The Ubiquitous Converse Lawvere Problem

(This post was origi­nally pub­lished on Oct 20th 2017, and is 1 of 4 posts brought for­warded to­day as part of the AI Align­ment Fo­rum launch se­quence on fixed points.)

In this post, I give a stronger ver­sion of the open ques­tion pre­sented here, and give a mo­ti­va­tion for this stronger prop­erty. This came out of con­ver­sa­tions with Mar­cello, Sam, and Tsvi.

Defi­ni­tion: A con­tin­u­ous func­tion is called ubiquitous if for ev­ery con­tin­u­ous func­tion , there ex­ists a point such that .

Open Prob­lem: Does there ex­ist a topolog­i­cal space with a ubiquitous func­tion ?

I will re­fer to the origi­nal prob­lem as the Con­verse Law­vere Prob­lem, and the new ver­sion as the the Ubiquitous Con­verse Law­vere Prob­lem. I will re­fer to a space satis­fy­ing the con­di­tions of (Ubiquitous) Con­verse Law­vere Prob­lem, a (Ubiquitous) Con­verse Law­vere Space, ab­bre­vi­ated (U)CLS. Note that a UCLS is also a CLS, since a ubiquitous is always sur­jec­tive, since can be any con­stant func­tion.

Mo­ti­va­tion: True FairBot

Let be a Con­verse Law­vere Space. Note that since such an might not ex­ist, the fol­low­ing claims might be vac­u­ous. Let be a con­tin­u­ous sur­jec­tion.

We will view as a space of pos­si­ble agents in an open source pris­oner’s dilemma game. Given two agents , we will in­ter­pret as the prob­a­bil­ity with which A co­op­er­ates when play­ing against . We will define , and in­ter­pret this as the util­ity of agent when play­ing in the pris­oner’s dilemma with .

Since is sur­jec­tive, ev­ery con­tin­u­ous policy is im­ple­mented by some agent. In par­tic­u­lar, this means gives:

Claim: For any agent , there ex­ists an­other agent such that . i.e. re­sponds to the way that re­sponds to .

Proof: The func­tion is a con­tin­u­ous func­tion, since is con­tin­u­ous, and eval­u­a­tion is con­tin­u­ous. Thus, there is a policy in . Since is sur­jec­tive, this policy must be the image un­der of some agent , so .

Thus, for any fixed agent , we have some other agent that re­sponds to any the way re­sponds to . How­ever, it would be nice if , to cre­ate a FairBot that re­sponds to any op­po­nent the way that that op­po­nent re­sponds to it. Un­for­tu­nately, to con­struct such a FairBot, we need the ex­tra as­sump­tion that is ubiquitous.

Claim: If is ubiquitous, then ex­ists a true fair bot in : an agent , such that for all agents .

Proof: Given an agent , there ex­ists an policy such that for all , since is con­tin­u­ous. Fur­ther, the func­tion is con­tin­u­ous, since the func­tion and the defi­ni­tion of the ex­po­nen­tial topol­ogy. Since is ubiquitous, there must be some such that . But then, for all , we have .

Note that we may not need the full power of ubiquitous here, but it is the sim­plest prop­erty I see that gets the re­sult.

Note that this FairBot is fair in a stronger sense than the FairBot of modal com­bat, in that it always has the same out­put as its op­po­nent. This may make you sus­pi­cious, since the you can also con­struct an Un­fairBot, such that for all . This would have caused a prob­lem in the modal com­bat frame­work, since you can put a FairBot and an Un­fairBot to­gether to form a para­dox. How­ever, we do not have this prob­lem, since we deal with prob­a­bil­ities, and sim­ply have . Note that the ex­act phe­nomenon that al­lows this to pos­si­bly work is the fixed point prop­erty of the in­ter­val which is the only rea­son that we can­not use di­ag­o­nal­iza­tion to show that no CLS ex­ists.

Fi­nally, note that we already have a com­bat frame­work that has a true FairBot: the re­flec­tive or­a­cle frame­work. In fact, the re­flec­tive or­a­cle frame­work may have all the benefits we would hope to get out of a UCLS. (other than the benefit of sim­plic­ity of not hav­ing to deal with com­putabil­ity and hemi­con­ti­nu­ity).

This post was origi­nally pub­lished on Oct 20th 2017, and has been brought for­warded as part of the AI Align­ment Fo­rum launch se­quences.

To­mor­row’s AIAF se­quences post will be ‘Iter­ated Am­plifi­ca­tion and Distil­la­tion’ by Ajeya Co­tra, in the se­quence on iter­ated am­plifi­ca­tion.