Optimal lower bounds on regular expression size using communication complexity

  • Authors:

  • Hermann Gruber;Jan Johannsen

  • Affiliations:

  • Institut für Informatik, LMU München, München, Germany;Institut für Informatik, LMU München, München, Germany

  • Venue:
  • FOSSACS'08/ETAPS'08 Proceedings of the Theory and practice of software, 11th international conference on Foundations of software science and computational structures
  • Year:
  • 2008

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Abstract

The problem of converting deterministic finite automata into (short) regular expressions is considered. It is known that the required expression size is 2Θ(n) in the worst case for infinite languages, and for finite languages it is nΩ(log log n) and nO(log n), if the alphabet size grows with the number of states n of the given automaton. A new lower bound method based on communication complexity for regular expression size is developed to show that the required size is indeed nΘ(log n). For constant alphabet size the best lower bound known to date is Ω(n2), even when allowing infinite languages and nondeterministic finite automata. As the technique developed here works equally well for deterministic finite automata over binary alphabets, the lower bound is improved to nΩ(log n).