Editing Fuzzy logic (section)

====Fuzzy logic operators====

Fuzzy logic works with membership values in a way that mimics [[Boolean logic]]. To this end, replacements for basic [[operator (computer programming)|operator]]s ("gates") AND, OR, NOT must be available. There are several ways to this. A common replacement is called the ''{{vanchor|Zadeh operator}}s'':

{| class="wikitable"
|-
! Boolean
! Fuzzy
|-
| AND(x,y)
| MIN(x,y)
|-
| OR(x,y)
| MAX(x,y)
|-
| NOT(x)
| 1 – x
|-
|}

For TRUE/1 and FALSE/0, the fuzzy expressions produce the same result as the Boolean expressions.

There are also other operators, more linguistic in nature, called ''hedges'' that can be applied. These are generally adverbs such as ''very'', or ''somewhat'', which modify the meaning of a set using a [[mathematical formula]].<ref>{{Cite journal |last=Zadeh |first=L. A. |date=January 1972 |title=A Fuzzy-Set-Theoretic Interpretation of Linguistic Hedges |url=http://dx.doi.org/10.1080/01969727208542910 |journal=Journal of Cybernetics |volume=2 |issue=3 |pages= 4–34|doi=10.1080/01969727208542910 |issn=0022-0280}}</ref>

However, an arbitrary choice table does not always define a fuzzy logic function. In the paper (Zaitsev, et al),<ref>{{ Cite journal | last1 = Zaitsev | first1 = D. A. | author2 = Sarbei, V. G. | author3 = Sleptsov, A. I. | year = 1998 | title = Synthesis of continuous-valued logic functions defined in tabular form | journal = [[Cybernetics and Systems Analysis]] | volume = 34 | issue = 2 | pages = 190–195 | doi = 10.1007/BF02742068 | s2cid = 120220846 }}</ref> a criterion has been formulated to recognize whether a given choice table defines a fuzzy logic function and a simple algorithm of fuzzy logic function synthesis has been proposed based on introduced concepts of constituents of minimum and maximum. A fuzzy logic function represents a disjunction of constituents of minimum, where a constituent of minimum is a conjunction of variables of the current area greater than or equal to the function value in this area (to the right of the function value in the inequality, including the function value).

Another set of AND/OR operators is based on multiplication, where

<syntaxhighlight lang="text">
x AND y = x*y
NOT x = 1 - x

Hence, 
x OR y = NOT( AND( NOT(x), NOT(y) ) )
x OR y = NOT( AND(1-x, 1-y) )
x OR y = NOT( (1-x)*(1-y) )
x OR y = 1-(1-x)*(1-y)
x OR y = x+y-xy
</syntaxhighlight>

Given any two of AND/OR/NOT, it is possible to derive the third. The generalization of AND is an instance of a [[t-norm]].