PSPACE

First Let us review what is P, NP and NP-complete.

How to prove one problem is NP-complete?

• Just need to show that the problem of interest can be reduced from other NP-complete problem.

Example: Show Set-Cover is NP-complete.

Since Vertex Cover is NP-Complete, we just need to show Vertex Cover \(\leq_P\) Set Cover, means every Vertex Cover problem can be reduced to a Set Cover problem.

PSPACE

TQSAT True Quantified SAT is PSPACE-complete.

https://www.cs.princeton.edu/~wayne/kleinberg-tardos/pdf/09PSPACE.pdf

It is easy to see TQSAT is PSPACE, but how to prove it is PSPACE-complete?

For TQSAT, the space requirement is the number of quantifiers in the formula, assuming the space can be reused.
However, the space required to stored the solution is exponential!?