# PSPACE and NP complexity

I'm reading a definition of PSPACE and say: are the decision problems solvable in polynomial space on a Deterministic Turing Machine. My question is: Why NP is in PSPACE?. I have a doubt beccause for example a SAT problem is solvable in Non-Deterministic Turing Machine and SAT $\in NP$ and not in $P$.

• Just to elaborate on the last point, the PSPACE machine, one by one, writes down satisfying assignments for the SAT instance. Each on of these only takes $O(n)$ space - we just need True or False for each variable. After writing down the assignment, it checks whether it satisfies the formula or not. If it does, then it answers YES. If not, it wipes that assignment, and reuses the same space to write the next one. It only says no if it gets through all the assignments and nothing worked. There's no nondeterminism, it takes $O(2^{n})$ time but we don't care as it uses only $O(n)$ space. – Luke Mathieson Dec 6 '13 at 12:31