Very very cool. Thank you for this drocta. What would it take to map out the sizes of the volumes corresponding to each of these mappings? Also, could you perhaps compute the exact Kolmogorov complexity of these mappings in some particular description language, since they are so small? It would be super interesting to me to assemble a table of volumes and Kolmogorov complexities for each of these small mappings. It may then be possible to write some code that does the same for 3-input and 4-input mappings.
For the volumes, I suppose that because scaling all of these parameters by the same positive constant doesn’t change the function computed, it would make sense to compute the volumes of the corresponding regions of the cube, and this would handle the issues with these regions having unbounded size. (this would still work with more parameters, it would just be a higher dimensional sphere) Er, would that give the same thing as the limit if we took the parameters within a cube? Anyway, at least in this case, if we use the “projected onto the sphere” case, we could evaluate the areas by splitting the regions (which would be polygons of some kind, with edges being arcs of great circles) into triangles, and then using the formulas for the areas of triangles on a sphere. Actually, they might already be triangles, I’m not sure.
Would this work in higher dimensions? I don’t know of formulas for computing the measure of a n-simplex (with flat facets or whatever the right terminology is) within an n-sphere, but I suspect that they shouldn’t be too bad?
I’m not sure which is the more sensible thing to measure, the volumes of the intersection of the half spaces (intersected with a large cube centered at the origin and aligned with the coordinate axes), or the volume (one dimension lower) of that intersected-with/projected-onto the unit sphere.
Well, I guess if we assume that the coefficients are identically and independently distributed with a Gaussian distribution, then that would be a fairly natural choice, and should result in things being symmetric about rotations in the origin, which would seem to point to the choice of projecting it all to the (hyper-)sphere.
Well, I suppose in either case (whether on the sphere or in a cube), even before trying to apply some formulas about the area of a triangle on a sphere, there’s always the “just take the integral” option.
(in the cube option, this would I think be more straightforwards. Just have to do a triple integral (more in higher dimensions) of 1 with linear inequalities for the bounds. No real issues should show up.)
I’ll attempt it with the conditions for “and” for the “on the sphere” case, to check the feasibility. If we have x+y+z>0, x+z<0, y+z<0, then we necessarily also have z<0 , x>0, y>0 , in particular x<-z , y<-z . If we have x,y,z on the unit sphere, then we have x^2+y^2+z^2=1 . So, for each value of z (which must be strictly between −1 and 0) we have x^2 + y^2 = 1 - z^2 , and because we have x>0 and y>0 , for a given z, for each value of x there is exactly one value of y, and visa versa. So, y = sqrt(1 - z^2 - x^2) , and so we have x + sqrt(1 - z^2 - x^2) > -z , … this is somewhat more difficult to calculate than I had hoped. Still confident that it can be done, but I shouldn’t finish this calculation right now due to responsibilities. It looks like, at least in this case with 3 parameters, that it would probably be easier to use the formulas for the area of triangles on a sphere, but I wouldn’t be surprised if, when generalizing to higher dimensions, doing it that way becomes harder.
It looks like Chris Mingard’s reply has nice results which say much of what I think one would want from this direction? Well, it is less “enumerate them specifically”, and more “for functions which have a given proportion of outputs being 1″, but, still. (also I haven’t read it, just looked briefly at it)
I don’t know what particular description language you would want to use for this. I feel like this is such a small case that small differences in choice of description language might overwhelm any difference in complexity that these would have within the given description language?
Yes it does seem challenging to compute realistic complexity measures for such small functions. Perhaps we could just look at the mappings ordered by their volume in parameter space, and check whether the mappings at the top of that ordering “seem” less complex than the mappings at the bottom.
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.
Very very cool. Thank you for this drocta. What would it take to map out the sizes of the volumes corresponding to each of these mappings? Also, could you perhaps compute the exact Kolmogorov complexity of these mappings in some particular description language, since they are so small? It would be super interesting to me to assemble a table of volumes and Kolmogorov complexities for each of these small mappings. It may then be possible to write some code that does the same for 3-input and 4-input mappings.
For the volumes, I suppose that because scaling all of these parameters by the same positive constant doesn’t change the function computed, it would make sense to compute the volumes of the corresponding regions of the cube, and this would handle the issues with these regions having unbounded size.
(this would still work with more parameters, it would just be a higher dimensional sphere)
Er, would that give the same thing as the limit if we took the parameters within a cube?
Anyway, at least in this case, if we use the “projected onto the sphere” case, we could evaluate the areas by splitting the regions (which would be polygons of some kind, with edges being arcs of great circles) into triangles, and then using the formulas for the areas of triangles on a sphere. Actually, they might already be triangles, I’m not sure.
Would this work in higher dimensions? I don’t know of formulas for computing the measure of a n-simplex (with flat facets or whatever the right terminology is) within an n-sphere, but I suspect that they shouldn’t be too bad?
I’m not sure which is the more sensible thing to measure, the volumes of the intersection of the half spaces (intersected with a large cube centered at the origin and aligned with the coordinate axes), or the volume (one dimension lower) of that intersected-with/projected-onto the unit sphere.
Well, I guess if we assume that the coefficients are identically and independently distributed with a Gaussian distribution, then that would be a fairly natural choice, and should result in things being symmetric about rotations in the origin, which would seem to point to the choice of projecting it all to the (hyper-)sphere.
Well, I suppose in either case (whether on the sphere or in a cube), even before trying to apply some formulas about the area of a triangle on a sphere, there’s always the “just take the integral” option.
(in the cube option, this would I think be more straightforwards. Just have to do a triple integral (more in higher dimensions) of 1 with linear inequalities for the bounds. No real issues should show up.)
I’ll attempt it with the conditions for “and” for the “on the sphere” case, to check the feasibility.
If we have x+y+z>0, x+z<0, y+z<0, then we necessarily also have z<0 , x>0, y>0 , in particular x<-z , y<-z . If we have x,y,z on the unit sphere, then we have x^2+y^2+z^2=1 . So, for each value of z (which must be strictly between −1 and 0) we have x^2 + y^2 = 1 - z^2 , and because we have x>0 and y>0 , for a given z, for each value of x there is exactly one value of y, and visa versa.
So, y = sqrt(1 - z^2 - x^2) , and so we have x + sqrt(1 - z^2 - x^2) > -z , …
this is somewhat more difficult to calculate than I had hoped.
Still confident that it can be done, but I shouldn’t finish this calculation right now due to responsibilities.
It looks like, at least in this case with 3 parameters, that it would probably be easier to use the formulas for the area of triangles on a sphere, but I wouldn’t be surprised if, when generalizing to higher dimensions, doing it that way becomes harder.
It looks like Chris Mingard’s reply has nice results which say much of what I think one would want from this direction? Well, it is less “enumerate them specifically”, and more “for functions which have a given proportion of outputs being 1″, but, still. (also I haven’t read it, just looked briefly at it)
I don’t know what particular description language you would want to use for this. I feel like this is such a small case that small differences in choice of description language might overwhelm any difference in complexity that these would have within the given description language?
Yes it does seem challenging to compute realistic complexity measures for such small functions. Perhaps we could just look at the mappings ordered by their volume in parameter space, and check whether the mappings at the top of that ordering “seem” less complex than the mappings at the bottom.
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.