Abstract
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].
M. Hospodár et al.—Research supported by grant VEGA 2/0084/15 and grant APVV-15-0091. This work was conducted as a part of PhD study of Michal Hospodár and Peter Mlynárčik at the Faculty of Mathematics, Physics and Informatics of the Comenius University.
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Hospodár, M., Jirásková, G., Mlynárčik, P. (2017). On the Descriptive Complexity of \(\overline{\varSigma ^*\overline{L}}\) . In: Charlier, É., Leroy, J., Rigo, M. (eds) Developments in Language Theory. DLT 2017. Lecture Notes in Computer Science(), vol 10396. Springer, Cham. https://doi.org/10.1007/978-3-319-62809-7_16
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