Fuzzy rules are used within
fuzzy logic systems to infer an output based on input variables.
Modus ponens and
modus tollens
In propositional logic, ''modus tollens'' () (MT), also known as ''modus tollendo tollens'' (Latin for "method of removing by taking away") and denying the consequent, is a deductive argument form and a rule of inference. ''Modus tollens' ...
are the most important rules of inference. 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.
This is achieved by representing the linguistic variables ''A'' and ''B'' using
fuzzy set
In mathematics, fuzzy sets (a.k.a. uncertain sets) are sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh in 1965 as an extension of the classical notion of set.
At the same time, defined ...
s.
In a fuzzy rule, modus ponens is extended to ''generalised modus ponens:.
''
: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
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
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''
''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 set
In mathematics, fuzzy sets (a.k.a. uncertain sets) are sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh in 1965 as an extension of the classical notion of set.
At the same time, defined ...
s.
Fuzzy rule connectors
Rules can connect multiple variables through
fuzzy set operations
Fuzzy set operations are a generalization of crisp set operations for fuzzy sets. There is in fact more than one possible generalization. The most widely used operations are called ''standard fuzzy set operations''; they comprise: fuzzy compl ...
using
t-norm In mathematics, a t-norm (also T-norm or, unabbreviated, triangular norm) is a kind of binary operation used in the framework of probabilistic metric spaces and in multi-valued logic, specifically in fuzzy logic. A t-norm generalizes intersectio ...
s and
t-conorms.
T-norms are used as an ''AND'' connector.
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.
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
A complement is something that completes something else.
Complement may refer specifically to:
The arts
* Complement (music), an interval that, when added to another, spans an octave
** Aggregate complementation, the separation of pitch-clas ...
of a fuzzy set is used as a negator.
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.
See also
*
Fuzzy logic
References
Fuzzy logic
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