Abstract

This paper summarizes a series of three lectures the first author was invited to present at the NZMRI summer 2000 workshop, held in Kaikoura, New Zealand. Lecture 1 presents the goals of computational complexity theory. We discuss (a) what complexity provably can never deliver, (b) what it hopes to deliver but thus far has not, and finally (c) where it has been extremely successful in providing useful theorems. In so doing, we introduce nondeterministic Turing machines. Lecture 2 presents alternation, a surprisingly-useful generalization of nondeterminism. Using alternation, we define more complexity classes, and inject clarity into a confusing situation. In Lecture 3 we present a few of the most beautiful results in computational complexity theory. In particular, we discuss (a) the algebraic approach to circuit complexity, (b) circuit lower bounds, and (c) derandomization.

Citations