Why interval arithmetic is so useful (2007) [pdf]

I think this kind of thing is great, but it runs into some of the same considerations that make probabilistic graphical models hard.

If I have two variables defined as

    x = [0,1];
    y = [0,1];
    z = x - y;

Then we see that z should be the interval [-1,1]. But for a similar problem,

    a = [0,1];
    b = a - a;
    x = a;
    y = a;
    z = x - y;

b should clearly be the interval [0,0]. z is computed the same way as the first example, as the difference between two [0,1] intervals. But this time we want it to be [0,0]. That means we want to keep around some notion similar to the joint distribution of x and y. That makes arithmetic immensely harder than simple rules.

Not to say it's not worth it, just that it's hard.

Could such intervals be represented by random variables of uniform distributions?