Edit: This article has major flaws. See my comment below.

This idea was informed by discussions with Abram Demski, Scott Garrabrant, and the MIRIchi discussion group.

# Summary

For a logical inductor , define logical counterfactuals by

for a suitable and a random variable independent of with respect to . Using this definition, one can construct agents that perform well in ASP-like problems.

# Motivation

Recall the Agent Simulates Predictor problem:

Naively, we want to solve this by argmaxing:

Hopefully, , , and . Also, two-boxing should be less attractive than one-boxing:

However, if we make this well-defined with -exploration, we'll get

and then the agent will two-box, contradiction. Instead we'd like to use predictable exploration and set

for small enough that the right-hand side is sensible. Let's see how.

# Predictable exploration

Choose so that . Our agent decides whether to explore at stage , and uses its beliefs at stage as a substitute for counterfactuals:

Here are small positive numbers. It's easy to see that, under reasonable assumptions, this agent 1-boxes on Agent Simulates Predictor. But it can't use the full strength of in its counterfactual reasoning, and this is a problem.

# Differential privacy

To illustrate the problem, add a term to the utility function that sometimes rewards two-boxing:

The agent should two-box if and only if . Assuming that's the case, and knows this, we have:

So if , two-boxing is the more attractive option, which is a contradiction. (I'm rounding to zero for simplicity.)

The problem is that the counterfactual has to rely on 's imperfect knowledge of . We want to combine 's ignorance of with 's knowledge of .

If is independent of conditioned on with respect to , then we can do this:

Then replace with :

This is more accurate than , and unbiased.

If is not independent of conditional on , we can introduce an auxilliary variable and construct a version of that is independent. This construction is a solution to the following differential privacy problem: Make a random variable that is a function of and independent randomness, maximizing the mutual conditional information , subject to the constraint that is independent of . Using the identity

we see that the maximum is attained when , which means that is a function of and .

Now here's the construction of :

Let be the finite set of possible values of , and let be the finite set of possible values of . We'll iteratively construct a set and define a random variable taking values in . To start with, let .

Now choose

and for each , choose some such that . Then make a random binary variable such that

Then let be the event defined by

and add to . After repeating this process times, we are done.

We can do this with a logical inductor as well. In general, to get a sentence such that , take .

Now given random variables and , and some informative sentences , let be the random variable encoding the values of . The above construction works approximately and conditional on to give us a random variable that is approximately independent of conditional on with respect to . Now we define

whenever .

This succeeds on the problem at the beginning of this section: Assume , and assume that knows this. Then:

which does not lead to contradiction. In fact, there are agents like this that do at least as well as any constant agent:

## Theorem

Let be a utility function defined with metasyntactic variables , , and . It must be computable in polynomial time as a function of , , and , where can be any polytime functions that doesn't grow too slowly and such that . Then there exists a logical inductor such that for every , there exists , , and a pseudorandom variable such that the agent defined below performs at least as well on as any constant agent, up to a margin of error that approaches as :

## Proof sketch

Choose smaller than the strength parameter of the weakest predictor in . If is the best constant policy for , assume . Since can compute , our agent's factual estimate is accurate, and the counterfactual estimate for is an accurate estimate of the utility assigned to the constant policy , as long as we make rich enough. So the agent will choose . Thus we have an implication of the form "if believes , then is true", and so we can create a logical inductor that always believes that for every by adding a trader with a large budget that bids up the price of .

# Isn't this just UDTv2?

This is much less general than UDTv2. If you like, you can think of this as an agent that at time chooses a program to run, and then runs that program at time , except the program always happens to be "argmax over this kind of counterfactual".

Also, it doesn't do policy selection.

# Next steps

Instead of handing the agent a pseudorandom variable that captures everything important, I'd like to have traders inside a logical inductor figure out what should be on their own.

Also, I'd rather not have to hand the agent an optimal value of .

Also, I hope that these counterfactuals can be used to do policy selection and win at counterfactual mugging.

This doesn't quite work. The theorem and examples only work if you maximize the unconditional mutual information, H(X;Y), not H(X;Y|A). And the choice of X is doing a lot of work — it's not enough to make it "sufficiently rich".