Finite two-way relay channel and causal code

A finite two-way relay channel is the three-node DMN of Finite discrete memoryless networks and causal codes with terminal inputs $X_1,X_2$, relay input $X_r$, terminal outputs $Y_1,Y_2$, relay observation $Y_r$, and transition law $W(y_1,y_2,y_r\mid x_1,x_2,x_r)$. Terminals 1 and 2 have independent uniform messages $M_1$ and $M_2$ and wish to exchange them. A causal code uses $X_{1,i}=f_{1,i}(M_1,Y_1^{i-1})$, $X_{2,i}=f_{2,i}(M_2,Y_2^{i-1})$, and $X_{r,i}=f_{r,i}(Y_r^{i-1})$; terminal 1 estimates $M_2$ from $(M_1,Y_1^n)$ and terminal 2 estimates $M_1$ from $(M_2,Y_2^n)$. The probability of error is the probability either estimate is wrong. This is a full-duplex model: all three nodes choose an input at every use, and each input depends only on that node's message and past observations.