Let some deterministic finite automata be given. There is a problem of determining whether the intersection of these DFA is empty, and I want to show its PSPACE-completeness.
It seems to me that I understand, why this problem lies in PSPACE: if $ n_1,\dots, n_k $ are the numbers of states, then one can take the product of these DFA and get that if there is a recognisable string, then there is a recognisable string of length $ \le n_1\dots n_k$. Then we can non-deterministically guess this string and use Savitch theorem.
So, the question is, how to prove PSPACE-hardness? Should I reduce any given TM to this problem, or maybe some known problem like QBF?