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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