Re: Consistency of fuzzy rules

From: Pim van den Broek (pimvdb@cs.utwente.nl)
Date: Mon Oct 15 2001 - 19:16:56 MET DST

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    WSiler@aol.com wrote:

    > I see no reason why different consequents mean necessarily that the
    > rules are inconsistent.

    I agree.
    I even would go one step further: rules are never inconsistent.
    Inconsistency only arises when rules contradict domain knowledge.
    This is irrespective of whether the rules are interpreted as
    Mamdani-like
    rules or as implications.
    Consider the following trivial example with crisp singleton sets, where
    the
    rules are supposed to contribute to a relation Is_Married_With:

    IF X = {George} THEN Y = {Anna}
    IF X = {George} THEN Y = {Sue}

    Interpreted as Mamdani-like rules, the first rule says that George is
    married
    with Anna, and the second rule says that he is marrried with Sue.
    Together
    the rules say that George is married to both Anna and Sue. This is not
    an
    inconsistency. There might be a contradiction however with domain
    knowledge. Such a domain knowledge might be: no man is married with
    more than one woman.

    Interpreted as implications, the first rule says that George is not
    married
    with any woman other than Anna, and the second rule says says that he
    is not married with any woman other than Sue. Together the rules say
    that George is unmarried. No inconsistency. There might be a
    contradiction
    with domain knowledge, for instance with the knowledge that George is
    a married man.

    Pim van den Broek

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    <!doctype html public "-//w3c//dtd html 4.0 transitional//en">
    <html>
    &nbsp;
    <p>WSiler@aol.com wrote:
    <blockquote TYPE=CITE><font face="Arial"><font color="#000000"><font size=+0>I
    see no reason why different consequents mean necessarily that the rules
    are inconsistent.</font></font></font></blockquote>
    I agree.
    <br>I even would go one step further: rules are never inconsistent.
    <br>Inconsistency only arises when rules contradict domain knowledge.
    <br>This is irrespective of whether the rules are interpreted as Mamdani-like
    <br>rules or as implications.
    <br>Consider the following trivial example with crisp singleton sets, where
    the
    <br>rules are supposed to contribute to a relation Is_Married_With:
    <p>IF X = {George} THEN Y = {Anna}
    <br>IF X = {George} THEN Y = {Sue}
    <p>Interpreted as Mamdani-like rules, the first rule says that George is
    married
    <br>with Anna, and the second rule says that he is marrried with Sue. Together
    <br>the rules say that George is married to both Anna and Sue. This is
    not an
    <br>inconsistency. There might be a contradiction however with domain
    <br>knowledge. Such a domain knowledge might be: no man is married with
    <br>more than one woman.
    <p>Interpreted as implications, the first rule says that George is not
    married
    <br>with any woman other than Anna, and the second rule says says that
    he
    <br>is not married with any woman other than Sue. Together the rules say
    <br>that George is unmarried. No inconsistency. There might be a contradiction
    <br>with domain knowledge, for instance with the knowledge that George
    is
    <br>a married man.
    <p>Pim van den Broek</html>

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