NP: The Optimist's Paradise
For every YES instance, NP provides:
- A witness (like a suspect's alibi)
- That's short (polynomial in size)
- And quick to verify (polynomial time)
\[
x \in L \Rightarrow
\]
\[
\exists w \text{ with } |w| \leq p(|x|)
\]
\[
\text{and } V(x,w) = 1
\]
Translation: There exists a proof that's easy to check
Real-world case: Sudoku solutions are easy to verify but hard to find!