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Until now we have learned: Let us have expressions in the form if A then B i. A is called antecedent or front member and B is called consequent or back member or conclusion. However, it is not used in digital electronics unlike other Boolean operators having their own gate symbols image.

Often in engineering, especially in control, a much used and practical interpretation for implication is technical norm i. A premiss If A then B premiss B conclusion. This equality does not need to hold in fuzzy reasoning: Jos oliolla X on ominaisuus A 1 , niin oliolla Y on ominaisuus B 1 , muuten. The above shows a generalized implication that is often used in expert systems engineering. You might compare it to the if-then-else-if-then-else.. The generalized implication can be interpreted as two-place relations R as shown above.

One possibility is to use the so called Lukasiewicz's implication 1 where. Also other interpretetation are used especially in traditional artificial intelligence applications. When using fuzzy logic we do not end up with a similar contradiction, because the truth value of the implications will decrease , until we end up with. Let us represent the given values by fuzzy sets 1.

For the traditional least squares regression see e. In order to get a fast fuzzy rule based system the rule base is tried to keep minimum. For this optimisation can be used e. Also the individual rules can be optimised. What is the best shape of the membership functions to get the best results.

Also here evolutionary methods are used. In our examples we have had piecewise linear membership functions. This is not necessary but the shape can be any well bevaving function like polynomials, gaussians, trigonometric functions, etc.

Also the fuzzy operations themselves can be modified in certain limits if needed. This gives us yet another optimisation arena. Applications of fuzzy logic in renewable energy systems - A review. Solar tracking system with fuzzy reasoning applied to crisp sets.

All MicroExam problems of this file excl. Let us continue application of fuzzy logic. How can we make fuzzy reasoning.

We are deep in the area of artificial intelligence. We can say that a fuzzy rule is an atomic conclusion. The rules of logic is used to combine these rules into more complex reasonings.

Implication is frequently used in mathematical reasoning. The meaning of implication from truthtable: Another interpretation is that the antecent is " sufficient but not necessary condition" for the consequent. Equivalence arrow both ways is also " necessary condition" i. Tot X gives the truth value as a fuzzy set of expression X. In general the expressions for the calculation of truth values can be deduced from the corresponding inductive mappings i.

Observation A starts action B E. The simpliest deduction consist of one implication and the truth value of its antecent: The classical reasoning Modus Ponens requires that the first premiss and the antecent of the implication are identical.

R can also be interpreted as a rule base. Implication can be interpreted as a fuzzy relation: One possibility is to use the so called Lukasiewicz's implication 1 where x is the truth value of the antecent and y is the truth value of the consequent.

Let us next consider a fuzzy interpolation problem. Let us have the above fuzzy rule base. Then make a fuzzy mapping relation R from implications 2. Using min-max joining we get 1.

The more the given value 2. Now we get for the solution e. Let us try to find AUTOmatically the relation spline , regression , neural network , GP between the antecent and the consequent This is one research and development topic in computational artificial intelligence. From the inducing mapping: Reasoning in a fuzzy rule based system: However, these more general functions tend to be much harder to process with.