[Note: this webpage last modified Friday, 04-Feb-2011 19:44:51 EST]
Something cool about theory of computing, something you can do with it...
Or, is there a problem in the book you'd like to talk about?
Today's attendance.
Administrative...
Any questions about homework 3?
Today: another NP-completeness proof if time allows and hierarchy theorems.
Note: we have skipped over chapter 3, which does more about circuits and some generalizations of NP. Good stuff, but we're skipping it (for now at least).
Time hierarchy theorem. Recall definition of Dtime. Then give the proof. The basic idea is very simple - just pick an input for each other machine/algorithm to do different on, and do different. And the "diagonalizing" machine has enough time to do this because it is allowed more time. To nail down the precise statement needs to consider things more carefully, but that is the main idea.
Similar proof for "memory space" hierarchy.
Implications for P versus EXP. EXP contains problems that are not in P. Even with time nlog log log log n, there are problems that are not in P.
And a particular problem in EXP that is hard for P - just the "machine encoding" language, which is "complete" for EXP too.
Next time: maybe another NP-completeness proof, and "nondeterministic time" hierarchy theorem. A cool proof!