Minimizing Finite Automata Is Computationally Hard
- 325 Downloads
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 . 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, respectively . 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 nondeterminism to be used. 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.
Unable to display preview. Download preview PDF.
- 6.Hopcroft, J.E.: An n log n algorithm for minimizing states in a finite automaton. In: Kohavi, Z. (ed.): Theory of machines and computations. Academic Press, New York (1971) 189–196Google Scholar
- 13.Meyer, A.R., Fischer, M.J.: Economy of descriptions by automata, grammars, and formal systems. IEEE Symposium on Foundations of Computer Science (1971) 188–191Google Scholar
- 14.Stockmeyer, L., Meyer, A.R.: Word problems requiring exponential time: preliminary report. Fifth Annual ACM Symposium on Theory of Computing (1973) 1–9Google Scholar
- 15.Yu, S.: Regular languages. In: Rozenberg, G., Salomaa, A. (eds.): Handbook of Formal Languages Vol. 1. Springer-Verlag, Berlin Heidelberg (1997) 41–110Google Scholar