Abstract
We show that the state complexity of the star-complement-star operation is given by \(\frac{3}{2}f(n\,-\,1) \,+\, 2 f(n\,-\,2) \,+\, 2n \,-\,5\), where \(f(2)=2\) and \(f(n) = \sum _{i=1}^{n-2}{n\atopwithdelims ()i} f (n\,-\,i) \,+\,2\). The function f(n) counts the number of distinct resistances possible for n arbitrary resistors each connected in series or parallel with previous ones, or the number of labeled threshold graphs on n vertices, and \(f(n)\sim n!(1-\ln 2)/(\ln 2)^{n+1} =2^{n \log n - 0.91 n + o(n)}\). Our witness language is defined over a quaternary alphabet, and we strongly conjecture that the size of the alphabet cannot be decreased.
J. Jirásek—Research supported by VEGA grant 1/0056/18 and grant APVV-15-0091.
G. Jirásková—Research supported by VEGA grant 2/0084/15 and grant APVV-15-0091.
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Acknowledgement
We would like to thank Jeffrey Shallit for proposing such an interesting problem. The work on finding its solution was really funny for both of us, and it helped us to almost forget that our children eventually left our place.
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Jirásek, J., Jirásková, G. (2018). The Exact Complexity of Star-Complement-Star. In: Câmpeanu, C. (eds) Implementation and Application of Automata. CIAA 2018. Lecture Notes in Computer Science(), vol 10977. Springer, Cham. https://doi.org/10.1007/978-3-319-94812-6_19
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