## Thursday, January 15, 2009

### Lecture 2

The topics covered in Lecture 2 were Universal Turing Machines, the Time and Space Hierarchy Theorems, Nondeterminism and NP as poly-time verification, poly-time reductions and NP-completeness, the Cook-Levin Theorem and Circuit-Sat being NP-complete, Circuit-Val is P-complete under log-space reductions, coNP.

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A glaring omission I made in stating the time and space hierarchy theorems was requiring constructibility of the concerned time (or space) bounds. (This was surely used in the proof I gave!) For example, I should have said the following: If f(n) is a time constructible function, i.e., a representation of the integer f(n) can itself be computed within O(f(n)) steps, then TIME(f(n)) is strictly contained in TIME(f(n)^3). Likewise for space hierarchy theorem, where we require f(n) to be computable in O(f(n)) space.

ReplyDeleteThe Gap theorem states that the hierarchy theorem can be spectacularly false in the absence of such niceness restrictions on the complexity bounds. You can find a proof

here.