An n log n algorithm for hyper-minimizing states in a (minimized) deterministic automaton

Abstract

We improve a recent result [A. Badr: Hyper-Minimization in O(n 2). In Proc. CIAA, LNCS 5148, 2008] for hyper-minimized finite automata. Namely, we present an O(nlogn) 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 shows that the time-bound O(n logn) is optimal for hyper-minimization. © 2009 Springer Berlin Heidelberg.