  The paper completely characterizes the primality of acyclic DFAs, where a DFA
$\mathcal{A}$ is prime if there do not exist DFAs
$\mathcal{A}_1,\dots,\mathcal{A}_t$ with $\mathcal{L}(\mathcal{A}) =
\bigcap_{i=1}^{t} \mathcal{L}({\mathcal{A}_i})$ such that each $\mathcal{A}_i$
has strictly less states than the minimal DFA recognizing the same language as
$\mathcal{A}$. A regular language is prime if its minimal DFA is prime. Thus,
this result also characterizes the primality of finite languages.
  Further, the $\mathsf{NL}$-completeness of the corresponding decision problem
$\mathsf{PrimeDFA}_{\text{fin}}$ is proven. The paper also characterizes the
primality of acyclic DFAs under two different notions of compositionality,
union and union-intersection compositionality.
  Additionally, the paper introduces the notion of S-primality, where a DFA
$\mathcal{A}$ is S-prime if there do not exist DFAs
$\mathcal{A}_1,\dots,\mathcal{A}_t$ with $\mathcal{L}(\mathcal{A}) =
\bigcap_{i=1}^{t} \mathcal{L}(\mathcal{A}_i)$ such that each $\mathcal{A}_i$
has strictly less states than $\mathcal{A}$ itself. It is proven that the
problem of deciding S-primality for a given DFA is $\mathsf{NL}$-hard. To do
so, the $\mathsf{NL}$-completeness of $\mathsf{2MinimalDFA}$, the basic problem
of deciding minimality for a DFA with at most two letters, is proven.

Formal Languages and Automata Theory