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Syntactic Minimization Of Nondeterministic Finite Automata

Authors Robert S. R. Myers, Henning Urbat

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Author Details

Robert S. R. Myers
  • London, United Kingdom
Henning Urbat
  • Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany

Funding

  • Urbat, Henning: Supported by Deutsche Forschungsgemeinschaft (DFG) under project SCHR 1118/15-1.

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Robert S. R. Myers and Henning Urbat. Syntactic Minimization Of Nondeterministic Finite Automata. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 78:1-78:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.MFCS.2021.78

Abstract

Nondeterministic automata may be viewed as succinct programs implementing deterministic automata, i.e. complete specifications. Converting a given deterministic automaton into a small nondeterministic one is known to be computationally very hard; in fact, the ensuing decision problem is PSPACE-complete. This paper stands in stark contrast to the status quo. We restrict attention to subatomic nondeterministic automata, whose individual states accept unions of syntactic congruence classes. They are general enough to cover almost all structural results concerning nondeterministic state-minimality. We prove that converting a monoid recognizing a regular language into a small subatomic acceptor corresponds to an NP-complete problem. The NP certificates are solutions of simple equations involving relations over the syntactic monoid. We also consider the subclass of atomic nondeterministic automata introduced by Brzozowski and Tamm. Given a deterministic automaton and another one for the reversed language, computing small atomic acceptors is shown to be NP-complete with analogous certificates. Our complexity results emerge from an algebraic characterization of (sub)atomic acceptors in terms of deterministic automata with semilattice structure, combined with an equivalence of categories leading to succinct representations.

Subject Classification

ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
Keywords
  • Algebraic language theory
  • Nondeterministic automata
  • NP-completeness

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