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The fuzzy logical ( fuzzy logic , in English) is a technique used in Artificial intelligence. It was formalized by Lotfi Zadeh in 1965 and was used in fields as varied as the automatism (brakes ABS), the Robotique (pattern recognition), the management of the road traffic (red lights), the air Contrôle, the environment (Météorologie, Climatologie, Sismologie), the Médecine (assistance with the diagnosis), the Assurance (selection and prevention of the Risque S) and well of others. In fact, the simple fact of noting, already under Jules Ferry, a pupil in various disciplines and of calculating to him a row by application of coefficients to its notes was already to make fuzzy logic without the knowledge.

It rests on the mathematical theory of the vague units . This theory, introduced by Zadeh, is an extension of the traditional set theory for the taking into account of definite units in a vague way. It is a formal and mathematical theory in the direction where Zadeh, on the basis of the concept of function of membership to model the definition of a subset of a given universe, worked out a complete model of properties and formal definitions. It as showed as this theory of the fuzzy subsets is reduced indeed to the theory traditional subsets if the functions of membership considered take binary values ({0,1}).

It is as of the interest to be easier and cheaper to implement as a probabilistic logic , although the latter alone is strictly speaking coherent (see Théorème of Cox-Jaynes). For example the curve ''' Ev (p) ''' can be replaced by three segments of right-hand side without excessive loss of precision for much of applications considered above.

Principle

Contrary to the Boolean logical , fuzzy logic allows a condition of being in another state that true or false . There are degrees in the checking of a condition.

Let us consider for example the speed of a vehicle on a trunk road. Normal speed is of 90 km/h. A speed can be regarded high above 100 km/h, and as any more high in lower part of 80 km/h. Boolean logic would consider the things in the following way (see fig. 1 ):

speed is considered with 100 % like high starting from 100 km/h, and with 0 % in lower part.

Is fuzzy logic, contrary, allows degrees of checking of the condition “speed high? ” (see fig. 2 ):

speed is regarded as at all high in lower part of 80 km/h. One can thus say that in lower part 80 km/h, speed is high with 0 %.

speed is regarded as high above 100 km/h. Speed is thus high with 100 % above 100 km/h.
speed is thus high with 50 % to 90 km/h, and 25 % to 85 km/h.

Same manner, the function “is speed relatively low? ” will be evaluated in the following way (see fig. 3 ):

speed is regarded as relatively low in lower part of 80 km/h. It is thus relatively low with 100 %.

speed is regarded as at all relatively low above 100 km/h. It is thus relatively low with 0 %.
speed is thus relatively low with 50 % with 90km/h, and 75 % to 85 km/h.

One a function “is speed can also define average? ” (see fig. 4):

speed is average to 90 km/h. With this pace, speed is average with 100 %.

speed is not at all average in lower part of 80 km/h and above 100 km/h. Out of this interval, speed is average with 0 %.
speed is thus average with 50 % to 85 km/h and 95 km/h.

It is not obligatory that the transition is linear. Hyperbolic transitions (like a Sigmoid or a hyperbolic tangent ), Exponential, Gaussian (in the case of an average state) or of any other nature are usable (see fig. 5 ).

Combination of several entries

In the case of combination of several entries (“If the sky is blue and if I have time”), two cases arise:

the entries are bound by a switching function “AND”: in this case, one can regard as first approaches only the entry having the smallest degree of the checking. In fact, it is enough to choose an operator $\ top$ such as $\ signal (X, there) \ Leq \ min (X, there)$ where $\ top$ is called a T-standard . $\ min$ is more optimistic T-standards.
the entries are bound by a switching function “OR”: in this case, one can regard as first approaches only the entry having the highest degree of checking. In fact, it is enough to choose an operator $\ bot$ such as $\ club-footed (X, there) \ geq \ max (X, there)$ where $\ bot$ is called a t-conorme . $\ max$ is more pessimistic t-conormes.

It is technically possible to represent all the basic binary operations while being based on fuzzy logic. Indeed, starting from the operators AND , OR and NOT ( AND , GOLD , NOT ), one can represent the 8 basic operations:

OR (GOLD): WITH GOLD B = max (has, B);
AND (AND): WITH AND B = min (has, B);
NOT (NOT): NOT HAS = 1 - HAS;
OR EXCLUSIVE (XOR): WITH XOR B = (GOLD B) AND NOT (has AND B) = has + B HAS - 2 × min (has, B);
NON-OU (NOR): WITH NOR B = 1 - max (has, B);
NON-ET (NAND): WITH NAND B = 1 - min (has, B);
NON-XOR (NXR): WITH NXR B = 1 + 2 × min (has, B) - (has + B);
FOLLOWING (NOP): NOP has = A.

In addition, the decimal dimension of the variables of fuzzy logic makes it possible to carry out nonbinary combinations:

the product: A.B or has × B (equivalent into binary with operation AND)
the addition: With + B (equivalent into binary with the operation GOLD)

Fuzzy operators

The fuzzy operators (or fuzzy ) can be implemented in various ways and the same application can call upon judiciously selected different implementations besides according to the context.

The example below watch that, contrary to a sometimes expressed opinion, the choice of the operators to be used is not a simple matter of taste or inspiration of the moment.

Example of use

Here an example which shows how to combine fuzzy operators (fuzzy) of various types.

the example: to confirm the membership of a person to a group

A person will be more or less member (on a level fzMembre , ranging between 0 and 1, inclusively) of a group, say the friendly flowers , is if, for any reason, it forms already more or less part of it (on a level fzDejaMembre ), is if it has a rather good knowledge of the orchises ( fzConnaitOrchidees ) and that the knowledge of the orchises is a criterion enough determinant ( fzOrchideesAmis ) of membership of the group of the friendly of the flowers .

a solution which appears good is:

fzMembre = Zadeh_OR (fzDejaMembre, Multiply_AND (fzConnaitOrchidees, fzOrchideesAmis))

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