# 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.

## Meta

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)

• How long would it take some­body to go from ba­sic alge­bra/​stats to be­ing able to un­der­stand a tech­ni­cal MIRI pa­per?

Sup­pose the per­son:
- is a de­cent pro­gram­mer
- is ex­pe­rienced with effec­tive learn­ing and pro­duc­tivity meth­ods
- can ded­i­cate two hours of fo­cused study ev­ery­day—
has con­sumed a lot of non-tech­ni­cal re­sources, e.g. Su­perIn­tel­li­gence, 80K Hours in­ter­views, FHI pod­cast, AI-Zom­bies, GEB, etc.

• Sounds like me at the be­gin­ning of this year; I’m now able to make my way through log­i­cal in­duc­tion. I’d be happy to help, by the way—feel free to mes­sage me.

• Note: In mark­down, the spoiler syn­tax is as fol­lows:

``::: spoiler This is some spoiler text :::``

Which should ren­der like this:

This is some spoiler text

• Note to GreaterWrong users: GW now has full sup­port for spoiler blocks (they will ren­der cor­rectly—mouse over one, or se­lect its text, to re­veal—and there’s a new but­ton in the ed­i­tor that will in­sert the cor­rect spoiler syn­tax for you.)

• The cur­rent im­ple­men­ta­tion of spoiler tags is pretty ex­per­i­men­tal, and we will prob­a­bly change how it ren­ders, but the syn­tax should con­tinue work­ing for the in­definite fu­ture.

• At the time of writ­ing, for the two spoilers in the main post, hov­er­ing over ei­ther will re­veal both. Is that in­ten­tional? It does not seem de­sir­able.

• Nope, not in­ten­tional. Will see whether I can get around to fix­ing that to­day.

• Do you have any recom­mended read­ing for learn­ing enough math to do these ex­er­cises? I’m sort of us­ing these as a text­book-list-by-proxy (e.g. google “In­ter­me­di­ate value the­o­rem”, check which area of math it’s from, oh hey it’s Anal­y­sis, get an in­tro­duc­tory text­book in Anal­y­sis, re­peat), though I also have lit­tle knowl­edge of the field and don’t want to wan­der down sub­op­ti­mal paths.