Cognitive Interference Channel (C-IFC)[]{data-label="fig:channelmodel"}](fig_block_diag.eps){width="11cm"}

Consider the three-user C-IFC in Fig.\[fig:channelmodel\], which is denoted by (${\mathcal{X}}_1\times{\mathcal{X}}_2\times{\mathcal{X}}_3,p(y_1^n,y_2^n,y_3^n|x_1^n,x_2^n,x_3^n),{\mathcal{Y}}_1\times{\mathcal{Y}}_2\times{\mathcal{Y}}_3$), where $X_u\in{\mathcal{X}}_u$ is the channel input of Transmitter $u$ (Tx$u$) and $Y_u\in{\mathcal{Y}}_u$ is the channel output at Receiver $u$ (Rx$u$) for $u\in\{1,2,3\}$. Also, $p(y_1^n,y_2^n,y_3^n|x_1^n,x_2^n,x_3^n)$ is the channel transition probability distribution. In $n$ channel uses, each Tx$u$ desires to send a message $m_u$ to Rx$u$ where $u\in\{1,2,3\}$.

\[def:code\] A $(2^{nR_1},2^{nR_2},2^{nR_3},n)$ code for the three-user C-IFC consists of (i) three independent message sets ${\mathcal{M}}_u=\{1,...,2^{nR_u}\}$, where $u\in\{1,2,3\}$, (ii) two encoding functions at the primary transmitters, $f_1:{\mathcal{M}}_1\mapsto{\mathcal{X}}_1^n$ at Tx1 and $f_2:{\mathcal{M}}_2\mapsto{\mathcal{X}}_2^n$ at Tx2, (iii) an encoding function at the cognitive transmitter, $f_3:{\mathcal{M}}_1\times{\mathcal{M}}_2\times{\mathcal{M}}_3\mapsto{\mathcal{X}}_3^n$, and (iv) three decoding functions, $g_u:{\mathcal{Y}}_u^n\mapsto{\mathcal{M}}_u$ at Rx$u$ where $u\in\{1,2,3\}$. We assume that the channel is memoryless. Thus, the channel transition probability distribution is given by $$\label{eqn:pmf}
p(y_1^n,y_2^n,\,y_3^n|x_1^n,x_2^n,x_3^n)=\prod\limits_{i=1}^np(y_{1,i},y_{2,i},y_{3,i}|x_{1,i},x_{2,i},x_{3,i}).$$

The probability of error for this code is defined as $P_e=max\{P_{e,1},P_{e,2},P_{e,3}\}$, where we have

[l]{} P\_[e,u]{}=\_[m\_1,m\_2,m\_3]{}[P(g\_u(Y\^n\_[u]{})m\_u | (m\_1,m\_2,m\_3))]{}

for $u\in\{1,2,3\}$.

\[def:rate\] A rate triple $(R_1,R_2,R_3)$ is achievable if there exists a sequence of $(2^{nR_1},2^{nR_2},2^{nR_3},n)$ codes with $P_e\rightarrow 0$ as $n\rightarrow \infty$. The capacity region ${\mathcal{C}}$, is the closure of the set of all achievable rates.

Achievable Rate Regions for Discrete Memoryless three-user C-IFC {#sec:Ach}
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In this section, we 