Determine the number of reachable states in a subset construction?

Question

Is there a polynomial time algorithm to determine the number of reachable states in the subset construction of a NFA to a DFA without having to construct the entire DFA?

Answer 1

Start with a 3SAT instance $\phi$ on $n$ variables with $m$ clauses $C_1,\ldots,C_m$. For each clause $C_j$, we construct an NFA $A_j$ which reads a binary input string, interpreting its first $n$ symbols as encoding a truth assignment, and checks whether the clause is satisfied. If $C_j$ is satisfied, then on the $(n+1)$'th symbol it branches to $n$ new states.

Now consider an NFA consisting of the NFAs $A_1,\ldots,A_m$ together with an initial state with $\epsilon$-transitions to the starting states of $A_1,\ldots,A_m$ (alternatively, if you allow multiple initial states, you can just put all of them together). If $\phi$ is satisfiable then the number of reachable states in the corresponding DFA will be at least $n^m$, otherwise it will be at most $n^{m-1} + 2^m n + 1$.