   3.  An Untrollable Mathematician   post by Abram Demski 455 days ago  Alex Appel, Sam Eisenstat, Vadim Kosoy, Jack Gallagher, Jessica Taylor, Paul Christiano, Scott Garrabrant and Vladimir Slepnev like this  1 comment  
 Followup to All Mathematicians are Trollable.
It is relatively easy to see that no computable Bayesian prior on logic can converge to a single coherent probability distribution as we update it on logical statements. Furthermore, the nonconvergence behavior is about as bad as could be: someone selecting the ordering of provable statements to update on can drive the Bayesian’s beliefs arbitrarily up or down, arbitrarily many times, despite only saying true things. I called this wild nonconvergence behavior “trollability”. Previously, I showed that if the Bayesian updates on the provabilily of a sentence rather than updating on the sentence itself, it is still trollable. I left open the question of whether some other side information could save us. Sam Eisenstat has closed this question, providing a simple logical prior and a way of doing a Bayesian update on it which (1) cannot be trolled, and (2) converges to a coherent distribution.
 
  4.  Reflective oracles as a solution to the converse Lawvere problem   post by Sam Eisenstat 523 days ago  Alex Mennen, Alex Appel, Vadim Kosoy, Abram Demski, Jessica Taylor, Scott Garrabrant and Vladimir Slepnev like this  discuss  
 1 Introduction
Before the work of Turing, one could justifiably be skeptical of the idea of a universal computable function. After all, there is no computable function \(f\colon\mathbb{N}\times\mathbb{N}\to\mathbb{N}\) such that for all computable \(g\colon\mathbb{N}\to\mathbb{N}\) there is some index \(i_{g}\) such that \(f\left(i_{g},n\right)=g\left(n\right)\) for all \(n\). If there were, we could pick \(g\left(n\right)=f\left(n,n\right)+1\), and then \[g\left(i_{g}\right)=f\left(i_{g},i_{g}\right)+1=g\left(i_{g}\right)+1,\] a contradiction. Of course, universal Turing machines don’t run into this obstacle; as Gödel put it, “By a kind of miracle it is not necessary to distinguish orders, and the diagonal procedure does not lead outside the defined notion.” [1]
The miracle of Turing machines is that there is a partial computable function \(f\colon\mathbb{N}\times\mathbb{N}\to\mathbb{N}\cup\left\{ \bot\right\}\) such that for all partial computable \(g\colon\mathbb{N}\to\mathbb{N}\cup\left\{ \bot\right\}\) there is an index \(i\) such that \(f\left(i,n\right)=g\left(n\right)\) for all \(n\). Here, we look at a different “miracle”, that of reflective oracles [2,3]. As we will see in Theorem 1, given a reflective oracle \(O\), there is a (stochastic) \(O\)computable function \(f\colon\mathbb{N}\times\mathbb{N}\to\left\{ 0,1\right\}\) such that for any (stochastic) \(O\)computable function \(g\colon\mathbb{N}\to\left\{ 0,1\right\}\), there is some index \(i\) such that \(f\left(i,n\right)\) and \(g\left(n\right)\) have the same distribution for all \(n\). This existence theorem seems to skirt even closer to the contradiction mentioned above.
We use this idea to answer “in spirit” the converse Lawvere problem posed in [4]. These methods also generalize to prove a similar analogue of the ubiquitous converse Lawvere problem from [5]. The original questions, stated in terms of topology, remain open, but I find that the model proposed here, using computability, is equally satisfying from the point of view of studying reflective agents. Those references can be consulted for more motivation on these problems from the perspective of reflective agency.
Section 3 proves the main lemma, and proves the converse Lawvere theorem for reflective oracles. In section 4, we use that to give a (circular) proof of Brouwer’s fixed point theorem, as mentioned in [4]. In section 5, we prove the ubiquitous converse Lawvere theorem for reflective oracles.
 
   6.  Hyperreal Brouwer   post by Scott Garrabrant 564 days ago  Vadim Kosoy and Stuart Armstrong like this  2 comments  
 This post explains how to view Kakutani’s fixed point theorem as a special case of Brouwer’s fixed point theorem with hyperreal numbers. This post is just math intuitions, but I found them useful in thinking about Kakutani’s fixed point theorem and many things in agent foundations. This came out of conversations with Sam Eisenstat.  
     10.  Smoking Lesion Steelman   post by Abram Demski 660 days ago  Tom Everitt, Sam Eisenstat, Vadim Kosoy, Paul Christiano and Scott Garrabrant like this  10 comments  
 It seems plausible to me that any example I’ve seen so far which seems to require causal/counterfactual reasoning is more properly solved by taking the right updateless perspective, and taking the action or policy which achieves maximum expected utility from that perspective. If this were the right view, then the aim would be to construct something like updateless EDT.
I give a variant of the smoking lesion problem which overcomes an objection to the classic smoking lesion, and which is solved correctly by CDT, but which is not solved by updateless EDT.
 
  11.  A cheating approach to the tiling agents problem   post by Vladimir Slepnev 662 days ago  Alex Mennen, Vadim Kosoy and Abram Demski like this  2 comments  
 (This post resulted from a conversation with Wei Dai.)
Formalizing the tiling agents problem is very delicate. In this post I’ll show a toy problem and a solution to it, which arguably meets all the desiderata stated before, but only by cheating in a new and unusual way.
Here’s a summary of the toy problem: we ask an agent to solve a difficult math question and also design a successor agent. Then the successor must solve another math question and design its own successor, and so on. The questions get harder each time, so they can’t all be solved in advance, and each of them requires believing in Peano arithmetic (PA). This goes on for a fixed number of rounds, and the final reward is the number of correct answers.
Moreover, we will demand that the agent must handle both subtasks (solving the math question and designing the successor) using the same logic. Finally, we will demand that the agent be able to reproduce itself on each round, not just design a custommade successor that solves the math question with PA and reproduces itself by quining.
 
          20.  Entangled Equilibria and the Twin Prisoners' Dilemma   post by Scott Garrabrant 801 days ago  Vadim Kosoy and Patrick LaVictoire like this  2 comments  
 In this post, I present a generalization of Nash equilibria to nonCDT agents. I will use this formulation to model mutual cooperation in a twin prisoners’ dilemma, caused by the belief that the other player is similar to you, and not by mutual prediction. (This post came mostly out of a conversation with Sam Eisenstat, as well as contributions from Tsvi BensonTilsen and Jessica Taylor)  
      
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