Mamdani implication

The Mamdani implication or Mamdani implication operator belongs to the group of fuzzy implication operators from the field of fuzzy logic . The fuzzy implication represents an extension of the implication from classical logic . With this extension, logical inference , as is known from classical logic, is also possible on the basis of uncertain knowledge. This is also called fuzzy logical reasoning. With fuzzy logic implication operators, the truth content of the premise affects the truth content of the result.

The Mamdani implication operator is defined by the membership function: ${\displaystyle \mathrm {min} [x,y]}$

${\displaystyle \mu _{A\to B}(x,y):=\mathrm {min} \lbrace \mu _{A}(x),\mu _{B}(y)\rbrace }$.

This is intended to express that the truth value of the result will not be greater than the truth value of the premise.

For other applications, a different truth relationship between premise and result may be more suitable. This is modeled by other fuzzy implication operators, for example the Zadeh implication is defined as . ${\displaystyle \mathrm {max} [\mathrm {min} [x,y],1-x]}$

The Mamdani implication operator is used, for example, in control engineering.

literature

• Josef Börcsök: Fuzzy Control. Theory and industrial use . Verlag Technik Berlin, 2000. ISBN 3-341-01237-0

See also

• Related terms: blurring and uncertainty
• Trivalent logic
• Multi-valued logic