  In coded bi-directional cooperation, two nodes wish to exchange messages over
a shared half-duplex channel with the help of a relay. In this paper, we derive
performance bounds for this problem for each of three protocols.
  The first protocol is a two phase protocol were both users simultaneously
transmit during the first phase and the relay alone transmits during the
second. In this protocol, our bounds are tight and a multiple-access channel
transmission from the two users to the relay followed by a coded broadcast-type
transmission from the relay to the users achieves all points in the two-phase
capacity region.
  The second protocol considers sequential transmissions from the two users
followed by a transmission from the relay while the third protocol is a hybrid
of the first two protocols and has four phases. In the latter two protocols the
inner and outer bounds are not identical, and differ in a manner similar to the
inner and outer bounds of Cover's relay channel. Numerical evaluation shows
that at least in some cases of interest our bounds do not differ significantly.
  Finally, in the Gaussian case with path loss, we derive achievable rates and
compare the relative merits of each protocol in various regimes. This case is
of interest in cellular systems. Surprisingly, we find that in some cases, the
achievable rate region of the four phase protocol sometimes contains points
that are outside the outer bounds of the other protocols.
