An [formula omitted] algorithm for hyper-minimizing a (minimized) deterministic automaton

Abstract

We improve a recent result [A. Badr, Hyper-minimization in O ( n 2 ) , Internat. J. Found. Comput. Sci. 20 (4) (2009) 735–746] for hyper-minimized finite automata. Namely, we present an O ( n log n ) algorithm that computes for a given deterministic finite 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) 69–94] and by Badr. Moreover, we show that minimization linearly reduces to hyper-minimization, which shows that the time-bound O ( n log n ) is optimal for hyper-minimization. Independently, similar results were obtained in [P. Gawrychowski, A. Jeż, Hyper-minimisation made efficient, in: Proc. 34th Int. Symp. Mathematical Foundations of Computer Science, in: LNCS, vol. 5734, Springer, 2009, pp. 356–368].