In essentially all cases, we have no real reason to believe or disbelieve. We choose these bounds because we can't do better. It's often the case that we later prove them wrong with some extremely strange result. Here, we have weird bounds for 3-SAT, and a few weeks ago we had one for integer multiplication.
But this has also happened repeatedly throughout history. In the 50s Kolmogorov conjectured that n-bit integer multiplication was \Omega(n^2). Karatsuba produced O(n^{1.58}) during Kolmogorov's seminar on the topic. Strassen discovered matrix multiplication is O(n^{2.81}) while trying to prove it is O(n^3). And on and on and on.
Maybe P and NP are well-formulated and they do perfectly capture our intuition of "efficient" and "efficiently verifiable" algorithms. Or maybe they don't. We are probably going to keep finding extremely weird, cheesed-out bounds for famous problems, though. You can choose what you want to believe, but I think it would be more remarkable if the reality of P and NP turned out to be as clean as the bounds indicate.
[–] calf 1 day ago
It is true that CS had to make theoretical concessions. Just as Newton did. We don't criticize Newtonian mechanics (or, I don't know, Galileo?) for not being realistic about friction or idealized balls rolling down ramps. Their anti-empiricism was a starting point for physics.
To wit, the mathematical rationale for P-NP was precisely that that under polytime reduction this is still closed in P. That is a very reasonable starting point for a theory. This is also the reason why it is tripping up a large swath of the comment thread with people asking about subexponentials which do not preserve that closure
The opposite narrative is that in science, these idealizations yield profound intuitions and insights.
Furthermore, how far are you willing to go? Do you think Turing machines are invalid, that Turing-equivalent computation is pointless and unrealistic? So we are never allowed use that theory to convey the essential idea of in principle all computers being fundamentally functionally the same and that anything claimed to be more powerful than that had better have some incredible evidence, lest it be appeal to magic. Computability and complexity have a shared foundation so how much of that foundation are you going to reexamine?
Whereas if one wishes to be like a covert finitist/intuitionist, they ought to be open about that instead of accuse asymptotic complexity (or analgously, Turing computability as above) of trying to "cheese" complexity theory as a research programme. Meanwhile, it is true that P vs NP could have large exponents or constants in P, e.g. O(c*n^100...), literally this is the first thing taught in any complexity course or textbook. Students and practitioners remember this admonition well, maybe they just don't talk about it publicly as much.
One can go even further back. Cantor was marginalized for his ideas about infinities.
The difference with people like Lipton, and Knuth, is that the critique is coming from inside the house. In that they know the theory and the importance of it, but also hope/aspire for some new way to address the open question. They are keenly aware that P / NP is a problematic question, but not for lack of trying or stupidity on the part of other scientists. What is more likely is that the 21st/22nd century math to pose the better question does not exist yet (or may never be accessible to humans, another question well on the minds of many today).
Here's something to ponder. It is often observed that despite being NP-complete, many actual human instances are in fact tractable. E.g. routine airline scheduling depending on SAT solvers. But is this evidence that NP-completeness is a wrong theory or evidence that human-sized problems are a special case that becomes tractable (anthropocentrism), or some other more complicated reason?
[–] antics 2 days ago