Quotient Complexity of Closed Languages

  • Janusz Brzozowski
  • Galina Jirásková
  • Chenglong Zou
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6072)


A language L is prefix-closed if, whenever a word w is in L, then every prefix of w is also in L. We define suffix-, factor-, and subword-closed languages in an analogous way, where by subword we mean subsequence. We study the quotient complexity (usually called state complexity) of operations on prefix-, suffix-, factor-, and subword-closed languages. We find tight upper bounds on the complexity of the subword-closure of arbitrary languages, and on the complexity of boolean operations, concatenation, star, and reversal in each of the four classes of closed languages. We show that repeated application of positive closure and complement to a closed language results in at most four distinct languages, while Kleene closure and complement gives at most eight.


automaton closed factor language prefix quotient regular operation state complexity subword suffix upper bound 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Janusz Brzozowski
    • 1
  • Galina Jirásková
    • 2
  • Chenglong Zou
    • 1
  1. 1.David R. Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada
  2. 2.Slovak Academy of SciencesMathematical InstituteKošiceSlovakia

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