It is well-known that every non-trivial NFA of $k$ states (an NFA that does not accept all strings) rejects a string of length at most $2^k$.
But is this upper bound asymptotically tight ?
I found a rough sub-exponential bound in this strategy.
Let $p_i$ be an $i$-th prime.
On the singleton alphabet, $\Sigma = \{a\}$, the following NFA rejects only
$\{a^n:n>0, p_1\cdots p_k|n\}$. With $p_1+p_2+\cdots+p_k+O(1) = \Theta(k^2\log k)$ states.
Initial state $q$, is an acceptance state and it is branched into $k$ cycles with $\epsilon$ marks.
Every $i$-th cycle has $p_i$ states, and only rejecting state is the state directly linked into initial state $q$. All of its transition are marked by $a$.
It is clear that the complement of accepting language is $\{a^n:n>0, p_1\cdots p_k|n\}$ as given.
But $p_1\cdots p_k=O(e^{\Theta(k\log k)})$. which is sub-exponential with respect to $\Theta(k^2\log k)$.
In this unary and multiplication strategy it seems to be this is optimal because $\text{lcm}\{p_1,p_2,\cdots,p_n\}$ is probably maximized with constraint $p_1+\cdots+p_n=C$ under this prime summation partition.
It seems we should use multi-ary language to improve this bound. Is there any suggestion or results in research?
