Re: On combinatorial explosions

Scott Dick (dick@morden.csee.usf.edu)
Sun, 14 Jun 1998 21:56:40 +0200 (MET DST)

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I have also read the paper by Coombs and Andrews. What was disturbing to me
was a key proof that seemed to violate some basic tenets of fuzzy rulebases. I
refer to Appendix A of that paper, in which the relation (p and q) -> r is
transformed into (p -> r) or (q -> r). The transforms given are obviously
correct for Boolean logic. However, I am not convinced that these transforms
hold up under the more general definitions of fuzzy logic.
Additionally, fuzzy rulebases in practice do not use proper fuzzy
implications. The classic Mamdani or TSK fuzzy system use the compositional rule
of inference, not a fuzzy version of the implication operator.

Bob Dalton wrote:

> Ranier Holve writes that there is a combinatorial explosion with n
> inputs and m fuzzy sets => m^n rules:
>
> >> Actually, the number of possible rules grows exponentially with the
> >> number of inputs, i.e. n inputs with m fuzzy sets -> m^n rules.
>
> Coombs and Andrews, IEEE Trans. Fuzzy Sys 6:1, pp 1-11, 1998 say
> otherwise, i. e. that the "curse of dimensionality" is no more. I believe
> this was ther reference Siler was referring to.
>
> Reading the Coombs paper, I can see how the exponentiality is addressed,
> but what I don't see is how the fidelity of the implications of the
> original (exponential) rulebase is retained: i. e. how the mn "new"
> rules imply the same thing the m^n "old" rules implied.
>
> Neither Coombs or Holve provides a reference for [Bellman] and the
> binary tree representation. It might be important.
>
> --------------------------------------------------------------------
> --------------------------------------------------------------------
>
> Robert E. Dalton, Ph. D. email: red@swl.msd.ray.com
>
> Raytheon Co. Mailstop: T3MN35 voice: (978) 858-5927
>
> 50 Apple Hill Drive, Tewksbury, MA 01876
>
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***********************************************************************
*                               *                                     *
*  Scott Dick                   *                                     *
*  Research Assistant           *       Cool & brilliant thought      *
*  USF Computer Science         *       still under contruction       *
*  dick@morden.csee.usf.edu     *                                     *
*                               *                                     *
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    I have also read the paper by Coombs and Andrews. What was disturbing to me was a key proof that seemed to violate some basic tenets of fuzzy rulebases. I refer to Appendix A of that paper, in which the relation (p and q) -> r is transformed into (p -> r) or (q -> r). The transforms given are obviously correct for Boolean logic. However, I am not convinced that these transforms hold up under the more general definitions of fuzzy logic.
    Additionally, fuzzy rulebases in practice do not use proper fuzzy implications. The classic Mamdani or TSK fuzzy system use the compositional rule of inference, not a fuzzy version of the implication operator.
 
 

Bob Dalton wrote:

Ranier Holve writes that there is a combinatorial explosion with n
inputs and m fuzzy sets => m^n rules:

>> Actually, the number of possible rules grows exponentially with the
>> number of inputs, i.e. n inputs with m fuzzy sets -> m^n rules.

Coombs and Andrews, IEEE Trans. Fuzzy Sys 6:1, pp 1-11, 1998 say
otherwise, i. e. that the "curse of dimensionality" is no more.  I believe
this was ther reference Siler was referring to.

Reading the Coombs paper, I can see how the exponentiality is addressed,
but what I don't see is how the fidelity of the implications of the
original (exponential) rulebase is retained:  i. e. how the mn "new"
rules imply the same thing the m^n "old" rules implied.

Neither Coombs or Holve provides a reference for [Bellman] and the
binary tree representation.  It might be important.

--------------------------------------------------------------------
--------------------------------------------------------------------

      Robert E. Dalton, Ph. D.          email:  red@swl.msd.ray.com

      Raytheon Co.  Mailstop:  T3MN35   voice:  (978) 858-5927

      50 Apple Hill Drive, Tewksbury, MA  01876

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This message was posted through the fuzzy mailing list.
(1)  To subscribe to this mailing list, send a message body of
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(2)  To unsubscribe from this mailing list, send a message body of
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to listproc@dbai.tuwien.ac.at
(3)  To reach the human who maintains the list, send mail to
fuzzy-owner@dbai.tuwien.ac.at
(4)  WWW access and other information on Fuzzy Sets and Logic see
http://www.dbai.tuwien.ac.at/ftp/mlowner/fuzzy-mail.info
(5) WWW archive: http://www.dbai.tuwien.ac.at/marchives/fuzzy-mail/index.html

 
-- 
***********************************************************************
*                               *                                     *
*  Scott Dick                   *                                     *
*  Research Assistant           *       Cool & brilliant thought      *         
*  USF Computer Science         *       still under contruction       *
*  dick@morden.csee.usf.edu     *                                     *
*                               *                                     *
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  --------------F0E9CA236593765FA807A48D-- ############################################################################ This message was posted through the fuzzy mailing list. (1) To subscribe to this mailing list, send a message body of "SUB FUZZY-MAIL myFirstName mySurname" to listproc@dbai.tuwien.ac.at (2) To unsubscribe from this mailing list, send a message body of "UNSUB FUZZY-MAIL" or "UNSUB FUZZY-MAIL yoursubscription@email.address.com" to listproc@dbai.tuwien.ac.at (3) To reach the human who maintains the list, send mail to fuzzy-owner@dbai.tuwien.ac.at (4) WWW access and other information on Fuzzy Sets and Logic see http://www.dbai.tuwien.ac.at/ftp/mlowner/fuzzy-mail.info (5) WWW archive: http://www.dbai.tuwien.ac.at/marchives/fuzzy-mail/index.html