Optimal Hyper-Minimization

Abstract

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 log n), 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 log n 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.

Hyper-optimization for deterministic tree automata

“…It is shown in [27] how to reduce DTA hyper-minimization to hyper-minimization for deterministic finite-state string automata (DFAs), for which efficient algorithms exist [18,19]. Similarly, efficient DFA hyper-optimization algorithms exist [24]. However, the reduction mentioned in [27] fails to work for hyper-optimization.…”

“…This approach was also successful in the string case [24], however in the tree case it is complicated by the fact that a single error tree can use such exclusively-preamble-states-to-kernel-state-transitions, also called special transitions, several times as the next example demonstrates.…”

“…The main application of the characterization is the calculation of the number of errors committed by a given hyperminimal DTA inspired by the approach of [24]. In this sense we deliver a much more refined, quantitative analysis of hyper-minimization.…”

“…We chose to select the left-most special transition, but other selections are possible. Fortunately, the restriction to left-most special transitions can be incorporated easily into the approach of [24]. Overall, we obtain an algorithm that computes the number of errors committed by a hyper-minimal DTA N in comparison to a reference DTA M (that recognizes a finitely different tree language) in time O(m · n), where m is the size of M and n is the number of states of M. Since our association and attribution of errors is such that the choices do not impact another, we can simply optimize each choice such that it causes the least number of errors locally.…”

Hyper-Minimization for Deterministic Tree Automata

“…This yields that several foundational results for dfa hyperminimization [2] do not faithfully generalize to dta. Nevertheless, we borrow much of the surrounding infrastructure from the existing hyper-minimization algorithms [8,14] and despite the theoretical dierences, we obtain an ecient hyper-minimization algorithm following the approach of [1].…”