Fixed Point Exercises

Sometimes people ask me what math they should study in order to get into agent foundations. My first answer is that I have found the introductory class in every subfield to be helpful, but I have found the later classes to be much less helpful. My second answer is to learn enough math to understand all fixed point theorems. These two answers are actually very similar. Fixed point theorems span all across mathematics, and are central to (my way of) thinking about agent foundations.

This post is the start of a sequence on fixed point theorems. It will be followed by several posts of exercises that use and prove such theorems. While these exercises aren’t directly connected to AI safety, I think they’re quite useful for preparing to think about agent foundations research. Afterwards, I will discuss the core ideas in the theorems and where they’ve shown up in alignment research.

The math involved is not much deeper than a first course in the various subjects (logic, set theory, topology, computability theory, etc). If you don’t know the terms, a bit of googling, wikipedia and math.stackexchange should easily get you most of the way. Note that the posts can be tackled in any order.

Here are some ways you can use these exercises:

  • You can host a local MIRIx group, and go through the exercises together. This might be useful to give a local group an affordance to work on math rather than only reading papers.

  • You can work on them by yourself for a while, and post questions when you get stuck. You can also post your solutions to help others, let others see an alternate way of doing a problem, or help you realize that there is a problem with your solution.

  • You can skip to the discussion (which has some spoilers), learn a bunch of theorems from Wikipedia, and use this as a starting point for trying to understand some MIRI papers.

  • You can use answering these questions as a goalpost for learning a bunch of introductory math from a large collection of different subfields.

  • You can show off by pointing out that some of the questions are wrong, and then I will probably fix them and thank you.

The first set of exercises is here.

Thanks to Sam Eisenstat for helping develop these exercises, Ben Pace for helping edit the sequence, and many AISFP participants for testing them and noticing errors.


Read the following.

Please use the (new) spoilers feature—the symbol ‘>’ followed by ‘!’ followed by space—in your comments to hide all solutions, partial solutions, and other discussions of the math. The comments will be moderated strictly to cover up spoilers!

I recommend putting all the object level points in spoilers and leaving metadata outside of the spoilers, like so:

Here’s my solution /​ partial solution /​ confusion for question #5:

And put your idea in here! (reminder: LaTex is cmd-4 /​ ctrl-4)