  The causal cognitive relay channel (CRC) has two sender-receiver pairs, in
which the second sender obtains information from the first sender causally and
assists the transmission of both senders. In this paper, we study both the
full- and half-duplex modes. In each mode, we propose two new coding schemes
built successively upon one another to illustrate the impact of different
coding techniques. The first scheme called partial decode-forward binning
(PDF-binning) combines the ideas of partial decode-forward relaying and
Gelfand-Pinsker binning. The second scheme called Han-Kobayashi partial
decode-forward binning (HK-PDF-binning) combines PDF-binning with Han-Kobayashi
coding by further splitting rates and applying superposition coding,
conditional binning and relaxed joint decoding.
  In both schemes, the second sender decodes a part of the message from the
first sender, then uses Gelfand-Pinsker binning technique to bin against the
decoded codeword, but in such a way that allows both state nullifying and
forwarding. For the Gaussian channels, this PDF-binning essentializes to a
correlation between the transmit signal and the binning state, which
encompasses the traditional dirty-paper-coding binning as a special case when
this correlation factor is zero. We also provide the closed-form optimal
binning parameter for each scheme.
  The 2-phase half-duplex schemes are adapted from the full-duplex ones by
removing block Markov encoding, sending different message parts in different
phases and applying joint decoding across both phases. Analysis shows that the
HK-PDF-binning scheme in both modes encompasses the Han-Kobayashi rate region
and achieves both the partial decode-forward relaying rate for the first sender
and interference-free rate for the second sender. Furthermore, this scheme
outperforms all existing schemes.
