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arXiv · 2605.07031

Deciding DFA-Primality is NP-Hard

Daniel Alexander Spenner
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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.

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Daniel Alexander Spenner. 2026-05-07. Deciding DFA-Primality is NP-Hard. https://arxiv.org/abs/2605.07031

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