Donald Knuth on P=NP, Algorithm Existence, and the Robertson-Seymour Theorem

The discussion profoundly explores the P=NP problem, with Donald Knuth articulating his intuition that P may indeed equal NP, but that such a proof would not directly lead to practical algorithms for solving difficult problems. This perspective is rooted in the fundamental distinction between the mathematical existence of an algorithm and its discoverability or comprehensibility by humans. Knuth posits that the sheer number of possible algorithms is so astronomically vast that many polynomial-time algorithms could exist for NP-complete problems, yet remain entirely beyond human capacity to conceive, understand, or practically implement, making their discovery an insurmountable challenge.\n\nA central illustration of this concept is the game of Hex, where a winning strategy for the first or second player is mathematically guaranteed to exist due to the game's finite nature and lack of draws, yet no general algorithm for finding this strategy is known. This non-constructive existence proof is further elaborated through the powerful Robertson-Seymour Theorem in graph theory. This theorem states that for any class of graphs closed under taking minors, a polynomial-time algorithm exists to determine if a given graph belongs to that class. This algorithm relies on identifying a finite set of "forbidden minors" or "obstructions." However, the theorem does not provide a method to find these obstructions for an arbitrary class, meaning the algorithm's existence is proven, but its specific form remains unknown and practically inaccessible.\n\nKnuth's intuition for P=NP's potential truth, despite the lack of discovered algorithms, is bolstered by the idea of "random coincidences" within the immense space of possible algorithms. He suggests that some complex, seemingly arbitrary combination of operations might, by chance, constitute a polynomial-time solution to a difficult problem. This is compared to the statistical certainty of coincidences in large populations, such as two people having the same number of hairs on their head. He also symmetrically counters the argument that P≠NP because smart people haven't found a solution, by noting that smart people have also failed to prove P≠NP, highlighting the limitations of human effort in such an expansive problem domain.\n\nThe broader implications of this discussion touch upon the philosophical divide between mathematical existence and constructive proof, the inherent limits of human cognition when confronted with combinatorial explosion, and the very nature of scientific and mathematical discovery. The analogy to the search for extraterrestrial intelligence reinforces the idea that the absence of discovered evidence does not equate to evidence of absence. Knuth's viewpoint challenges the common assumption that all existing mathematical truths are eventually discoverable by humans, suggesting that some fundamental algorithms might forever remain theoretically proven but practically elusive, residing in a space beyond human reach.

if somebody does prove that P equals NP it will not directly lead to an actual algorithm to solve difficult problems

many more algorithms exist that anybody can end understand or ever make your discovery yeah because they're just way beyond human comprehension

we know that algorithm exists but we don't know we don't the foggiest idea what the algorithms are

there are cases where you can prove the existence of the solution but we but nobody knows any way how to find it

Robertson to Seymour proved that any such family of graph there is a finite number of minimum graphs that are obstructions

all we know is that the number of minors is finite we don't know what we we might only know one or two of those minors but we don't know that if we got if we got 20 of them we don't know there might be 20 125

here we have a polynomial time algorithm that we don't know

the space of possibility is so huge you have to rule them all out and so that's the reason for my intuition

P must be equal to NP because there's so many smart people out here been trying to prove it unequal NP and they've all failed

this kind of reminds me of the discussion about the search for aliens that we've been trying to look for them and we haven't found them yet therefore they don't exist