Deciding DFA-Primality is NP-Hard

TL;DR

This paper proves that deciding whether a DFA is prime is NP-hard, marking progress in understanding the computational complexity of DFA primality testing.

Contribution

It establishes NP-hardness of Prime-DFA, the first such result narrowing the previously known complexity bounds.

Findings

Prime-DFA is NP-hard to decide.

The proof uses a reduction from SAT.

This advances the complexity understanding of DFA primality.

Abstract

A DFA $\mathcal{A}$ is composite if there 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 deciding $\mathcal{L}(\mathcal{A})$. 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.