  I proposed rate tolerance and discussed its relation to rate distortion in my
book "A Generalized Information Theory" published in 1993. Recently, I examined
the structure function and the complexity distortion based on Kolmogorov's
complexity theory. It is my understanding now that complexity-distortion is
only a special case of rate tolerance while constraint sets change from fuzzy
sets into clear sets that look like balls with the same radius. It is not true
that the complexity distortion is generally equivalent to rate distortion as
claimed by the researchers of complexity theory. I conclude that a rate
distortion function can only be equivalent to a rate tolerance function and
both of them can be described by a generalized mutual information formula where
P(Y|X) is equal to P(Y|Tolerance). The paper uses GPS as an example to derive
generalized information formulae and proves the above conclusions using
mathematical analyses and a coding example. The similarity between the formula
for measuring GPS information and the formula for rate distortion function can
deepen our understanding the generalized information measure.
