  Half-duplex communication complexity with adversary was defined in [Hoover,
K., Impagliazzo, R., Mihajlin, I., Smal, A. V. Half-Duplex Communication
Complexity, ISAAC 2018.] Half-duplex communication protocols generalize
classical protocols defined by Andrew Yao in [Yao, A. C.-C. Some Complexity
Questions Related to Distributive Computing (Preliminary Report), STOC 1979].
It has been unknown so far whether the communication complexities defined by
these models are different or not. In the present paper we answer this
question: we exhibit a function whose half-duplex communication complexity with
adversary is strictly less than its classical communication complexity.
