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Deciding DFA-Primality Is NP-Hard

Author Daniel Alexander Spenner

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LIPIcs.ICALP.2026.192.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Regular languages
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Deterministic finite automaton (DFA)
  • Regular languages
  • Finite languages
  • Decomposition
  • Primality
  • NP-Hardness

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Abstract

A DFA 𝒜 is composite if there exist DFAs 𝒜_1,…,𝒜_t with ℒ(𝒜) = ⋂_{i=1}^t ℒ(𝒜_i) such that each 𝒜_i has strictly less states than the minimal DFA deciding ℒ(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.

Cite As Get BibTex

Daniel Alexander Spenner. Deciding DFA-Primality Is NP-Hard. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 192:1-192:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.192

Author Details

Daniel Alexander Spenner
  • Technische Universität Dortmund, Germany

References

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