Suppose we have a channel with memory with two states:
In state 0, the channel has three inputs $(A,B,C)$ and two outputs $(0,1)$...
the transition probabilities are $A \to 0$ with prob $1$, $B\to(0 \, \, \text{or} \, \, 1)$ with prob $\frac{1}{2}$ and $C \to 1$ with prob $1$.
In state 1, the roles of $A$ and $B$ are reversed.
We move from state $i \to i$ with prob $\alpha$ and state $i \to (i+1)\, \text{mod}\, 2$ with prob $1-\alpha$
How then would I begin to calculate the channel capacity with and without feedback? I have tried to begin by calculating $H(Y|X)$ and $H(Y)$, but these seem quite messy. I'm thinking that brute force may not be the best idea, but I'm not sure how to get started with some nice symmetry or other. Thanks for any help
Edit: Pretty sure the capacity with feedback is $1-H(.0001)$
($H(.0001)$ is the entropy of the binary distribution with $p=.0001$), but still unsure how to begin the capacity without feedback (I tried to compute something like the marginals but failed).