The success of neural networks on problems like Go and protein folding, long considered intractable NP-hard problems, is profound. It suggests our formal understanding of computational hardness, which focuses on worst-case scenarios, may be an incomplete model for how to find useful, approximate solutions in practice.

P represents problems we can solve, while NP represents problems where a solution can be easily verified. If P=NP, any problem with a verifiable solution could be efficiently solved, implying we can know everything we want to know. It's a question about the ultimate limits of discovery.

The NEX vs. EX problem is equivalent to solving highly compressed, structured SAT instances. Because real-world SAT instances are also highly structured (not random), the possibility that NEX=EX implies that structure might be the key to efficiently solving certain exponential-time problems.

Zero-knowledge proofs are universal for any problem in NP (any problem with a verifiable proof). The method involves reducing the original problem to an NP-complete problem like graph 3-coloring. By proving knowledge of the graph coloring without revealing it, one indirectly proves the original theorem without revealing its substance.

The popular O(n^2) solution for the 'Threesome' problem can be beaten. The advanced technique involves sorting, grouping elements into small blocks, and using pre-processed data structures to accelerate the search, treating groups as single units.

Professor Williams assigns only 80% confidence to P != NP, lower than his peers. His rationale is that our intuition about computational limits is frequently proven wrong by surprising new algorithms. The vast, unexplored space of algorithms makes a definitive conclusion more uncertain than widely believed.

Humanity's intellectual pursuits, from science to engineering, inherently focus on problems where a potential solution can be verified upon discovery. We wouldn't begin searching for something if we couldn't recognize it once found, which is the definition of an NP problem.

While any NP-complete problem can be reduced to another, SAT solvers are the practical choice because of the immense effort poured into developing heuristics that efficiently handle the structured instances arising in real-world applications. Their advantage lies in engineering, not pure theory.

While problems like protein folding are NP-hard in theory, the instances found in nature have structural properties that allow for efficient solutions. Real-world cases of NP-hard problems aren't the adversarial, worst-case scenarios used in complexity proofs, explaining the gap between theory and practice.