It seems obvious to me that $PSPACE \neq EXPTIME$. I, however, do not believe that my seemingly obvious logic would not be picked up by more intelligent people if it was so simple, so I'm assuming there is some kind of a false assumption I'm making.
We know that there are $EXPTIME$-complete problems; i.e. any problem in $EXPTIME$ has a polynomial-time reduction to some problem $K$. So here's my take on this:
Assume we have an oracle for $K$. Now assume that $PSPACE = EXPTIME$. That means there must be some algorithm in $PSPACE$ which solves $K$. But now note, that with an oracle for $K$, we could solve any $EXPTIME$ problem as follows:
- Reduce the problem to $K$, which takes polynomial time.
- Query the oracle to solve $K$. Since we've assumed that $PSPACE = EXPTIME$, the answer must therefore fit in polynomial space; so it will only take polynomial time to write the answer to the tape.
Both steps together still take polynomial time, so $EXPTIME^K = P^K$. However, we know by the time hierarchy theorem, which relativizes, there cannot be an oracle such that $P^X = EXPTIME^X$, so our original assumption that $PSPACE = EXPTIME$ is wrong. Therefore, $PSPACE \neq EXPTIME$.
Where's the error in my reasoning?