Syntactic Minimization of Nondeterministic Finite Automata

TL;DR

This paper investigates the complexity of minimizing nondeterministic finite automata by focusing on subatomic and atomic automata, revealing NP-completeness results for their minimization problems through algebraic and categorical characterizations.

Contribution

It introduces a new algebraic framework for analyzing subatomic and atomic nondeterministic automata, establishing NP-completeness of their minimization tasks.

Findings

Minimizing subatomic automata is NP-complete.

Minimizing atomic automata is NP-complete.

Algebraic and categorical methods underpin the complexity results.

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.