Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.07031 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915991114481664 |
|---|---|
| author | Spenner, Daniel Alexander |
| author_facet | Spenner, Daniel Alexander |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_07031 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Deciding DFA-Primality is NP-Hard Spenner, Daniel Alexander Formal Languages and Automata Theory F.4.3 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. |
| title | Deciding DFA-Primality is NP-Hard |
| topic | Formal Languages and Automata Theory F.4.3 |
| url | https://arxiv.org/abs/2605.07031 |