Friday, January 30

The Hierarchy Theorem

The hierarchy theorem sets the stage for attempting to understand the boundaries between complexity classes and therefore forms part of the groundwork for dealing with the P vs NP question.  We looked at it in detail, but its intuition is simply this:  If you give me enough extra time to work with, I can solve more problems.

How much extra time is enough?  If you give me time f(n), certainly cf(n) is not enough extra time, because, as the speedup theorem indicates, one can always speed a solution up by a constant factor (using Turing machines).  The hierarchy theorem lets us know that with a polynomial increase in time we can solve more problems.  That's all we need for studying the P vs NP question.  However those pursuing the time hierarchy further have found much tighter hierarchies than polynomial.

The Gap Theorem

We just discussed the intuition behind this theorem.