On the Descriptive Complexity of \(\overline{\varSigma ^*\overline{L}}\)

  • Michal HospodárEmail author
  • Galina Jirásková
  • Peter Mlynárčik
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10396)


We examine the descriptive complexity of the combined unary operation \(\overline{\varSigma ^*\overline{L}}\) and investigate the trade-offs between various models of finite automata. We consider complete and partial deterministic finite automata, nondeterministic finite automata with single or multiple initial states, alternating, and boolean finite automata. We assume that the argument and the result of this operation are accepted by automata belonging to one of these six models. We investigate all possible trade-offs and provide a tight upper bound for 32 of 36 of them. The most interesting result is the trade-off from nondeterministic to deterministic automata given by the Dedekind number \({{\mathrm{M}}}(n-1)\). We also prove that the nondeterministic state complexity of \(\overline{\varSigma ^*\overline{L}}\) is \(2^{n-1}\) which solves an open problem stated by Birget [1996, The state complexity of \(\overline{\varSigma ^*\overline{L}}\) and its connection with temporal logic, Inform. Process. Lett. 58, 185–188].


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© Springer International Publishing AG 2017

Authors and Affiliations

  • Michal Hospodár
    • 1
    Email author
  • Galina Jirásková
    • 1
  • Peter Mlynárčik
    • 1
  1. 1.Mathematical InstituteSlovak Academy of SciencesKošiceSlovakia

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