Minimizing finite automata is computationally hard

  • Authors:

  • Andreas Malcher

  • Affiliations:

  • Institut für Informatik, Johann Wolfgang Goethe Universität, D-60054 Frankfurt am Main, Germany

  • Venue:
  • Theoretical Computer Science - Developments in language theory
  • Year:
  • 2004

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Abstract

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 (in: Z. Kohavi (Ed.), Theory of Machines and Computations, Academic Press, New York, 1971, pp. 189-196). On the other hand, it is known that unambiguous finite automata and nondeterministic finite automata can be algorithmically minimized too, but their minimization problems turn out to be NP-complete and PSPACE-complete, respectively (SIAM J. Comput. 22(6) (1993) 1117-1141). 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.