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Square, Power, Positive Closure, and Complementation on Star-Free Languages

  • Sylvie Davies
  • Michal HospodárEmail author
Conference paper
  • 150 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11612)

Abstract

We examine the deterministic and nondeterministic state complexity of square, power, positive closure, and complementation on star-free languages. For the state complexity of square, we get a non-trivial upper bound \((n-1)2^n - 2(n-2)\) and a lower bound of order \({\Theta }(2^n)\). For the state complexity of the k-th power in the unary case, we get the tight upper bound \(k(n-1)+1\). Next, we show that the upper bound kn on the nondeterministic state complexity of the k-th power is met by a binary star-free language, while in the unary case, we have a lower bound \(k(n-1)+1\). For the positive closure, we show that the deterministic upper bound \(2^{n-1}+2^{n-2}-1\), as well as the nondeterministic upper bound n, can be met by star-free languages. We also show that in the unary case, the state complexity of positive closure is \(n^2-7n+13\), and the nondeterministic state complexity of complementation is between \((n-1)^2+1\) and \(n^2-2\).

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© IFIP International Federation for Information Processing 2019

Authors and Affiliations

  1. 1.David R. Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada
  2. 2.Mathematical InstituteSlovak Academy of SciencesKošiceSlovakia

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