An nlogn Algorithm for Hyper-minimizing States in a (Minimized) Deterministic Automaton

Authors:
Markus Holzer;Andreas Maletti
Affiliations:
Institut für Informatik, Universität Giessen, Giessen, Germany 35392;Departament de Filologies Romàniques, Universitat Rovira i Virgili, Tarragona, Spain 43002
Venue:
CIAA '09 Proceedings of the 14th International Conference on Implementation and Application of Automata
Year:
2009

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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 (n logn ) 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.