The PCP Theorem, and computational complexity

One reason I like computational complexity is because it’s mysterious: you see surprising results that don’t agree with your common sense. The first example was the equality NL = co-NL, which I saw a few years ago, and has a very elegant proof. And the second example was the PCP e, i.e. NP = PCP (log n, 1), which is quite surprising, and the proof is perhaps very complicated. That’s why you can never be sure that NP\P is nonempty, even if  everyone believes it.

For almost the same reason, I liked probability theory when I was in high-school: it is mysterious, and there were lots of paradoxical problems related to probability. Well, when I studied formal probability theory (Kolmogorov axioms etc.) there wasn’t any paradoxes anymore (fortunately?), but I still like it, because of its surprisingly strong applications in graph theory and algorithms.


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