Reversal Hierarchies for Small 2DFAs
A two-way deterministic finite automaton with r(n) reversals performs ≤ r (n) input head reversals on every n-long input. Let 2D[r(n)] be all families of problems solvable by such automata of size polynomial in the index of the family. Then the reversal hierarchy 2D ⊆ 2D ⊆ 2D ⊆ ⋯ is strict, but 2D[O(1)] = 2D[o(n)]. Moreover, the inner-reversal hierarchy 2D(0) ⊆ 2D(1) ⊆ 2D(2) ⊆ ⋯ , where now the bound is only for reversals strictly between the input end-markers, is also strict.
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