Intelligent Agent Foundations Forumsign up / log in
by Jonathan Lee 892 days ago | Jessica Taylor and Patrick LaVictoire like this | link | parent

It looks like Theorem 1 can be improved slightly, by dropping the “only if” condition on \(p_{CD} > 0\). We can then code up something like Kolmogorov complexity by adding a probability \(\frac{1}{2}\) transition from every site to our chosen UTM.

If you only want the weaker statement that there is no stationary distribution, it looks like there’s a cheaper argument: Since \(\Phi\) is aperiodic and irreducible the hypothetical stationary distribution \(\pi\) is unique. \(\Phi\) is closed under the action of \(\Delta\), and (2) implies that for any \(g \in \Delta\), the map \(\Gamma_g\) is an automorphism of the Markov chain. If the (infinite) transition matrix is \(T\), then \(\Gamma_g\) can be considered as a permutation matrix with (abusing notation) \(\Gamma_g^{-1}T\Gamma_g = T\). Then \(T\Gamma_g\pi = \Gamma_g\pi\) and so \(\Gamma_g\pi = \pi\) by uniqueness. So \(\pi\) is constant on orbits of \(\Gamma_{\Delta}\), which are all countably infinite. Hence \(\pi\) is everywhere \(0\), a contradiction.

The above still holds if (2) is restricted to only hold for a group \(G < \Delta\) such that every orbit under \(\Gamma_G\) is infinite.

I think the above argument shows why (2) is too strong; we shouldn’t expect the world to look the same if you pick a “wrong” (ie. complicated) UTM to start off with. Weakening (2) might mean saying something like asserting only \(p_{CD} = \sum \mu(\Gamma) p_{\Gamma(C)\Gamma(D)}\). To do this, we might define the measures \(\mu\) and \(p\) together (ie. finding a fixed point of a map from pairs \((p, \mu)\) to \((p', \mu')\)). In such a model, \(\mu\) constraints the transition probabilities, \(p'\) is stationary; it’s not clear how one might formalise a derivation of \(\mu'\) from \(p'\) but it seems plausible that there is a canonical way to do it.



NEW LINKS

NEW POSTS

NEW DISCUSSION POSTS

RECENT COMMENTS

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

I think the point I was
by Abram Demski on Predictable Exploration | 0 likes

(also x-posted from
by Sören Mindermann on The Three Levels of Goodhart's Curse | 0 likes

(x-posted from Arbital ==>
by Sören Mindermann on The Three Levels of Goodhart's Curse | 0 likes

>If the other players can see
by Stuart Armstrong on Predictable Exploration | 0 likes

Thinking about this more, I
by Abram Demski on Predictable Exploration | 0 likes

> So I wound up with
by Abram Demski on Predictable Exploration | 0 likes

Hm, I got the same result
by Alex Appel on Predictable Exploration | 1 like

Paul - how widely do you want
by David Krueger on Funding opportunity for AI alignment research | 0 likes

I agree, my intuition is that
by Abram Demski on Smoking Lesion Steelman III: Revenge of the Tickle... | 0 likes

RSS

Privacy & Terms