Some constructions for proof-based cooperation without Löb

5James Payor

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Something I'm now realizing, having written all these down: the core mechanism really does echo Löb's theorem! Gah, maybe these are more like Löb than I thought.

(My whole hope was to generalize to things that Löb's theorem doesn't! And maybe these ideas still do, but my story for why has broken, and I'm now confused.)

As something to ponder on, let me show you how we can prove Löb's theorem following the method of ideas #3 and #5:

- is assumed
- We consider the loop-cutter
- We verify that if activates then must be true:
- Then, can satisfy by finding the same proof.
- So activates, and is true.

In english:

- We have who is blocked on
- We introduce to the loop cutter , who will activate if activation provably leads to being true
- encounters the argument "if activates then is true, and this causes to activate"
- This satisfies 's requirement for some , so becomes true.

This post presents five closely-related ways to achieve proof-based cooperation without using Löb's theorem, and muses on legible cooperation in the real world.

(Edit: maybe they're closer to just-use-Löb's-theorem than I originally thought! See this comment. If these constructions somehow work better, I'm more confused than before about why.)

I'm writing this as a follow-up to Andrew Critch's recent post, to share more of my perspective on the subject.

We're going to dive straight into the weeds. (I'm planning to also write a more accessible explainer post soon.)

## The ideas

## Idea #1: try to prove A→B

I claim the following are sufficient for robust cooperation:

A tries to prove that A→B, and B tries to prove A. The reason this works is that B can prove that A→□A, i.e. A only cooperates in ways legible to B. (Proof sketch: A↔□X→□□X↔□A.)

The flaw in this approach is that we

neededto know that A won't cooperate for illegible reasons. Otherwise we can't verify that B will cooperate whenever A does.This indicates to me that "A→B" isn't the right "counterfactual". It shouldn't matter if A

couldcooperate for illegible reasons, if A isactually cooperatingfor a legible one.## Idea #2: try to prove □A→B

We can weaken the requirements with a simple change:

Note that this form is close to the lemma discussed in Critch's post.

In this case, the condition □A→B is trivial. And when the condition activates, it also ensures that □A is true, which discharges our assumption and ensures B is true.

I still have the sense that the condition for cooperation should talk about

itself activating, not A. Because we want it to activate whenthatis sufficient for cooperaion. But I do have to admit that □A→B works for mostly the right reasons, comes with a simple proof, and is the cleanest two-agent construction I know.## Idea #3: factor out the loop-cutting gadget

We can factor the part that is trying to cut the loop out from A, like so:

This gives the loop-cutting logic a name, X. Now X can refer to itself, and roughly says "I'll legibly activate if I can verify this will cause B to be true".

Like with idea #2, we just need A to reveal a mechanism by which it can be compelled to cooperate. The mechanism X is designed to be self-contained and easy for B to verify.

## Idea #4: everyone tries to prove □me→them

What about three people trying to cooperate? We can try applying lots of idea #2:

And, this works! Proof sketch:

The proof simplifies the group one person at a time, since each person is asking "what would happen if everyone else could tell I cooperate". This lets us prove the whole thing by induction. It's neat that it works, though it's not the easiest thing to see.

## Idea #5: the group agrees to a shared mechanism or leader

What if we factor out the choosing logic in a larger group? Here's one way to do it:

This is the cleanest idea I know for handling the group case. The group members agree on some trusted leader or process X. They set things up so X activates legibly, verifies things in a way trusted by everyone, and only activates when it verifies this will cause cooperation.

We've now localized the choice-making in one place. Everyone can watch our process X verify that □X→A∧B∧C, which causes X to activate, and everyone cooperates.

## Closing remarks on groups in the real world

Centralizing the choosing like in idea #5 make the logic simpler, but this sort of approach is prone to manipulation and other problems when the verification is not reliably done. This means I don't unambiguously prefer idea #5 to idea #4, in which everyone is doing their own legwork.

One solid takeaway from idea #5 though is that if you can actually trust other people/processes to do some of the verification, this can really simplify things. For example, if you believe that your friend Bob's verification is mostly trustworthy, then what Bob thinks can inform your read of the group.

Here's a sketch for something that might work in practice:

cantrust themselves.