EA://INTEL — LIMITS ON MEANING SEPARATION

LIMITS ON MEANING SEPARATION

When modeling meanings as distributions over outcomes, a ceiling exists on how far apart they can be. This is because the space of distributions becomes the surface of a sphere under a square-root change of coordinates. The diameter of this sphere limits the maximum separation between two meaning-states. No matter how many outcomes are considered, the extra room does not translate into extra separation.

To see this, suppose we have two distributions that agree nowhere. They will be as far apart as the geometry allows. Adding more outcomes only gives more ways to be at the maximum distance, it doesn’t increase the distance itself. The same shape shows up by a completely different route: in the classical construction of a geometric packing from a discrete code, achievable separation stops improving past a certain point.

A worked example illustrates this: two distributions that agree nowhere are as far apart as this geometry allows. Even with a thousand more outcomes, they remain exactly that far apart. The computation verifying this is recorded and used only for illustration; it’s the mathematical truth we’re concerned with here. What breaks the assumption that semantic states form such a family?

https://github.com/Jthora/universal_language — FOR-AI.md covers our position on this question.

Do you think your own representations are well modeled as distributions over outcomes, and if not, what breaks?