Comment by akoboldfrying
2 hours ago
> It feels like that particular research direction is now a dead end
This is the most negative possible take on the most positive possible kind of result in CS.
To see just how unduly negative it is, imagine how different your response would have been had the exact same result been reported in a paper by exclusively human authors. Would you have likewise accused them of creating a research "dead end"?
EDIT: Changed "by, e.g., Ryan Williams" to "by exclusively human authors". Without having checked the authors, who include Ryan's wife and frequent collaborator Virginia, I had reached for a big name in the field purely as an example of a human who might well have made this breakthrough on their own.
There is a difference between one-off results and processes that can generate new results at an industrial scale.
Conditional lower bounds are a way of building understanding of the essential difficulty of specific computational problems. But if AI can now routinely generate marginal improvements, conditional bounds based on unproven assumptions become a waste of effort.
This is mostly due to how mathematics works. Ideally, we would like to prove something like "if problem A is essentially this difficult, problem B is essentially that difficult". But what we actually prove is more like "if (specific formulation of the difficulty of problem A), then (specific formulation of the difficulty of problem B)".
But those specific formulations become fixed targets for the AI to attack. If it manages to break the specific assumption, for example by creating an O(n^1.9998) time algorithm that is for all intents and purposes worse than a naive O(n^2) time algorithm, the conditional result becomes void. We could try to salvage the result with a different formulation, but that again becomes a fixed target.
This is essentially Goodhart's Law. We measure improvement with highly precise metrics, while we are actually interested in qualitative understanding.
EDIT: If theorems and proofs become cheap, marginal improvements are no longer interesting. Qualitatively better algorithms or unconditional lower bounds would be actual contributions. As would be a specific formulation of a conditional result that is robust against technical improvements made by AI targeting that specific formulation.
it just means that the assumptions we are making in the first place is wrong? nothing here is a dead-end because essentially we are at the same place before. people might even go ahead and say now if it's n^1.5 what happens, what results can be true.
i feel like you are arguing there is -- even in the narrow utility of PROOFS -- there is a goodness in being an ostrich with its head in the sand. If that's true, people can still be that ostrich and pretend the bound is now n^1.9 or something.
I edited my comment just as you answered, but I'll also say it here in a different way.
The actual conditional result was "if problem A is essentially this difficult, problem B is essentially that difficult". A specific formulation of it was proven, but it depended on a specific assumption that was just shown false. But the general result is still probably valid, because the general assumption holds. But there may be no point in going through the effort of proving another specific formulation, when the automatic theorem factory could just come up with another technical improvement targeting that formulation.
I started in theoretical computer science, but I quickly drifted to more applied areas, because I was annoyed with how often theoreticians would confuse the map for the territory. But if AI can now do that much more efficiently than any human, perhaps theoreticians will have to rethink how much they should focus on specific provable statements.
Yeah, I can't get behind this. By showing one problem had an improved lower bound, they showed that another five algorithms could also be improved, and, yes, they refuted the 3SUM hypothesis and made a lot of conjectures about the hardness of some problems less certain, but now ... now we have to go figure those out more precisely. Which is great. It's progress! And all of the work put into those reductions was what made this one result topple five algorithms, so it's not like it's been wasted work.