A DFA 𝒜 is composite if there exist DFAs 𝒜_1,…,𝒜_t with ℒ(𝒜) = ⋂_{i=1}^t ℒ(𝒜_i) such that each 𝒜_i has strictly less states than the minimal DFA deciding ℒ(𝒜). Otherwise, it is prime. Prime-DFA is the problem of deciding primality for a given DFA. It was defined by Kupferman and Mosheiff in 2015 and it was shown to be NL-hard and in ExpSpace. This paper proves the NP-hardness of Prime-DFA, thereby making the first progress in closing this doubly-exponential gap. It proves the NP-hardness by a reduction from the propositional logic satisfiability problem. The correctness of the reduction relies on an involved characterization of primality for a class of DFAs which contains those that can occur in the reduction.

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.