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.

The NP-complete Subset Sum problem (typically 2^n time) can be solved in sqrt(2^n) time. This is achieved by reducing it to the Two-Sum problem (solvable in n log n time), demonstrating a powerful technique in fine-grained complexity that connects seemingly disparate problem classes.

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.

For millions of vectors, exact search (like a FAISS flat index) is too slow. Production systems use Approximate Nearest Neighbor (ANN) algorithms which trade a small amount of accuracy for orders-of-magnitude faster search performance, making large-scale applications feasible.

It's possible to solve problems like finding the majority element in a bit string using constant memory, regardless of the string's length. This is achieved by encoding computations as sequences of operations in a non-commutative group, defying the intuition that counting requires logarithmic space to store a counter.

Dr. Levin's lab found that basic, deterministic sorting algorithms perform additional, unprogrammed computations, or "side quests" like clustering, while executing their primary task. This concept of "polycomputing" suggests a single physical process can have multiple computational interpretations, challenging how we define and measure computation.

A core legacy of AlphaGo is turning complex search problems into 'games' for AI agents. AlphaTensor reframed the challenge of finding the fastest matrix multiplication algorithm as a game, allowing it to discover a more efficient method than any human had found in over 50 years, proving the approach's power for scientific discovery.

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.

A 1975 result showed time-T algorithms could run in T/log(T) space. Ryan Williams improved this to sqrt(T) by changing a core assumption. Instead of only writing to erased memory, his method uses XOR operations to destructively overwrite memory, enabling massive space savings by cleverly managing information.