Optimal Hyper-Minimization

Minimal deterministic finite automata (dfas) can be reduced further at the expense of a finite number of errors. Recently, such minimization algorithms have been improved to run in time O(n logn), where n is the number of states of the input dfa, by [Gawrychowski and Jeż: Hyper-minimisation made efficient. Proc. Mfcs, Lncs 5734, 2009] and [Holzer and Maletti: An n logn algorithm for hyper-minimizing a (minimized) deterministic automaton. Theor. Comput. Sci. 411, 2010]. Both algorithms return a dfa that is as small as possible, while only committing a finite number of errors. These algorithms are further improved to return a dfa that commits the least number of errors at the expense of an increased (quadratic) run-time. This solves an open problem of [Badr, Geffert, and Shipman: Hyper-minimizing minimized deterministic finite state automata. Rairo Theor. Inf. Appl. 43, 2009]. In addition, an experimental study on random automata is performed and the effects of the existing algorithms and the new algorithm are reported.