**Stephen Paul King** (*stephenk1@home.com*)

*Fri, 13 Aug 1999 07:30:23 GMT*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Matti Pitkanen: "[time 535] Re: [time 533] Re: [time 530] Surreal numbers"**Previous message:**Stephen P. King: "[time 533] Re: [time 530] Surreal numbers"

Hi All,

More info on Fuzzy operators! "One little step at a time" :-)

On 09 Aug 1999 14:46:16 GMT,wsiler@aol.com (WSiler) wrote:

*>Several recent messages have been concerned with methods of choosing among the
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*>great variety of definitions for the AND and OR operators in multivalued logic.
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*>The following is derived from J.J. Buckley and W. Siler, "A new t-norm", in
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*>Fuzzy Sets and Systems 100:283-290, November, 1998.
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*>
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*>We are given proposition A with truth value a, and proposition B with truth
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*>value b.
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*>The method is based on a model for the truth value of a fuzzy proposition which
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*>asserts that a truth value may be modeled as the average of a large number of
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*>binary (0 or 1) values. Given two such propositions A and B, we may model the
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*>truth value of A AND B (A OR B) by the average of the binary operations of the
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*>0/1 components of A and B. Examples follow:
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*>
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*>(1) Truth values of A and B maximally positively associated; yields Zadehian
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*>min(a,b), max(a,b) logic:
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*>
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*>A B A AND B A OR B
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*>Underlying binary process:
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*>0 0 0 0
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*>0 1 0 1
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*>0 1 0 1
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*>1 1 1 1
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*>Observed truth values:
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*>0.25 0.75 0.25 0.75
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*>
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*>(2) Truth values of A and B maximally negatively associated; yields max(0,
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*>1-(a+b)), min(1, a+b) logic:
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*>
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*>A B A AND B A OR B
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*>Underlying binary process:
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*>0 1 0 1
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*>0 1 0 1
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*>0 1 0 1
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*>1 0 0 1
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*>Observed truth values:
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*>0.25 0.75 0 1
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*>
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*>(2) Truth values of A and B have zero association; yields probabilistic a*b,
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*>a+b-a*b logic:
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*>
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*>A B A AND B A OR B
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*>Underlying binary process:
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*>0 0 0 0
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*>0 1 0 1
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*>1 0 0 1
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*>1 1 1 1
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*>Observed truth values:
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*>0.5 0.5 0.25 0.75
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*>
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*>This approach leads to the following family of multivalued logics:
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*>
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*>A AND B = a*b - p*sqrt(a*(1-a)*b*(1-b))
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*> (1)
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*>A OR B = a + b - a*b - p*sqrt(a*(1-a)*b*(1-b)) (2)
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*>
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*>in which p is the correlation coefficient between the binary values of a and b
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*>in the underlying binary process.
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*>
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*>Any inaccuracies in the estimates of a, b and p require computational
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*>restrictions to be placed on formulas (1) and (2) above. Given specific values
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*>for a and b, the upper and lower possible values for p are:
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*>
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*>pu = (min(a,b) - a*b)) / sqrt(a*(1-a)*b*(1-b)), a,b < 1, a,b > 0
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*>pl = (max(a+b-1, 0) - a*b) / sqrt(a*(1-a)*b*(1-b)), a,b < 1, a,b > 0
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*>
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*>Computationally, pu and pl should be calculated from the specified truth values
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*>a and b. If the specified prior correlation coefficient is outside these
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*>limits, p should be replaced by pu if p > pu or by pl if p < pl. If a or b = 0
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*>or 1, pu and pl are undefined. However, in this case the correlation
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*>coefficient is computationally irrelevant, and the specified prior correlation
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*>coefficient may be used.
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*>
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*>The net result is that if A and B are known a prior to be strongly positively
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*>associated, the Zadehian min-max operators should be used; if strongly
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*>negatively associated, the max(0, 1-a+b), min(a+b, 1) operators should be used;
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*>if a and b are statistically independent, the probabilisitic a*b, a+b-a*b
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*>operators should be used.
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*>
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*>If we have no prior knowledge of prior association between a and b, any desired
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*>operator can be used. We suggest that the Zadehian max-m in operators have
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*>properties that may make them a desirable default.
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*>
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*>We also have cases where the propositions A and B are semantically associated.
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*>For example. In A AND A we have maximally positive association, and the
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*>Zadehian operators should be used; if A AND NOT A we have maximally negative
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*>association, and the max(0, 1-a+b), min(a+b, 1) operators should be used. The
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*>effect of this is to restore the laws of excluded middle and contradiction to
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*>multivalued logics, even when the Zadehian operators are used as the default,
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*>eliminating the annoying notch found when ORing fuzzy numbers which are closely
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*>adjacent using max-min logic. Also, Elkan's well-known "proof" that fuzzy logic
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*>will not work falls apart using the above choice of operators with max-min as
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*>the default.
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*>
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*>William Siler
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*>
*

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