Tuesday, January 20, 2009

Lecture 3

Lecture 3 covered the following topics: NP vs. coNP and the theme of proof complexity; nondeterministic time hierarchy via lazy diagonalization; Ladner's Theorem statement; Randomized TMs, BPP, RP, ZPP; problems in BPP but not known to be in P (Berlekamp's algorithm vs. factoring cubics mod p) and primality; lack of hierarchy and complete problems for BPP; polynomial identity testing; error amplification for BPP (and statement about randomness reduction); poly-size circuit families; Adleman's Theorem BPP in P/poly.

2 comments:

  1. Problem set 1 was handed out in today's lecture and is posted on the course webpage. It is due in two weeks on Feb 3.

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  2. When we discussed the problem of finding the roots a_1,a_2,a_3 of a polynomial f(X) = x^3 + b X^2 + c X + d that factors as (X-a_1) (X-a_2) (X-a_3) modulo p, the following question came up: Can we detect (in deterministic polynomial time) if f(X) has this property?

    The answer is Yes, and method is very simple: Compute gcd(f(X),X^p-X). If the result equals f(X), f(X) splits into 3 such factors, otherwise not.

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