Stephen Paul King (stephenk1@home.com)
Fri, 13 Aug 1999 07:30:23 GMT
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
>great variety of definitions for the AND and OR operators in multivalued logic.
>The following is derived from J.J. Buckley and W. Siler, "A new t-norm", in
>Fuzzy Sets and Systems 100:283-290, November, 1998.
>
>We are given proposition A with truth value a, and proposition B with truth
>value b.
>The method is based on a model for the truth value of a fuzzy proposition which
>asserts that a truth value may be modeled as the average of a large number of
>binary (0 or 1) values. Given two such propositions A and B, we may model the
>truth value of A AND B (A OR B) by the average of the binary operations of the
>0/1 components of A and B. Examples follow:
>
>(1) Truth values of A and B maximally positively associated; yields Zadehian
>min(a,b), max(a,b) logic:
>
>A B A AND B A OR B
>Underlying binary process:
>0 0 0 0
>0 1 0 1
>0 1 0 1
>1 1 1 1
>Observed truth values:
>0.25 0.75 0.25 0.75
>
>(2) Truth values of A and B maximally negatively associated; yields max(0,
>1-(a+b)), min(1, a+b) logic:
>
>A B A AND B A OR B
>Underlying binary process:
>0 1 0 1
>0 1 0 1
>0 1 0 1
>1 0 0 1
>Observed truth values:
>0.25 0.75 0 1
>
>(2) Truth values of A and B have zero association; yields probabilistic a*b,
>a+b-a*b logic:
>
>A B A AND B A OR B
>Underlying binary process:
>0 0 0 0
>0 1 0 1
>1 0 0 1
>1 1 1 1
>Observed truth values:
>0.5 0.5 0.25 0.75
>
>This approach leads to the following family of multivalued logics:
>
>A AND B = a*b - p*sqrt(a*(1-a)*b*(1-b))
> (1)
>A OR B = a + b - a*b - p*sqrt(a*(1-a)*b*(1-b)) (2)
>
>in which p is the correlation coefficient between the binary values of a and b
>in the underlying binary process.
>
>Any inaccuracies in the estimates of a, b and p require computational
>restrictions to be placed on formulas (1) and (2) above. Given specific values
>for a and b, the upper and lower possible values for p are:
>
>pu = (min(a,b) - a*b)) / sqrt(a*(1-a)*b*(1-b)), a,b < 1, a,b > 0
>pl = (max(a+b-1, 0) - a*b) / sqrt(a*(1-a)*b*(1-b)), a,b < 1, a,b > 0
>
>Computationally, pu and pl should be calculated from the specified truth values
>a and b. If the specified prior correlation coefficient is outside these
>limits, p should be replaced by pu if p > pu or by pl if p < pl. If a or b = 0
>or 1, pu and pl are undefined. However, in this case the correlation
>coefficient is computationally irrelevant, and the specified prior correlation
>coefficient may be used.
>
>The net result is that if A and B are known a prior to be strongly positively
>associated, the Zadehian min-max operators should be used; if strongly
>negatively associated, the max(0, 1-a+b), min(a+b, 1) operators should be used;
>if a and b are statistically independent, the probabilisitic a*b, a+b-a*b
>operators should be used.
>
>If we have no prior knowledge of prior association between a and b, any desired
>operator can be used. We suggest that the Zadehian max-m in operators have
>properties that may make them a desirable default.
>
>We also have cases where the propositions A and B are semantically associated.
>For example. In A AND A we have maximally positive association, and the
>Zadehian operators should be used; if A AND NOT A we have maximally negative
>association, and the max(0, 1-a+b), min(a+b, 1) operators should be used. The
>effect of this is to restore the laws of excluded middle and contradiction to
>multivalued logics, even when the Zadehian operators are used as the default,
>eliminating the annoying notch found when ORing fuzzy numbers which are closely
>adjacent using max-min logic. Also, Elkan's well-known "proof" that fuzzy logic
>will not work falls apart using the above choice of operators with max-min as
>the default.
>
>William Siler
>
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