by Stuart Armstrong 256 days ago | Scott Garrabrant likes this | link | parent A small note: it’s not hard to construct spaces that are a bit too big, or a bit too small (raising the possibility that a true $$X$$ lies between them). For instance, if $$I$$ is the unit interval, then we can map $$I$$ onto the countable-dimensions hypercube $$I^\omega$$ ( https://en.wikipedia.org/wiki/Space-filling_curve#The_Hahn.E2.80.93Mazurkiewicz_theorem ). Then if we pick an ordering of the dimensions of the hypercube and an ordering of $$\mathbb{Q}\cap I$$, we can see any element of $$I^\omega$$ - hence any element of $$I$$ - as a function from $$\mathbb{Q}\cap I$$ to $$I$$. Let $$C(I)$$ be the space of continuous functions $$I \to I$$. Then any element of $$C(I)$$ defines a unique function $$\mathbb{Q}\cap I \to I$$ (the converse is not true - most functions $$\mathbb{Q}\cap I \to I$$ do not correspond to continuous functions $$I \to I$$). Pulling $$C(I)$$ back to $$I$$ via $$I^\omega$$ we define the set $$Y \subset I$$. Thus $$Y$$ maps surjectively onto $$C(I)$$. However, though $$C(I)$$ maps into $$C(Y)$$ by restriction (any function from $$I$$ is a function from $$Y$$), this map is not onto (for example, there are more continuous functions from $$I - \{1/2\}$$ than there are from $$I$$, because of the potential discontinuity at $$1/2$$). Now, there are elements of $$I-Y$$ that map (via $$I^\omega$$) to functions in $$C(Y)$$ that are not in $$C(I)$$. So there’s a hope that there may exist an $$X$$ with $$Y \subset X \subset I$$, $$C(I) \subset C(X) \subset C(Y)$$, and $$X$$ mapping onto $$C(X)$$. Basically, as $$X$$ `gets bigger’, its image in $$C(Y)$$ grows, while $$C(X)$$ itself shrinks, and hopefully they’ll meet.

NEW DISCUSSION POSTS

This is exactly the sort of
 by Stuart Armstrong on Being legible to other agents by committing to usi... | 0 likes

When considering an embedder
 by Jack Gallagher on Where does ADT Go Wrong? | 0 likes

The differences between this
 by Abram Demski on Policy Selection Solves Most Problems | 0 likes

Looking "at the very
 by Abram Demski on Policy Selection Solves Most Problems | 0 likes

 by Paul Christiano on Policy Selection Solves Most Problems | 1 like

>policy selection converges
 by Stuart Armstrong on Policy Selection Solves Most Problems | 0 likes

Indeed there is some kind of
 by Vadim Kosoy on Catastrophe Mitigation Using DRL | 0 likes

Very nice. I wonder whether
 by Vadim Kosoy on Hyperreal Brouwer | 0 likes

Freezing the reward seems
 by Vadim Kosoy on Resolving human inconsistency in a simple model | 0 likes

Unfortunately, it's not just
 by Vadim Kosoy on Catastrophe Mitigation Using DRL | 0 likes

>We can solve the problem in
 by Wei Dai on The Happy Dance Problem | 1 like

Maybe it's just my browser,
 by Gordon Worley III on Catastrophe Mitigation Using DRL | 2 likes

At present, I think the main
 by Abram Demski on Looking for Recommendations RE UDT vs. bounded com... | 0 likes

In the first round I'm
 by Paul Christiano on Funding opportunity for AI alignment research | 0 likes

Fine with it being shared
 by Paul Christiano on Funding opportunity for AI alignment research | 0 likes