NOTES ON LAST YEAR'S COLLECTING

The last hundred years.

Notes on Complexity Theory Last updated : November , 2011 Lecture 25

Randomization provides unconditional benefits in many settings; examples include cryptography (where random keys are used to provide protection against an adversary) and distributed computing (where randomness can be used as a means to break symmetry between parties). Randomness also appears to help in algorithm design. But is it possible that, from a complexity-theoretic perspective, randomnes...

Notes on Complexity Theory Last updated : November , 2011 Lecture 21

Work on interactive proof systems motivates further exploration of non-interactive proof systems (e.g., the class NP). One specific question is: how many bits of the proof does the verifier need to read? Note that in the usual certificate-based definition of NP, the deterministic “verifier” reads the entire certificate, and correctness and soundness hold with probability 1. If we allow the veri...

Notes on Complexity Theory Last updated : December , 2011 Lecture 2

The running time of a Turing machine M on input x is the number of “steps” M takes before it halts. Machine M is said to run in time T (·) if for every input x the running time of M(x) is at most T (|x|). (In particular, this means it halts on all inputs.) The space used by M on input x is the number of cells written to by M on all its work tapes1 (a cell that is written to multiple times is on...

Notes on Complexity Theory Last updated : December , 2011 Lecture 27

We now prove Theorem 1. Let M be a probabilistic machine running in space S (and time 2S), using R ≤ 2S random bits, and deciding a language L with two-sided error. (Note that S, R are functions of the input length n, and the theorem requires S = Ω(log n).) We will assume without loss of generality that M always uses exactly R random bits on all inputs; recall also that M has read-once access t...