The paper completely characterizes the primality of acyclic DFAs, where a DFA
A is prime if there do not exist DFAs
A1​,…,At​ with L(A)=⋂i=1t​L(Ai​) such that each Ai​
has strictly less states than the minimal DFA recognizing the same language as
A. A regular language is prime if its minimal DFA is prime. Thus,
this result also characterizes the primality of finite languages.
Further, the NL-completeness of the corresponding decision problem
PrimeDFAfin​ 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
A is S-prime if there do not exist DFAs
A1​,…,At​ with L(A)=⋂i=1t​L(Ai​) such that each Ai​
has strictly less states than A itself. It is proven that the
problem of deciding S-primality for a given DFA is NL-hard. To do
so, the NL-completeness of 2MinimalDFA, the basic problem
of deciding minimality for a DFA with at most two letters, is proven