Complementation and determinization are two fundamental notions in automata theory. The close relationship between the two has been well observed in the literature. In the case of nondeterministic finite automata on finite words (NFA), complementation and determinization have the same state complexity, namely Θ(2) where n is the state size. The same similarity between determinization and comple...

We call F : {0, 1}n×{0, 1}n → {0, 1} a symmetric XOR function if for a function S : {0, 1, ..., n} → {0, 1}, F (x, y) = S(|x⊕ y|), for any x, y ∈ {0, 1}n, where |x⊕ y| is the Hamming weight of the bit-wise XOR of x and y. We show that for any such function, (a) the deterministic communication complexity is always Θ(n) except for four simple functions that have a constant complexity, and (b) up ...