I’ve now computed the volumes within the [-a,a]^3 cube for and, or, and the constant 1 function. I was surprised by the results. (I hadn’t considered that the ratios between the volumes will not depend on the size of the cube) If we select x,y,z uniformly at random within this cube, the probability of getting the and gate is 1⁄48, the probability of getting the or gate is 2⁄48, and the probability of getting the constant 1 function is 13⁄48 (more than 1⁄4). This I found quite surprising, because of the constant 1 function requiring 4 half planes to express the conditions for it.
So, now I’m guessing that the ones that required fewer half spaces to specify, are the ones where the individual constraints are already implying other constraints, and so actually will tend to have a smaller volume.
On the other hand, I still haven’t computed any of them for if projecting onto the sphere, and so this measure kind of gives extra weight to the things in the directions near the corners of the cube, compared to the measure that would be if using the sphere.
I’ve now computed the volumes within the [-a,a]^3 cube for and, or, and the constant 1 function. I was surprised by the results.
(I hadn’t considered that the ratios between the volumes will not depend on the size of the cube)
If we select x,y,z uniformly at random within this cube, the probability of getting the and gate is 1⁄48, the probability of getting the or gate is 2⁄48, and the probability of getting the constant 1 function is 13⁄48 (more than 1⁄4).
This I found quite surprising, because of the constant 1 function requiring 4 half planes to express the conditions for it.
So, now I’m guessing that the ones that required fewer half spaces to specify, are the ones where the individual constraints are already implying other constraints, and so actually will tend to have a smaller volume.
On the other hand, I still haven’t computed any of them for if projecting onto the sphere, and so this measure kind of gives extra weight to the things in the directions near the corners of the cube, compared to the measure that would be if using the sphere.