Polynomial complementation of two-way nondeterministic finite automata

Theorem 1.1: for every $n\ge4$ there is an $n$-state 2NFA $A_n$ for relation-product liveness on $n-2$ points such that every 2NFA for its complement has at least $\tfrac12 2^{\lfloor(n-4)/127\rfloor}-1$ states. Corollary: every 2DFA for the same language needs $\tfrac12 2^{\lfloor(n-4)/127\rfloor}$ states when $n\ge131$. The alphabet has $2^{(n-2)^2}$ letters, so the result says nothing about complementation over a fixed alphabet, about transition-table size, or about uniform logarithmic-space classes.

Theorem 1.1: for every $n\ge4$ there is an $n$-state 2NFA $A_n$ for relation-product liveness on $n-2$ points such that every 2NFA for its complement has at least $\tfrac12 2^{\lfloor(n-4)/127\rfloor}-1$ states. Corollary: every 2DFA for the same language needs $\tfrac12 2^{\lfloor(n-4)/127\rfloor}$ states when $n\ge131$. The alphabet has $2^{(n-2)^2}$ letters, so the result says nothing about complementation over a fixed alphabet, about transition-table size, or about uniform logarithmic-space classes.