In this paper we study the capacity of the half-duplex wireless butterfly network, in which 
a relay node facilitates the communication between two interfering transmitter-receiver pairs. 
We examine this problem in the context of the deterministic channel interaction model, which 
eliminates the channel noise and allows us to focus on the interaction between signals. We 
derive a new outer bound on the capacity of this network which is in general tigher than the 
cut-set upper bound. We also propose a transmission strategy and show that for symmetric channel 
gains it achieves the outer bound, hence characterize the capacity region in the symmetric case. 
Furthermore, these results provide insights to new outer bounds and transmission strategies for 
the noisy (Gaussian) case and enable us to find an approximate capacity region of the symmetric 
half-duple Gaussian butterfly network.