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For all \(h,n\ge 2\), the problem of one-way liveness for height h and length n captures the task of simulating one-way nondeterministic finite automata with h states on inputs with \(n{-}2\) symbols. We prove that the number of states in a smallest two-way deterministic finite automaton which decides this problem is \(\varTheta (h)\), if \(n=2\); and \(\varTheta (h^2/\log h)\), if \(n=3\).

Optimal 2DFA Algorithms for One-Way Liveness on Two and Three Symbols