of truth that fan speed is fast.

T-conorms are used as an OR connector.^[5] For example,

IF temperature is hot OR humidity is high

THEN fan speed is fast

The result of a t-conorm operation on these two degrees is used as the degree of truth that fan speed is fast.

The complement of a fuzzy set is used as a negator.^[5] For example,

IF temperature is NOT hot

THEN fan speed is slow

The fuzzy set not hot is the complement of hot. The degree of truth assigned to temperature is not hot is used as the degree of truth that fan speed is slow.

T-conorms are less commonly used as rules can be represented by AND and OR connectors exclusively.

References

^ B., Enderton, Herbert (2001). A mathematical introduction to logic (2nd ed.). San Diego, Calif.: Academic Press. ISBN 978-0122384523. OCLC 45830890.{{cite book}}: CS1 maint: multiple names: authors list (link)
^ ^a ^b ^c Mendel, Jerry M. (2001). Uncertain rule-based fuzzy logic systems : introduction and new directions. Upper Saddle River, NJ: Prentice Hall PTR. ISBN 978-0130409690. OCLC 45314121.
^ Martin Larsen, P. (1980). "Industrial applications of fuzzy logic control". International Journal of Man-Machine Studies. 12 (1): 3–10. doi:10.1016/s0020-7373(80)80050-2. ISSN 0020-7373.
^ Mamdani, E.H. (1974). "Application of fuzzy algorithms for control of simple dynamic plant". Proceedings of the Institution of Electrical Engineers. 121 (12): 1585. doi:10.1049/piee.1974.0328. ISSN 0020-3270.
^ ^a ^b ^c H.-J., Zimmermann (1991). Fuzzy Set Theory - and Its Applications (Second, revised ed.). Dordrecht: Springer Netherlands. ISBN 9789401579490. OCLC 851369348.