On the average state and transition complexity of finite languages

  • Authors:

  • Hermann Gruber;Markus Holzer

  • Affiliations:

  • Institut für Informatik, Ludwig-Maximilians-Universität München, Oettingenstraβe 67, D-80538 München, Germany;Institut für Informatik, Technische Universität München, Boltzmannstraβe 3, D-85748 Garching bei München, Germany

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2007

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Abstract

We investigate the average-case state and transition complexity of deterministic and nondeterministic finite automata, when choosing a finite language of a certain ''size'' n uniformly at random from all finite languages of that particular size. Here size means that all words of the language are either of length n, or of length at most n. It is shown that almost all deterministic finite automata accepting finite languages over a binary input alphabet have state complexity @Q(2^nn), while nondeterministic finite automata are shown to perform better, namely the nondeterministic state complexity is in @Q(2^n). Interestingly, in both cases the aforementioned bounds are asymptotically like in the worst case. However, the nondeterministic transition complexity is shown to be again @Q(2^nn). The case of unary finite languages is also considered. Moreover, we develop a framework that allows us to investigate the average-case complexity of operations like, e.g., union, intersection, complementation, and reversal, on finite languages in this setup.