Minimizing finite automata is computationally hard

  • It is known that deterministic finite automata (DFAs) can be algorithmically minimized, i.e., a DFA M can be converted to an equivalent DFA M' which has a minimal number of states. The minimization can be done efficiently [6]. On the other hand, it is known that unambiguous finite automata (UFAs) and nondeterministic finite automata (NFAs) can be algorithmically minimized too, but their minimization problems turn out to be NP-complete and PSPACE-complete [8]. In this paper, the time complexity of the minimization problem for two restricted types of finite automata is investigated. These automata are nearly deterministic, since they only allow a small amount of non determinism to be used. On the one hand, NFAs with a fixed finite branching are studied, i.e., the number of nondeterministic moves within every accepting computation is bounded by a fixed finite number. On the other hand, finite automata are investigated which are essentially deterministic except that there is a fixed number of different initial states which can be chosen nondeterministically. The main result is that the minimization problems for these models are computationally hard, namely NP-complete. Hence, even the slightest extension of the deterministic model towards a nondeterministic one, e.g., allowing at most one nondeterministic move in every accepting computation or allowing two initial states instead of one, results in computationally intractable minimization problems.

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Author:Andreas Malcher
Parent Title (German):Frankfurter Informatik-Berichte ; Nr. 02,6
Series (Serial Number):Frankfurter Informatik-Berichte (02, 6)
Publisher:Johann Wolfgang Goethe-Univ., Fachbereich Biologie und Informatik, Inst. für Informatik
Place of publication:Frankfurt am Main
Document Type:Working Paper
Year of Completion:2002
Year of first Publication:2002
Publishing Institution:Universitätsbibliothek Johann Christian Senckenberg
Release Date:2009/10/16
Page Number:IV, 16 S.
Institutes:Informatik und Mathematik / Informatik
Dewey Decimal Classification:0 Informatik, Informationswissenschaft, allgemeine Werke / 00 Informatik, Wissen, Systeme / 004 Datenverarbeitung; Informatik
Licence (German):License LogoDeutsches Urheberrecht