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1.Reflective oracles as a solution to the converse Lawvere problem
post by Sam Eisenstat 388 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.

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2.A cheating approach to the tiling agents problem
post by Vladimir Slepnev 527 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 custom-made successor that solves the math question with PA and reproduces itself by quining.

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3.Some Criticisms of the Logical Induction paper
link by Tarn Somervell Fletcher 528 days ago | Alex Mennen, Sam Eisenstat and Scott Garrabrant like this | 10 comments
4.Loebian cooperation in the tiling agents problem
post by Vladimir Slepnev 533 days ago | Alex Mennen, Vadim Kosoy, Abram Demski, Patrick LaVictoire and Stuart Armstrong like this | 4 comments

The tiling agents problem is about formalizing how AIs can create successor AIs that are at least as smart. Here’s a toy model I came up with, which is similar to Benya’s old model but simpler. A computer program X is asked one of two questions:

  • Would you like some chocolate?

  • Here’s the source code of another program Y. Do you accept it as your successor?

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5.Two Major Obstacles for Logical Inductor Decision Theory
post by Scott Garrabrant 601 days ago | Alex Mennen, Sam Eisenstat, Abram Demski, Jessica Taylor, Patrick LaVictoire and Tsvi Benson-Tilsen like this | 3 comments

In this post, I describe two major obstacles for logical inductor decision theory: untaken actions are not observable and no updatelessness for computations. I will concretely describe both of these problems in a logical inductor framework, but I believe that both issues are general enough to transcend that framework.

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