Decomposing Finite Languages

TL;DR

This paper characterizes the primality of acyclic DFAs and finite languages, introduces S-primality, and proves complexity results including NL-completeness and NL-hardness for related decision problems.

Contribution

It provides a complete characterization of primality for acyclic DFAs and finite languages, introduces the novel concept of S-primality, and establishes complexity bounds for these problems.

Findings

Characterizes primality of acyclic DFAs and finite languages.

Proves NL-completeness of the PrimeDFA_fin decision problem.

Establishes NL-hardness of deciding S-primality and minimality for 2-letter DFAs.

Abstract

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.