Hyper-Minimizing States in a (Minimized) Deterministic Automaton

We improve a recent result [A. Badr: Hyper-Minimization in O(n). In Proc. CIAA, LNCS 5148, 2008] for hyper-minimized finite automata. Namely, we present an O(n log n) algorithm that computes for a given finite deterministic automaton (dfa) an almost equivalent dfa that is as small as possible—such an automaton is called hyper-minimal. Here two finite automata are almost equivalent if and only if the symmetric difference of their languages is finite. In other words, two almost-equivalent automata disagree on acceptance on finitely many inputs. In this way, we solve an open problem stated in [A. Badr, V. Geffert, I. Shipman: Hyper-minimizing minimized deterministic finite state automata. RAIRO Theor. Inf. Appl. 43(1), 2009] and by Badr. Moreover, we show that minimization linearly reduces to hyper-minimization, which is good evidence that our algorithm performs reasonably well.