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anhinga_anhinga) wrote2009-08-05 11:30 am
anhinga_anhinga) wrote2009-08-05 11:30 am
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Duality between metric and logical viewpoints
The metric viewpoint: how far two objects are from each other. The logical viewpoint: to what degree two objects overlap.
Fuzzy mathematics is traditionally done from the logical viewpoint, so the first step in introducing fuzzy metrics is often the transformation f(x,y) = exp(-d(x,y)).
Then we have the following correspondences:
d(x,y) = 0 if and only if f(x,y) = 1.
d(x,y) is plus infinity if and only if f(x,y) = 0.
d(x1, y1) < d(x2, y2) if and only if f(x1, y1) > f(x2, y2).
The axiom d(x,x) = 0 becomes f(x,x) = 1.
The axiom d(x,z) < d(x,y) + d(y,z) becomes f(x,y) * f(y,z) < f(x,z).
Non-expansive maps become maps which respect overlap by not letting it decrease.
Etc..
However, this seems to be a rather superficial duality: basically two equivalent ways to write the same things using different notation.
The question is whether there is also a natural deeper duality here (of a contravariant nature, where function arrows would reverse direction when one switches between these two viewpoints).
Fuzzy mathematics is traditionally done from the logical viewpoint, so the first step in introducing fuzzy metrics is often the transformation f(x,y) = exp(-d(x,y)).
Then we have the following correspondences:
d(x,y) = 0 if and only if f(x,y) = 1.
d(x,y) is plus infinity if and only if f(x,y) = 0.
d(x1, y1) < d(x2, y2) if and only if f(x1, y1) > f(x2, y2).
The axiom d(x,x) = 0 becomes f(x,x) = 1.
The axiom d(x,z) < d(x,y) + d(y,z) becomes f(x,y) * f(y,z) < f(x,z).
Non-expansive maps become maps which respect overlap by not letting it decrease.
Etc..
However, this seems to be a rather superficial duality: basically two equivalent ways to write the same things using different notation.
The question is whether there is also a natural deeper duality here (of a contravariant nature, where function arrows would reverse direction when one switches between these two viewpoints).