# Re: Basic Theory

From: Paul Victor Birke (nonlinear@home.com)
Date: Tue Apr 03 2001 - 06:59:53 MET DST

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Yasser wrote:
>
> "AlanMcRob" <alanmcrob@aol.com> wrote in message
> news:20010324213723.01436.00000539@ng-de1.aol.com...
> > Can anyone explain the basic theory behind fuzzy logic to me.
>
> Basically, fuzzy logic is about the fuzziness of logic. Meaning that in
> real life, things are not always black or white, true or false, etc. They
> actually come in various "grey levels". For example, there is a tall man
> and a short man, but is a 175 cm man short or tall? The answer is fuzzy. He
> is tall compared to a 150 cm man, but short compared to a basket ball
> player. It all depends on the context and perception.
>
> In logic, we deal with "sets". For example, the set of numbers {1,2,3}.
> Now any number can either belong to that set (eg. number 2) or does not
> belong to that set (e.g. number 4). This is called "crisp" logic and has
> been in use for thousands of years. Now, consider the set of all tall men.
> Does a 175 cm man belong to this set or not? The answer depends on what you
> mean by "tall men", it is vague, it is fuzzy.
>
> Hence, fuzzy logic is the logic that deals with situations where you can't
> give a clear yes/no type of answer. This turns out to be a very common
> situation in many human decision making processes. It also has significant
> applications for systems that are highly non-linear and very complex to
> model mathematically.
>
> In Zadeh's fuzzy logic, the gradual transition from Yes to No, True to
> False, is expressed by a "membership function", graphically describing the
> degree of truth of a certain proposition. Crisp logic is thus just a
> special case of fuzzy logic. In crisp logic, membership functions are just
> one vertical line, whereas in fuzzy logic, they have a distribution. For
> practical purposes, you can think of membership functions like probability
> distributions (although conceptually, there is a difference).
>
> Once, you accept this "membership function" issue, the rest is just standard
> logic operation (AND, OR, NOT, UNION, INTERSECTION, etc) and standard
> arithmetic ( +, -, x, etc), only that these operations have been extended to
> include the effect of membership functions. This turns out to be handy in
> many practical situations, for example: automatic control. Why spend time
> solving complex non-linear equations, and testing stability etc, when you
> can achieve the same commercial outcome by means of few "fuzzy rules"?
>
> Conceptually, fuzzy logic is a nice new idea; mathematically, it is
> revolutionary; commercially it means more money for your product. However,
> to be fair, the outcome you get from FL is always achievable by other means.
> In my opinion, Fuzzy logic, Neural Networks, Genetic Algorithms,
> Probability, Fractals, etc are all faces of the same daemon: a unifying
> piece of science yet to be discovered.
>
> Finally, Fuzzy Logic seems to be in contradiction with all mono-theistic
> religions including Christianity, and in good conformity with Buddhism.
> That is why half the Fuzzy Logic population is in China.
> --
> Yasser Ali
> Mechanical Engineering Dept., J07
> Sydney University
************************************

Dear Yasser

This was a wonderfully concise yet full explanation!

Only to add by saying the membership function itself, by this I mean its
special shape we hold so dear, usually is somekind of "S" shape,
normally symetric but not necessarily so I have claimed its form could
be generally in 3-parameter Weibull in shape. I mean the "S" shape
would correspond to the CDF of a Weibull PDF Distribution with variable
to the modeller the full SCALE, SLOPE & __LOCATION__ parameters!

Paul

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