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Mildly Infuriating
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Would you trust the output of an A.I. swarm?
The idea is to pick some task that is very difficult to calculate, but easy to verify. E.g., confirming a crossword is correct is much faster than filling one in.
Protein folding is one such example, and it's useful to boot. The results can be used in medical research, for example.
Donating compute via Folding@Home was quite popular for a while for this exact reason.
Folding@Home being built into the PS3 might have been one of the last tech-utopian hurrahs I can think of experiencing. As opposed to everything tech now feeling too dystopian for one of any number of reasons
The problem being the challenge needs to be unique and issued by the server on demand, otherwise a computer can have a bunch of pre-processed challenge answers ready to fire out a huge burst of requests. (Or re-use the solutions on other websites)
I don't know how protein folding works, but if it's easy to check it's correct, then doesn't that violate that mathematical principle where verifying an answer is as difficult as computing the answer in the first place? I forget the name of it.
I guess what I'm trying to say is, volunteers at home are already self-selecting to be mostly honest and donating correct answers for a good cause. Their output trustworthiness is moderate-high. Spammers, bots, A.I. swarms, ddos networks, random users, etc… not so much. You can't really trust their output at all, you have to verify it very carefully and treat it as malicious or just incorrect junk by default.
You're thinking, as someone else said, of P = NP. This is technically an unsolved problem. We don't know if P = NP is true, there is no proof that guarantees it. However, it's looking pretty unlikely.
This problem would have huge consequences for a wide range of fields, and it would break most cryptography and encryption algorithms we have (assuming it is proven true).
Anyway, this is what Anubis is already doing. That's why OP complained about being stuck for 3 minutes with the CPU at full blast. Anubis gave the PC some problem that was difficult to compute. The problem itself was not important, it's just to keep the computer busy for a bit. (The idea is that this significantly slows down spam bots and other automated access, I believe).
Of course Anubis has to verify the answer is correct, and that the client didn't just send garbage. So the problem Anubis already uses is one that is easier to verify than to compute.
Btw, it is possible to verify solutions where checking is as expensive as solving. You provide each client with at least 2 problems. One where you already know the answer, and one where you don't. You have the client solve both. If the known one is good, you let them through, and add the unknown one to the known set. You can even verify it by sending it to two other clients, see if they all get the same answer. But I believe that this solution wouldn't be required, because protein folding 'violates' the P=NP problem.
You're thinking of the P vs NP problem, but you got the details wrong. There are known cases where the answer is easy to verify.
What the distributed computing stuff does is check many many potential answers. You're only interested in the right answers, and they're easy to verify. If there's a concern about junk answers, the answers could probably be compared across multiple clients to verify.