In this section, we characterize the capacity region of MA-CIFC under specific conditions. First, we consider a class of degraded MA-CIFC and derive conditions under which the inner bound in Theorem 1 achieves the outer bound of Theorem 4. Next, we investigate the strong interference regime by deriving two sets of strong interference conditions under which the region of Theorem 3 achieves capacity. We also compare these two sets of conditions and identify the weaker set. Finally, we extend the strong interference results to a network with *k* primary users.

### A. Degraded MA-CIFC

Now, we characterize the capacity region for a class of MA-CIFC with a *degraded* primary receiver. We define MA-CIFC with a degraded primary receiver as a MA-CIFC where *Y*_{1} and *X*_{3} are independent given *Y*_{3}, *X*_{1}, *X*_{2}. More precisely, the following Markov chain holds:

{X}_{3}|{X}_{1},{X}_{2}\to {Y}_{3}|{X}_{1},{X}_{2}\to {Y}_{1}|{X}_{1},{X}_{2},

(29)

or equivalently, *X*_{3} → (*X*_{1}, *X*_{2}, *Y*_{3}) → *Y*_{1} forms a Markov chain. This means that the primary receiver (Rx1) observes a degraded or noisier version of the cognitive user's signal (Tx3) compared with the cognitive receiver (Rx3).

Assume that the following conditions are satisfied for MA-CIFC over all p.m.fs that factor as (11):

I\left(U,{X}_{1};{Y}_{1}|T,V,{X}_{2}\right)\le I\left(U,{X}_{1};{Y}_{3}|T,V,{X}_{2}\right)

(30)

I\left(V,{X}_{2};{Y}_{1}|T,U,{X}_{1}\right)\le I\left(V,{X}_{2};{Y}_{3}|T,U,{X}_{1}\right)

(31)

I\left(U,{X}_{1},V,{X}_{2};{Y}_{1}|T\right)\le I\left(U,{X}_{1},V,{X}_{2};{Y}_{3}|T\right)

(32)

I\left(T,U,{X}_{1},V,{X}_{2};{Y}_{1}\right)\le I\left(T,U,{X}_{1},V,{X}_{2};{Y}_{3}\right)

(33)

Under these conditions, the cognitive receiver (Rx3) can decode the messages of the primary users with no rate penalty. If MA-CIFC with a degraded primary receiver satisfies conditions (30)-(33), the region of Theorem 1 coincides with {\mathcal{R}}_{o}^{1} and achieves capacity, as stated in the following theorem.

*Theorem 5:* The capacity region of MA-CIFC with a degraded primary receiver, defined in (29), satisfying (30)-(33) is given by the union of rate regions satisfying (2)-(6) over all joint p.m.fs (11).

*Remark 2:* The messages of the primary users (*m*_{0}, *m*_{1}, *m*_{2}) can be decoded at Rx3 under conditions (30)-(33). Therefore, Rx3-Tx3 achieves the rate in (2). Moreover, we can see that due to the degradedness condition in (29), treating interference as noise at the primary receiver (Rx1) achieves capacity. We show in Section 6 that, in the Gaussian case the capacity is achieved by using the region of Theorem 2 based on DPC (or GP binning), where the cognitive receiver (Rx3) does not decode the primary messages and conditions (30)-(33) are not necessary.

*Proof:* __Achievability:__ The proof follows from the region of Theorem 1. Using the condition in (30), the sum of the bounds in (2) and (3) makes the bound in (7) redundant. Similarly, conditions (31)-(33), along with the bound in (2), make the bounds in (8)-(10) redundant and the region reduces to (2)-(6).

__Converse:__ To prove the converse part, we evaluate {\mathcal{R}}_{o}^{1} of Theorem 4 with the degradedness condition in (29). It is noted that the p.m.f of Theorem 5 is the same as the one for {\mathcal{R}}_{o}^{1}. Moreover, the bounds in (3)-(6) are equal for both regions. Hence, it is only necessary to show the bound in (2). Considering (19), we obtain:

\begin{array}{c}{R}_{3}\le I\left({X}_{3};{Y}_{3},{Y}_{1}|T,U,{X}_{1},V,{X}_{2}\right)\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}=I\left({X}_{3};{Y}_{3}|T,U,{X}_{1},V,{X}_{2}\right)+I\left({X}_{3};{Y}_{1}|T,U,{X}_{1},V,{X}_{2},{Y}_{3}\right)\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\stackrel{\left(a\right)}{=}I\left({X}_{3};{Y}_{3}|T,U,{X}_{1},V,{X}_{2}\right)\end{array}

where (a) is obtained by applying the degradedness condition in (29). This completes the proof.

### B. Strong interference regime

Now, we derive two sets of strong interference conditions under which the region of Theorem 3 achieves capacity. First, assume that the following set of strong interference conditions, referred to as *Set1*, holds for all p.m.fs that factor as (18):

I\left({X}_{3};{Y}_{3}|{X}_{1},{X}_{2},T\right)\le I\left({X}_{3};{Y}_{1}|{X}_{1},{X}_{2},T\right)

(34)

I\left({X}_{1},{X}_{3};{Y}_{1}|{X}_{2},T\right)\le I\left({X}_{1},{X}_{3};{Y}_{3}|{X}_{2},T\right)

(35)

I\left({X}_{2},{X}_{3};{Y}_{1}|{X}_{1},T\right)\le I\left({X}_{2},{X}_{3};{Y}_{3}|{X}_{1},T\right)

(36)

I\left({X}_{1},{X}_{2},{X}_{3};{Y}_{1}\right)\le I\left({X}_{1},{X}_{2},{X}_{3};{Y}_{3}\right).

(37)

In fact, under these conditions, interfering signals at the receivers are strong enough that all messages can be decoded by both receivers. Condition (34) implies that the cognitive user's message (*m*_{3}) can be decoded at Rx1, while conditions (35)-(37) guarantee the decoding of the primary messages (*m*_{0}, *m*_{1}, *m*_{2}) along with *m*_{3} at Rx3 in a MAC fashion.

*Theorem 6:* The capacity region of MA-CIFC satisfying (34)-(37) is given by:

\begin{array}{l}{\mathcal{C}}_{1}^{\text{str}}={\displaystyle \underset{p(t)p({x}_{1}|t)p({x}_{2}|t)p({x}_{3}|{x}_{1},{x}_{2},t)}{\cup}\{({R}_{0},{R}_{1},{R}_{2},{R}_{3}):}\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{R}_{0},{R}_{1},{R}_{2},{R}_{3}\ge 0\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{R}_{3}\le I({X}_{3};{Y}_{3}|{X}_{1},{X}_{2},T)\hfill & (38)\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{R}_{1}+{R}_{3}\le I({X}_{1},{X}_{3};{Y}_{1}|{X}_{2},T)\hfill & (39)\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{R}_{2}+{R}_{3}\le I({X}_{2},{X}_{3};{Y}_{1}|{X}_{1},T)\hfill & (40)\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{R}_{0}+{R}_{1}+{R}_{2}+{R}_{3}\le I({X}_{1},{X}_{2},{X}_{3};{Y}_{1})\}.\hfill & (41)\hfill \end{array}

*Remark 3:* The message of the cognitive user (*m*_{3}) can be decoded at Rx1, under condition (34) and (*m*_{0}, *m*_{1}, *m*_{2}) can be decoded at Rx3 under conditions (35)-(37). Hence, the bound in (38) gives the capacity of a point-to-point channel with message *m*_{3} with side-information *X*_{1}, *X*_{2} at the receiver. Moreover, (38)-(41) with condition (34) give the capacity region for a three-user MAC with common information where *R*_{1} and *R*_{2} are the common rates, *R*_{3} is the private rate for Tx3, and the private rates for Tx1 and Tx2 are zero.

*Remark 4:* If we omit Tx2, i.e., {X}_{2}=\varnothing, and Tx2 has no message to transmit, i.e., *R*_{2} = 0, the model reduces to a CIFC, and {\mathcal{C}}_{1}^{\mathsf{\text{str}}} coincides with the capacity region of the strong interference channel with unidirectional cooperation (or CIFC), which was characterized in [8, Theorem 5]. It is noted that in this case, the common message can be ignored, i.e., T=\varnothing and *R*_{0} = 0.

*Proof:* __Achievability:__ Considering (35)-(37), the proof follows from Theorem 3.

__Converse:__ Consider a \left({2}^{n{R}_{0}},{2}^{n{R}_{1}},{2}^{n{R}_{2}},{2}^{n{R}_{3}},n\right) code with an average error probability of {P}_{e}^{n}\to 0. Define the following RV for *i* = 1, ..., *n*:

It is noted that due to the encoding functions *f*_{1}, *f*_{2} and *f*_{3}, defined in Definition 1, the independence of messages, and the above definitions for *T*^{n}, RVs satisfy the p.m.f (18) of Theorem 6. First, we provide a useful lemma which we need in the proof of the converse part.

*Lemma 1:* If (34) holds for all distributions that factor as (18), then

I\left({X}_{3}^{n};{Y}_{3}^{n}|{X}_{1}^{n},{X}_{2}^{n},{T}^{n},U\right)\le I\left({X}_{3}^{n};{Y}_{1}^{n}|{X}_{1}^{n},{X}_{2}^{n},{T}^{n},U\right).

(43)

*Proof:* The proof relies on the results in [24, Proposition 1] and [25, Lemma]. By redefining *X*_{2} = *X*_{3}, *Y*_{2} = *Y*_{3}, *X*_{1} = (*X*_{1}, *X*_{2}, T) in [8, Lemma 5], the proof follows.

Now, using Fano's inequality [23], we derive the bounds in Theorem 6. Using (23) provides:

\begin{array}{ll}\hfill n{R}_{3}-n{\delta}_{3n}& \le I\left({M}_{3};{Y}_{3}^{n}|{M}_{0},{M}_{1},{M}_{2}\right)\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{2.77695pt}{0ex}}\stackrel{\left(a\right)}{=}I\left({M}_{3},{X}_{3}^{n};{Y}_{3}^{n}|{T}^{n},{M}_{1},{M}_{2},{X}_{1}^{n},{X}_{2}^{n}\right)\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{2.77695pt}{0ex}}\stackrel{\left(b\right)}{\le}I\left({X}_{3}^{n};{Y}_{3}^{n}|{T}^{n},{X}_{1}^{n},{X}_{2}^{n}\right)\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{2.77695pt}{0ex}}\stackrel{\left(c\right)}{=}\sum _{i=1}^{n}I\left({X}_{3}^{n};{Y}_{3,i}|{T}^{n},{X}_{1}^{n},{X}_{2}^{n},{Y}_{3}^{i-1}\right)\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{2.77695pt}{0ex}}\stackrel{\left(d\right)}{\le}\sum _{i=1}^{n}I\left({X}_{3,i};{Y}_{3,i}|{X}_{1,i},{X}_{2,i},{T}_{i}\right)\phantom{\rule{2em}{0ex}}\end{array}

(44)

where (a) is due to (42) and the encoding functions *f*_{1}, *f*_{2} and *f*_{3}, defined in Definition 1, (b) follows from two facts; conditioning does not increase entropy and \left({M}_{1},{M}_{2},{M}_{3}\right)\to \left({X}_{1}^{n},{X}_{2}^{n},{X}_{3}^{n}\right)\to {Y}_{3}^{n} forms a Markov chain, (c) is obtained from the chain rule, and (d) follows from the memoryless property of the channel and the fact that conditioning does not increase entropy.

Now, applying Fano's inequality and the independence of the messages, we can bound *R*_{1} + *R*_{3} as

\begin{array}{c}n\left({R}_{1}+{R}_{3}\right)-n\left({\delta}_{1n}+{\delta}_{3n}\right)\le \\ \phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}I\left({M}_{1};{Y}_{1}^{n}|{M}_{0},{M}_{2}\right)+I\left({M}_{3};{Y}_{3}^{n}|{M}_{0},{M}_{1},{M}_{2}\right)\\ \phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}\stackrel{\left(a\right)}{=}I\left({M}_{1},{X}_{1}^{n};{Y}_{1}^{n}|{M}_{0},{M}_{2},{X}_{2}^{n}\right)\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}+I\left({M}_{3},{X}_{3}^{n};{Y}_{3}^{n}|{M}_{0},{M}_{1},{M}_{2},{X}_{1}^{n},{X}_{2}^{n}\right)\\ \phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}\stackrel{\left(b\right)}{=}I\left({M}_{1},{X}_{1}^{n};{Y}_{1}^{n}|{T}^{n},{M}_{2},{X}_{2}^{n}\right)\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}+I\left({X}_{3}^{n};{Y}_{3}^{n}|{T}^{n},{M}_{1},{M}_{2},{X}_{1}^{n},{X}_{2}^{n}\right)\\ \phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}\stackrel{\left(c\right)}{\le}I\left({M}_{1},{X}_{1}^{n};{Y}_{1}^{n}|{T}^{n},{M}_{2},{X}_{2}^{n}\right)\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}+I\left({X}_{3}^{n};{Y}_{1}^{n}|{T}^{n},{M}_{1},{M}_{2},{X}_{1}^{n},{X}_{2}^{n}\right)\\ \phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}=I\left({M}_{1},{X}_{1}^{n},{X}_{3}^{n};{Y}_{1}^{n}|{T}^{n},{M}_{2},{X}_{2}^{n}\right)\\ \phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}\stackrel{\left(d\right)}{=}\sum _{i=1}^{n}I\left({M}_{1},{X}_{1}^{n},{X}_{3}^{n};{Y}_{1,i}|{T}^{n},{M}_{2},{X}_{2}^{n},{Y}_{1}^{i-1}\right)\\ \phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}\stackrel{\left(e\right)}{\le}\sum _{i=1}^{n}I\left({X}_{1,i},{X}_{3,i};{Y}_{1,i}|{X}_{2,i},{T}_{i}\right)\end{array}

(45)

where (a) follows from encoding functions *f*_{1}, *f*_{2} and *f*_{3}, (b) follows from (42) and the fact that {M}_{3}\to \left({X}_{1}^{n},{X}_{2}^{n},{X}_{3}^{n}\right)\to {Y}_{3}^{n} forms a Markov chain, (c) is obtained from (43), (d) follows from the chain rule, and (e) follows from the memoryless property of the channel and the fact that conditioning does not increase entropy.

Applying similar steps, we can show that,

n\left({R}_{2}+{R}_{3}\right)-n\left({\delta}_{2n}+{\delta}_{3n}\right)\le \sum _{i=1}^{n}I\left({X}_{2,i},{X}_{3,i};{Y}_{1,i}|{X}_{1,i},{T}_{1}\right).

(46)

Finally, the sum-rate bound can be obtained as

\begin{array}{c}n\left({R}_{0}+{R}_{1}+{R}_{2}+{R}_{3}\right)-n\left({\delta}_{0n}+{\delta}_{1n}+{\delta}_{2n}+{\delta}_{3n}\right)\\ \phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}\le I\left({M}_{0},{M}_{1},{M}_{2};{Y}_{1}^{n}\right)+I\left({M}_{3};{Y}_{3}^{n}|{M}_{0},{M}_{1},{M}_{2}\right)\\ \phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}=I\left({M}_{0},{M}_{1},{M}_{2},{X}_{1}^{n},{X}_{2}^{n};{Y}_{1}^{n}\right)\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}+I\left({M}_{3},{X}_{3}^{n};{Y}_{3}^{n}|{M}_{0},{M}_{1},{M}_{2},{X}_{1}^{n},{X}_{2}^{n}\right)\\ \phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}\stackrel{\left(a\right)}{\le}I\left({T}^{n},{M}_{1},{M}_{2},{X}_{1}^{n},{X}_{2}^{n};{Y}_{1}^{n}\right)\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}+I\left({X}_{3}^{n};{Y}_{1}^{n}|{T}^{n},{M}_{1},{M}_{2},{X}_{1}^{n},{X}_{2}^{n}\right)\\ \phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}=I\left({T}^{n},{M}_{1},{M}_{2},{X}_{1}^{n},{X}_{2}^{n},{X}_{3}^{n};{Y}_{1}^{n}\right)\\ \phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}\stackrel{\left(b\right)}{=}I\left({T}^{n},{X}_{1}^{n},{X}_{2}^{n},{X}_{3}^{n};{Y}_{1}^{n}\right)\\ \phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}\stackrel{\left(c\right)}{=}\sum _{i=1}^{n}I\left({X}_{1,i},{X}_{2,i},{X}_{3,i};{Y}_{1,i}\right)\end{array}

(47)

where (a) follows from steps (a)-(c) in (45), (b) is due to the fact that \left({M}_{1},{M}_{2}\right)\to \left({X}_{1}^{n},{X}_{2}^{n},{X}_{3}^{n}\right)\to {Y}_{1}^{n} forms a Markov chain, and (c) follows from the memoryless property of the channel and the fact that conditioning does not increase entropy. Using a standard time-sharing argument for (44)-(47) completes the proof.

Next, we derive the second set of strong interference conditions, called *Set2*, under which the region of Theorem 3 is the capacity region. For all p.m.fs that factor as (18), *Set2* includes (34) and the following conditions:

I\left({X}_{1};{Y}_{1}|{X}_{2},T\right)\le I\left({X}_{1};{Y}_{3}|{X}_{2},T\right)

(48)

I\left({X}_{2};{Y}_{1}|{X}_{1},T\right)\le I\left({X}_{2};{Y}_{3}|{X}_{1},T\right)

(49)

I\left({X}_{1},{X}_{2};{Y}_{1}\right)\le I\left({X}_{1},{X}_{2};{Y}_{3}\right).

(50)

*Remark 5:* Similar to the condition *Set1*, under these conditions interfering signals at the receivers are strong enough that all messages can be decoded by both receivers. The first condition in (34) is equal in the two sets under which the cognitive user's message (*m*_{3}) can be decoded at Rx1. However, conditions (48)-(50) imply that the primary messages (*m*_{0}, *m*_{1}, *m*_{2}) can be decoded at Rx3 in a MAC fashion, while in *Set1*, they can be decoded along with *m*_{3}.

*Theorem 7:* The capacity region of MA-CIFC, satisfying (34) and (48)-(50), referred to as {\mathcal{C}}_{2}^{\mathsf{\text{str}}}, is given by the union of rate regions satisfying (14)-(17) over all p.m.fs that factor as (18).

*Proof:* See "Appendix B".

*Remark 6:* Similar to Remark 4, by omitting \mathsf{\text{Tx}}2\phantom{\rule{2.77695pt}{0ex}}\left(T={X}_{2}=\varnothing ,\phantom{\rule{2.77695pt}{0ex}}{R}_{0}={R}_{2}=0\right), the model reduces to a CIFC. Moreover, {\mathcal{C}}_{2} and *Set2* reduce to the capacity region and strong interference conditions which have been derived in [13] for non-causal CIFC.

*Remark 7 (Comparison of two sets of conditions):* In the strong interference conditions of *Set1*, the first condition in (34) is used in the converse part, while (35)-(37) are used to reduce the inner bound to {\mathcal{C}}_{1}^{\mathsf{\text{str}}}. However, all the conditions of *Set2* are utilized to prove the converse part. Now, we compare the conditions in these two sets. We can write (35) as

\begin{array}{c}I\left({X}_{1};{Y}_{1}|{X}_{2},T\right)+\underset{{I}_{\mathsf{\text{diff}}}}{\underset{\u23df}{\left[I\left({X}_{3};{Y}_{1}|{X}_{1},{X}_{2},T\right)-I\left({X}_{3};{Y}_{3}|{X}_{1},{X}_{2},T\right)\right]}}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\le I\left({X}_{1};{Y}_{3}|{X}_{2},T\right).\end{array}

Considering (34), it can be seen that *I*_{diff} ≥ 0. Hence, condition (35) implies condition (48), but not vice versa. Similar conclusions can be drawn for other conditions of these two sets. Therefore, *Set1* implies *Set2*, and the conditions of *Set2* are weaker compared to those of *Set1*.

### C. Multiple access-cognitive interference network (MA-CIFN)

Now, we extend the result of Theorem 6 to a network with *k* + 1 transmitters and two receivers; a *k*-user MAC as a primary network and a point-to-point channel with a cognitive transmitter. We call it Multiple Access-Cognitive Interference Network (MA-CIFN). Consider MA-CIFN in Figure 3, denoted by \left({\mathcal{X}}_{1}\times {\mathcal{X}}_{2}\times \cdot \cdot \cdot \times {\mathcal{X}}_{k}\times {\mathcal{X}}_{k+1},p\left({y}_{1}^{n},{y}_{k+1}^{n}|{x}_{1}^{n},{x}_{2}^{n},\cdot \cdot \cdot {x}_{k}^{n},{x}_{k+1}^{n}\right),{\mathcal{Y}}_{1}\times {\mathcal{Y}}_{k+1}\right), where {X}_{j}\in {\mathcal{X}}_{j} is the channel input at Transmitter *j* (Tx*j*), for j\in \left\{1,...,k+1\right\};{Y}_{1}\in {\mathcal{Y}}_{1} and {Y}_{k+1}\in {\mathcal{Y}}_{k+1} are channel outputs at the primary and cognitive receivers, respectively, and p\left({y}_{1}^{n},{y}_{k+1}^{n}|{n}_{1}^{n},{x}_{2}^{n},...,{x}_{k}^{n},{x}_{k+1}^{n}\right) is the channel transition probability distribution. In *n* channel uses, each Tx*j* desires to send a message pair *m*_{
j
}to the primary receiver where *j* ∈ {1, ..., *k*}, and Tx*k* + 1 desires to send a message *m*_{k+1}to the cognitive receiver. We ignore the common information for brevity. Definitions 1 and 2 can be simply extended to the MA-CIFN. Therefore, we state the result on the capacity region under strong interference conditions.

*Corollary 1:* The capacity region of the MA-CIFN, satisfying

I\left({X}_{k+1};{Y}_{k+1}|X\left(\left[1:k\right]\right)\right)\le I\left({X}_{k+1};{Y}_{1}|X\left(\left[1:k\right]\right)\right)

(51)

I\left({X}_{k+1},X\left(S\right);{Y}_{1}|X\left({S}^{c}\right)\right)\le I\left({X}_{k+1},X\left(S\right);{Y}_{k+1}|X\left({S}^{c}\right)\right)

(52)

for all *S* ⊆ [1: *k*] and for every *p*(*x*_{1})*p*(*x*_{2})*...p*(*x*_{
k
}) *p*(*x*_{k+ 1}*|x*_{1},*x*_{2},...,*x*_{
k
})*p*(*y*_{1},*y*_{k+1}|*x*_{1},*x*_{2},...,*x*_{
k
},*x*_{k+1}), is given by

\begin{array}{l}{\mathcal{C}}_{net}^{\text{str}}={\displaystyle \underset{p({x}_{1})p({x}_{2})\dots p({x}_{k})p({x}_{k+1}|{x}_{1},{x}_{2},\dots ,{x}_{k})}{\cup}\{}\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}({R}_{1},{R}_{2},\dots ,{R}_{k},{R}_{k+1}):{R}_{1},{R}_{2},\dots ,{R}_{k},{R}_{k+1}\ge 0\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{R}_{k+1}\le I({X}_{k+1};{Y}_{k+1}|X([1:k]))\hfill & (53)\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{R}_{k+1}+{\displaystyle \sum _{j\in S}{R}_{j}\le I({X}_{k+1},X(S);{Y}_{1}|X({S}^{c}))}\}\hfill & (54)\hfill \end{array}

for all *S* ⊆ [1: *k*], where *X*(*S*) is the ordered vector of *X*_{
j
}, *j* ∈ *S*, and *S*^{c}denotes the complement of the set *S.*

*Proof:* Following the same lines as the proof of Theorem 6, the proof is straightforward. Therefore, it is omitted for the sake of brevity.

*Remark 8:* Under condition (51), the message of the cognitive user (*m*_{k+1}) can be decoded at the primary receiver (*Y*_{1}). Also, the cognitive receiver (*Y*_{k+1}), under condition (52), can decode *m*_{
j
}; *j* ∈ {1,...,*k*} in a MAC fashion. Therefore, the bound in (53) gives the capacity of a point-to-point channel with message *m*_{k+1}with side-information *X*_{
j
}; *j* ∈ {1,..., *k*} at the cognitive receiver. Moreover, (53) and (54) with condition (51), give the capacity region for a *k +* 1-user MAC with common information at the primary receiver.