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Sweeping Permutation Automata
This paper introduces sweeping permutation automata, which move over an input
string in alternating left-to-right and right-to-left sweeps and have a
bijective transition function. It is proved that these automata recognize the
same family of languages as the classical one-way permutation automata
(Thierrin, "Permutation automata", Mathematical Systems Theory, 1968). An
n-state two-way permutation automaton is transformed to a one-way permutation
automaton with F(n)=\max_(k+l=n, m <= l) k (l \choose m) (k - 1 \choose l - m)
(l - m)! states. This number of states is proved to be necessary in the worst
case, and its growth rate is estimated as F(n) = n^(n/2 - (1 + \ln 2)/2 \cdot
n/(\ln n) \cdot (1 + o(1))).Comment: In Proceedings NCMA 2023, arXiv:2309.0733