Fuzzy operator

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This article is definitively not a tutorial on fuzzy logic. It's simply refers a category of usefull images to help writing wiki articles on fuzzy logic operators. Only, very short comments are thus provided here.

Contents 1 Fuzzyfication 1.1 linear 1.2 non linear 2 Monadic Operators 2.1 NOP, NOT 2.2 CONCENTRATE, DILATE 2.3 Something AND it's Contrary 3 Zadeh Dyadic Operators 4 Dyadic Operators based on a Hyperbolic Paraboloid 5 Yager Dyadic Operators 6 Ternary Operators 6.1 Ternary Operator applied to fuzzy values 6.2 Ternary Operator applied to fuzzy operations 6.2.1 Search Oriented 6.2.2 Miscellaneous 7 The 10 basic Operations 7.1 NOP [0 1] => [0 1] 7.2 NOT [0 1] => [1 0] 7.3 [0 0 1 1] AND [0 1 0 1] => [0 0 0 1] 7.4 [0 0 1 1] OR [0 1 0 1] => [0 1 1 1] 7.5 [0 0 1 1] XOR [0 1 0 1] => [0 1 1 0] (UNEQUAL) 7.6 [0 0 1 1] IMPLIES [0 1 0 1] => [1 1 0 1] 7.7 [0 0 1 1] NAND [0 1 0 1] => [1 1 1 0] 7.8 [0 0 1 1] NOR [0 1 0 1] => [1 0 0 0] 7.9 [0 0 1 1] NXR [0 1 0 1] => [1 0 0 1] (EQUAL) 7.10 [0 0 1 1] NOT_IMPLIES [0 1 0 1] => [0 0 1 0]

[edit] Fuzzyfication

[edit] linear

one segment

two segments

[edit] non linear

one segment 5,15,45... 3645

two segments 10 − 5 to 103

two segments − 103 to 107

[edit] Monadic Operators

[edit] NOP, NOT

NOP (2D view)

NOP (3D view)

NOT (2D view)

NOT (3D view)

[edit] CONCENTRATE, DILATE

CONCENTRATE (2D view)

CONCENTRATE (3D view)

DILATE (2D view)

DILATE (3D view)

[edit] Something AND it's Contrary

A AND (NOT A) (2D view)

A AND (NOT A) (3D view)

[edit] Zadeh Dyadic Operators

The standard fuzzy operators are given the name of Prof. Dr. Lofti A. Zadeh

These operators use very few resources and are thus suitable a.o. for embedded intelligent systems.

Zadeh AND

Zadeh OR

Zadeh XOR (exclusive OR)

Zadeh IMPLIES

Zadeh NAND (NOT AND)

Zadeh NOR (NOT OR)

Zadeh NXR (NOT XOR)

Zadeh NOT IMPLIES

[edit] Dyadic Operators based on a Hyperbolic Paraboloid

HyperbolicParaboloid (by Mathematica)

fuzzy NXR (NOT XOR)

With the increasing power of computers, these operators become usable.

AND (Hyperbolic Paraboloid)

OR (Hyperbolic Paraboloid)

XOR (Hyperbolic Paraboloid)

IMPLIES (Hyperbolic Paraboloid)

NAND (Hyperbolic Paraboloid)

NOR (Hyperbolic Paraboloid)

NXR (Hyperbolic Paraboloid)

NOT IMPLIES (Hyperbolic Paraboloid)

[edit] Yager Dyadic Operators

The images are related to the Yager-2 operators (with exponent = 2).

The operators are given the name of Prof. Dr. [R. Yager]

Yager-2 AND (with exponent=2)

Yager-2 OR (with exponent=2)

[edit] Ternary Operators

Some applications of these operators may need important computer power.

[edit] Ternary Operator applied to fuzzy values

MEAN 50%, 50%

WEIGHTED 75%, 25%

WEIGHTED (french: pondération)

[edit] Ternary Operator applied to fuzzy operations

These operators and similar surfaces are developed by [Marc W.F. Meurrens] (User:Marc.M)

AND-OR 75%, 25%

[edit] Search Oriented

Such fuzzy operators may be usefull a.o. for fuzzy searches. Mind that such operators may be time comsuming.

MEAN-AND-OR

MEAN-OR-AND

WEIGHTED-AND-OR 75%, 25%

WEIGHTED-OR-AND 75%, 25%

[edit] Miscellaneous

The representation of these (generally unused) operators are provided here for the completeness of the Ternary section.

AND-OR-AND (a.k.a. AND)

OR-AND-OR (a.k.a. AND)

XOR-OR-AND (a.k.a. AND)

XOR-AND-OR (a.k.a. AND)

[edit] The 10 basic Operations

[edit] NOP [0 1] => [0 1]

NOP (2D view)

NOP (3D view)

[edit] NOT [0 1] => [1 0]

NOT (2D view)

NOT (3D view)

[edit] [0 0 1 1] AND [0 1 0 1] => [0 0 0 1]

Zadeh AND

Hyperbolic Paraboloid AND

Yager-2 AND

compare fzX == fzY

compare fzY == (1 - fzX)

compare fzX == 75%

compare fzY == 25%

[edit] [0 0 1 1] OR [0 1 0 1] => [0 1 1 1]

Zadeh OR

Hyperbolic Paraboloid OR

Yager-2 OR

compare fzX == fzY

compare fzY == (1 - fzX)

[edit] [0 0 1 1] XOR [0 1 0 1] => [0 1 1 0] (UNEQUAL)

Zadeh XOR

Hyperbolic Paraboloid XOR

[edit] [0 0 1 1] IMPLIES [0 1 0 1] => [1 1 0 1]

Boolean IMPLIES

Zadeh IMPLIES

Hyperbolic Paraboloid IMPLIES

[edit] [0 0 1 1] NAND [0 1 0 1] => [1 1 1 0]

Zadeh NAND

Hyperbolic Paraboloid NAND

[edit] [0 0 1 1] NOR [0 1 0 1] => [1 0 0 0]

Zadeh NOR

Hyperbolic Paraboloid NOR

[edit] [0 0 1 1] NXR [0 1 0 1] => [1 0 0 1] (EQUAL)

Zadeh NXR

Hyperbolic Paraboloid NXR

[edit] [0 0 1 1] NOT_IMPLIES [0 1 0 1] => [0 0 1 0]

Boolean NOT_IMPLIES

Zadeh NOT_IMPLIES

Hyperbolic Paraboloid NOT_IMPLIES