Two-way Networks: when Adaptation is Useless
\IEEEauthorblockN{Zhiyu Cheng
Natasha Devroye
zcheng3
ISIT 2012 \cite{zcheng_ISIT}
and Allerton 2012 \cite{zcheng_Allerton2012}. The work of Z. Cheng and N. Devroye was partially supported by NSF under award 1053933. The contents of this article are solely the responsibility of the authors and do not necessarily represent the official views of the NSF. } }
Most wireless communication networks are two-way, where nodes act as both sources and destinations of messages. This allows for ``adaptation'' at or ``interaction'' between the nodes -- a node's channel inputs may be functions of its message(s) and previously received signals, in contrast to feedback-free one-way channels where inputs are functions of messages only. How to best adapt, or cooperate, is key to two-way communication, rendering it complex and challenging. However, examples exist of channels where adaptation is {not} beneficial from a capacity perspective; it is known that for the point-to-point two-way modulo 2 adder and Gaussian channels, adaptation does not increase capacity. We ask whether analogous results hold for several multi-user two-way networks. We first consider deterministic two-way channel models: the binary modulo-2 addition channel and a generalization of this, and the linear deterministic channel which models Gaussian channels at high SNR. For these deterministic models we obtain the capacity region for the two-way multiple access/broadcast channel, the two-way Z channel and the two-way interference channel (under certain ``partial'' adaptation constraints in some regimes). We permit all nodes to adapt their channel inputs to past outputs (except for portions of the linear high-SNR two-way interference channel where we only permit 2 of the 4 nodes to fully adapt). However, we show that this adaptation is useless from a capacity region perspective. That is, the two-way fully or partially adaptive capacity region consists of two parallel ``one-way'' regions operating simultaneously in opposite directions, achieved by strategies where the channel inputs at each use do not adapt to previous inputs. We next consider two noisy channel models: first, the Gaussian two-way MAC/BC, where we show that adaptation can at most increase the sum-rate by $\frac{1}{2}$ bit in each direction. Next, for the two-way interference channel, partial adaptation is shown to be useless when the interference is very strong. In the strong and weak interference regimes, we show that the non-adaptive Han and Kobayashi scheme utilized in parallel in both directions achieves to within a constant gap for the symmetric rate of the fully (for some regimes) or partially (for the remaining regimes) adaptive models. The central technical contribution is the derivation of new, computable outer bounds which allow for adaptation. Inner bounds follow from known, non-adaptive achievability schemes of the corresponding one-way channel models.
Cheng_Devroye_revision_final.tex
\documentclass[12pt,onecolumn]{IEEEtran}
%\linespread{1.5}
\usepackage{graphicx}
%\usepackage{amsfonts, amsmath, amssymb, epsfig, subfigure, graphicx}
\usepackage{amssymb, amsmath, graphicx, cite, color, epsfig, subfigure}
\newtheorem{thm}{Theorem}[section]
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lem}[thm]{Lemma}

\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{definition}{Definition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{example}{Example}
\newtheorem{case}{Case}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{remark}[definition]{Remark}

%\newtheorem{theorem}{Theorem}
%\newtheorem{lemma}[theorem]{Lemma}
%\newtheorem{proposition}[theorem]{Proposition}
%\newtheorem{definition}[theorem]{Definition}
%\newtheorem{corollary}[theorem]{Corollary}
%\newtheorem{example}[theorem]{Example}
%\newtheorem{case}[theorem]{Case}
%\newtheorem{conjecture}[theorem]{Conjecture}
%\newtheorem{comment}[theorem]{Comment}
%\newtheorem{fact}[theorem]{Fact}
%\newtheorem{remark}[theorem]{Remark}

\newcommand{\nrd}[1]{\textcolor{red}{#1}}
\newcommand{\nbl}[1]{\textcolor{blue}{#1}}

\begin{document}

\title{Two-way Networks: when Adaptation is Useless}
\author{\IEEEauthorblockN{Zhiyu Cheng, Natasha Devroye\\
University of Illinois at Chicago \\
zcheng3, devroye@uic.edu
\thanks{Portions of this work appeared in Allerton 2011 \cite{zcheng_Allerton},  ISIT 2012 \cite{zcheng_ISIT}, and Allerton 2012 \cite{zcheng_Allerton2012}. The work of Z. Cheng and N. Devroye was partially supported by NSF under award 1053933. The contents of this article are solely the responsibility of the authors and do not necessarily represent the official views of the NSF. 
}
}}

\maketitle

\begin{abstract}
Most wireless communication networks are two-way, where nodes act as both sources and destinations of messages. This allows for ``adaptation'' at or ``interaction'' between the nodes --  a node's channel inputs may be  functions of its message(s) and previously received signals, in contrast to feedback-free one-way channels where inputs are functions of messages only. 
How to best  adapt, or cooperate,  is key to two-way communication, rendering it complex and challenging. 
However, examples exist of channels where adaptation is {\it not} beneficial from a capacity perspective;  it is known that for the point-to-point two-way modulo 2 adder and Gaussian channels,  adaptation does not increase capacity. We ask whether analogous results hold for several multi-user two-way networks. 

We first consider deterministic two-way channel models: the binary modulo-2 addition channel and a generalization of this, and the linear deterministic channel  which models Gaussian channels at high SNR. For these deterministic models we obtain the capacity region for the two-way multiple access/broadcast channel, the two-way Z channel and the two-way interference channel (under certain ``partial'' adaptation constraints in some regimes).  
We permit all nodes to adapt their channel inputs to past outputs (except for portions of the linear high-SNR two-way interference channel where we only permit 2 of the 4 nodes to fully adapt). However, we show that this adaptation is useless from a capacity region perspective. That is, 
the two-way fully or partially adaptive capacity region consists of two parallel ``one-way'' regions operating simultaneously in opposite directions, achieved by strategies where the channel inputs at each use do not adapt to previous inputs. 
We next consider two noisy channel models: first, the  Gaussian two-way MAC/BC, where we show that adaptation can at most increase the sum-rate by $\frac{1}{2}$ bit in each direction. Next, for the two-way interference channel, partial adaptation is shown to be useless when the interference is very strong. In the strong and weak interference regimes, we show that the non-adaptive Han and Kobayashi scheme utilized in parallel in both directions achieves to within a constant gap for the symmetric rate of the fully (for some regimes) or partially (for the remaining regimes) adaptive models.
The central technical contribution is the derivation of new, computable outer bounds which allow for adaptation. Inner bounds follow from known, non-adaptive achievability schemes of the corresponding one-way channel models. 
\end{abstract}

\section{Introduction}

Two-way communication, where users A and B wish to exchange a stream of information, is a natural form of communication of relevance in present and future wireless networks. Applications include two-way high data-rate tele-medicine over wireless broadband links, mobile video conferencing over next generation cellular networks, the synchronization of data among terminals, and communication between a base station and clients. Indeed, much of our current wireless communication is already two-way in nature, but it is not treated as such in practice. Rather, current channel coding schemes orthogonalize the two directions, rendering the two-way channel equivalent to two one-way communication links. 
While this is simple to implement, whether such non-adaptive two-way coding schemes are optimal from a capacity perspective remains an open question. 

What makes two-way communications, in which two (or more) users exchange messages over the same shared channel, challenging are the possibilities that stem from having  nodes be both  sources and destinations of messages. This permits them to adapt their channel inputs to their past received signals. Such two-way adaptation was first considered in the point-to-point two-way channel by Shannon \cite{Shannon:1961}. Shannon's inner and outer bounds \cite{Shannon:1961} are not tight in general, and a general computable\footnote{By computable we mean single-letter expression without the use of unbounded cardinality auxiliary random variables. Multi-letter formulas for the capacity of two-way channels exist, see the expressions involving directed information over code-trees of \cite{Kramer:thesis}.} formula for the capacity region of the point-to-point  two-way channel remains open. 

%\nrd{COULD CONSIDER ADDING A PICTURE OF TWO-WA MOD 2 AND TWO-WAY GAUSSIAN AND HOW THEY BREAK UP INTO PARALLEL CHANNELS.}
 \begin{figure*}[h]
\begin{center}
\includegraphics[width=12cm]{intro-new}
\caption{Examples of point-to-point two-way channels breaking up into two parallel one-way channels.} 
\label{fig:intro}
\end{center}
\end{figure*}
%Two-way communication refers to the exchange of two messages between two users. What distinguishes two-way communication from one-way communication is that nodes are both sources and destinations of messages and may both transmit and receive over the channel. This allows nodes to adapt their current channel inputs to the past received signals. Shannon first proposed and studied the point-to-point two-way channel in 1961 \cite{Shannon:1961}, where he provided inner and outer bounds which are not tight in general. While Shannon's bounds have been tightened \cite{Zhang:1986,Schalkwijk:1983, Kramer:DMN}, capacity of the two-way channel in general remains open, with the two-way (deterministic) binary multiplier channel being a notable example of a simple channel for which capacity remains elusive. 

However, encouragingly, capacity is known for several point-to-point two-way channel models where the interaction between one's own signal and that of the other user may be resolved.  For example, in the two-way modulo 2 binary adder channel where channel outputs  $Y_1 = Y_2 = X_1 \oplus  X_2$ for binary inputs $X_1,X_2$ and  $\oplus$ modulo 2 addition, the capacity region is one bit per user per channel use. Each user is able to ``undo'' the effect of the other as shown in Fig. \ref{fig:intro} (a), something that is not possible in one channel use for the  elusive binary multiplier channel with $Y_1=Y_2 = X_1 X_2$. In the binary modulo 2 adder channel, information independently flows in the $\rightarrow$ and the $\leftarrow$ ``directions''  and nodes need not interact, or adapt their current inputs to past outputs,  to achieve capacity.  In a similar fashion, the capacity of a two-way Gaussian point-to-point channel is equal to two parallel Gaussian channels as shown in Fig. \ref{fig:intro} (b), which may be achieved without the use of adaptation at the nodes \cite{Han:1984}. 

{\bf A note on terminology.} In this work, ``adaptation'' or ``interaction'' is said to take place when the next channel  input of a node is a non-trivial function of that node's past received signals. One may alternatively use the terms ``feedback'' or ``cooperation'' instead of adaptation or interaction. However, we feel that ``adaptation'' and ``interaction'' better highlights the nature of {\it two-way} communications where there is no real notion of feedback (which suggests backwards links which serve to aid communication in the forward direction) as all links may carry information for forward and backwards directions simultaneously. 
%That is, the term  ``feedback'' suggests that the backwards or cooperative links serve to improve communication in the forward direction, while in two-way channels, % there are two directions of communication and what is a feedback link for one direction is the desired or data link for the other. In fact, 
%each link can carry a combination of forward and backward traveling messages. 
``Cooperation'' may also be used and more accurately reflects the fact that nodes may help each other in multi-user two-way channels, but has been 
%. However, again -- cooperation is a term which has been 
used in many existing one-way communication scenarios. We feel fresh terminology to emphasize the fact that all nodes may adapt their transmission to each other is useful, as it does not contain any notion of directionality. 

%One may equivalently use the term ``cooperation'', though we feel ``adaptation'' emphasizes the fact that a node can adapt current inputs to past outputs.) 


%However, capacity is known for several channel models in which, essentially, the confusion between signals caused by the two-way nature of the problem, may be resolved by each receiver. For example, if we consider a two-way deterministic modulo 2 adder channel (the channel outputs $Y=Y_1=Y_2$ are all equal to the modulo 2 sum of the binary channel inputs, $X_1\oplus X_2$), we easily see that the capacity region is one bit per channel use per user, resulting in two parallel noise-free point-to-point channels. In essence, here one user's own signal does not affect reception of the other user's signal and both the $\rightarrow$ and the $\leftarrow$ ``directions'' may simultaneously carry information.  In this case, allowing for adaptation at the transmitters does not effect the capacity. In a similar fashion, it was shown that the capacity of a Gaussian two-way point-to-point channel is equal to two parallel Gaussian channels, which may be achieved without the use of adaptation at the nodes \cite{Han:1984}. 
%%\cite{Cadambe2008J}

\subsection{Contributions} We seek examples of  multi-user two-way channels rather than point-to-point two-way channels where, even though nodes may adapt current inputs to past outputs, this is {\it not} beneficial from a capacity region perspective. In two-way networks, one may expect adaptation to, in general, be useful and enlarge the capacity region. For example, in multi-user Gaussian channels one may  intuitively expect adaptation to allow for correlation between channel inputs which may translate to coherent gains, or allow for routing messages along different paths.  However, as we will see, there exist multi-user channels for which adaptation is useless.  In particular, we introduce three two-way channel models: 
\begin{enumerate}
\item the {\bf two-way Multiple Access / Broadcast  channel (MAC/BC)} in which there are 4 messages and 3 terminals forming a MAC channel in the $\rightarrow$ direction (2 messages) and a BC channel in the opposite $\leftarrow$ direction (2 messages); 

\item the {\bf two-way Z channel} in which there are 6 messages and  4 terminals forming a Z channel in the $\rightarrow$ direction (3 messages) and another Z channel in the opposite $\leftarrow$ direction (3 messages); 

\item the {\bf two-way interference channel (IC)} with 4 messages and 4 terminals forming an IC in the $\rightarrow$ direction (2 messages) and another IC in the $\leftarrow$ direction (2 messages).
\end{enumerate}

We emphasize that all nodes are permitted to adapt, i.e. channel inputs at node $j$ at time $i$ may be functions of the received signals at node $j$ from times 1 to $i-1$, and that data and ``feedback'' share the same links, i.e. there are no orthogonal feedback links. % anywhere in these networks. 
%For these channel models, 
Our central {contributions}  are the derivation of the exact, computable, or approximate (to within a constant gap) capacity region of several two-way networks in which adaptation is useless (or leads to bounded gaps) from a capacity perspective. Typically two-way problems/networks result in multi-letter expressions or auxiliary random variables; our results do not.  %may be summarized as follows:

\smallskip
\noindent $\bullet$ We consider {\bf deterministic binary modulo 2 adder channels} for each of the three above channel models. These are the simplest examples of multi-user two-way channels where one might intuitively expect adaptation to be useless. 
%All inputs and outputs are binary, there is no noise,  and the signals are added modulo 2. 
For these channel models, and slight generalizations thereof, we obtain outer bounds, and demonstrate that non-adaptive time-sharing schemes between nodes transmitting in the same direction achieves capacity. Nodes transmitting data in opposite directions simultaneously transmit. %may simultaneously transmit as in the point-to-point modulo 2 adder and Gaussian channel models. % and also employ non-adaptive time-sharing schemes. %, i.e. adaptation is useless.

\smallskip

\noindent $\bullet$ We next consider {\bf linear deterministic models} of the three two-way channels above which model Gaussian channels at high SNR  \cite{Avestimehr2009}
 and  again ask whether adaptation may increase the capacity regions beyond that of two parallel one-way multi-user channels in the $\rightarrow$ and $\leftarrow$ directions.  We will show that it does not for the first two channel models by obtaining their capacity regions. For the two-way interference channel, we show that {\it partial adaptation} where only two of the four nodes may adapt,  can ``block'' the two-way information flow and destroy the ability to relay / cooperate,  resulting in a capacity region equal to  two non-adaptive ICs. In addition, in some regimes of the relative link strengths, we obtain the capacity region for the symmetric model with {\it full adaptation} where all four nodes are permitted to adapt.
 % The central contribution here is the derivation of the capacity region which allows for fully (or partially for the two-way IC) adaptive channel models; typically two-way problems of this form end up having multi-letter expressions or auxiliary random variables.  %We are able to obtain capacity explicitly. 

\smallskip

\noindent $\bullet$ We next consider two noisy Gaussian networks. {First, for the {\bf Gaussian two-way MAC/BC} we demonstrate that adaptation may only increase the sum-rate in each direction by up to $\frac{1}{2}$ bit. } Next, we consider the {\bf symmetric two-way Gaussian IC} where all ``direct'' links are equal and all ``cross-over'' links are equal. We derive new, computable outer bounds for the symmetric sum-rates for this Gaussian channel model and show that: a) adaptation is useless in very strong interference for the partially adaptive model, b) in strong but not very strong interference, non-adaptive schemes perform to within 1 bit per user per direction of the fully adaptive capacity region, and c) the particular non-adaptive Han and Kobayashi scheme of \cite{etkin_tse_wang} employed in each direction, achieves to within a  constant gap (2 bits per user per direction maximally) of fully or partially adaptive outer bounds in all other regimes. 
%In general, when all nodes are permitted to adapt, we do not believe that a non-adaptive scheme will achieve to within a constant gap for all regimes but this is left open.  
We provide examples of {\bf non-symmetric Gaussian two-way ICs} where adaptation may provide unbounded gain over non-adaptation, and where perfect output feedback may provide unbounded gain over adaptation. 

The emphasis of this work is on demonstrating when {\it adaptive} schemes are useless, and when, even if adaptation is permitted, it does not significantly increase the capacity region. %\nrd{CHANGE ONCE SEE FINAL RESULTS FOR GAUSSIAN}



%We first consider several multi-user deterministic modulo 2 adder channels -- channels models which are of interest due to their simplicity and also bear relationship to the DOF of the multi-user channels \cite{cadambe2009degrees}. All inputs and outputs are binary, there is no noise,  and the signals are added modulo 2.  Specifically, we consider the two-way deterministic binary adder Multiple Access / Broadcast  channel (MAC/BC with 4 messages), the two-way deterministic modulo 2 adder interference channel (4 messages), and the two-way deterministic modulo 2 adder Z channel (6 messages). 
 %We ask whether adaptation may increase the capacity regions of these deterministic multi-user two-way modulo 2 adder  channels. We will show that it does not, and that capacity may actually be achieved by simple time-sharing schemes between nodes transmitting in the same ``direction'', while nodes in opposite directions may simultaneously transmit as in the point-to-point modulo 2 adder and Gaussian channel models. 
%In deriving outer bounds for these channel models, we use carefully chosen genies and  Markov chain structures.

%Furthermore, we introduce three  two-way multi-user linear deterministic channels which model Gaussian channels at high SNR  \cite{Avestimehr2009}. 
%Specifically, we consider the linear deterministic two-way a) Multiple Access / Broadcast  channel (MAC/BC with 4 messages), b)  Z channel (6 messages), and c) interference channel (IC, 4 messages). 
% We ask whether adaptation may increase the capacity regions beyond that of two parallel one-way multi-user channels in the $\rightarrow$ and $\leftarrow$ directions. 
% We will show that it does not for the first two channel models. For the two-way interference channel, we will show that partial adaptation where only 2 of the four nodes may adapt,  can ``block'' the two-way information flow and destroy the ability to relay / cooperate. Under this constraint, we obtain the capacity region which is equal to two non-adaptive interference channels. 
% In deriving outer bounds for these channel models, we use  carefully chosen genies and  Markov chain structures.
% 
%Finally, we consider Gaussian two-way interference channel, and show that adaptation is useless when the interference is very strong. For the strong interference and weak interference with partial adaptation, we show a constant gap result.
%\begin{figure*}
%\begin{center}
%\includegraphics[width=18cm]{channels}
%\caption{Three multi-user two-way deterministic modulo 2 adder channel models.  $M_{jk}$ denotes the message %from node $j$ to node $k$.} 
%\label{fig:model}
%\end{center}
%\end{figure*}



\subsection{Related Work} 
This work builds on: point-to-point two-way channels, one-way multi-user deterministic channels, and one-way multi-user channels with feedback. Little work exists thus far on two-way multi-user channels. 

The capacity region of the general point-to-point discrete memoryless  two-way channel may be written in terms as a limit of multi-letter expressions as in \cite[Section 15]{Shannon:1961}, or \cite[Theorem 4.1]{Kramer:thesis}. Given the complexity in computing this capacity region, it is not entirely satisfying and the capacity region of the two-way channel is  generally considered to be open. The binary multiplier channel (BMC) \cite{Dueck:1979, Schalkwijk:1982, Schalkwijk:1983, Zhang:1986, Hekstra:1989} is a nice example of a deterministic, binary, common output two-way channel where capacity is not exactly known, though its capacity may be expressed in terms of directed information as in \cite[Corollary 4.1]{Kramer:thesis}. However, the capacity regions of particular two-way channels shown in Fig. \ref{fig:intro} are known; in both examples adaptation is useless and the capacity region decomposes into two parallel one-way channels. 
These models were the inspiration for asking whether such examples exist in multi-user two-way networks. %, as mentioned in Fig. \ref{fig:intro}. %-- the binary modulo 2 adder two-way channel with inputs $X_1,X_2 \in \{0,1\}$ and outputs $Y_1=Y_2 = X_1\oplus X_2$ where $\oplus$ denotes modulo 2 addition, and the Gaussian two-way channel  \cite{Han:1984}. In both cases, the capacity region is that of two parallel one-way versions of the channel, i.e. 1 bit per channel use per direction for the modulo 2 adder channel and two parallel Gaussian channels in the forwards and backwards directions for the Gaussian two-way channel. 
%For both channel models, binary modulo 2 adder two-way channel and Gaussian two-way channel, adaptation is useless -- this inspired us to ask whether there exist multi-user two-way channels for which adaptation is useless and hence capacity may be easily determined as parallel channels in the $\rightarrow$ and $\leftarrow$ directions. 

The first of our three channel models is a two-way MAC/BC channel.   The capacity regions of the linear-deterministic one-way MAC and BC channels were obtained in \cite{bresler_tse}.  An achievable rate region and an outer bound of a similar two-way and adaptive multi-user half-duplex two-way channel is derived in  \cite{Dash:allerton,Dash:asilomar} for Gaussian and discrete memoryless channels (DMC), respectively.  In particular, the achievable rate region derived employs adaptation using Block Markov encoding, and the outer bound contains both auxiliary random variables and messages in its expression and is thus difficult to compute.  %The former \cite{Dash:allerton} considers a Gaussian channel model, while the latter \cite{Dash:asilomar} is derived for the general discrete memoryless channel. 
These works differ from our model in that we assume full-duplex operation, have 2 broadcast messages rather than a common one. %, and we will be considering deterministic channel models rather than general DMC and Gaussian models. 
Other than \cite{Dash:allerton,Dash:asilomar}, the two-way MAC/BC has not been considered, and bears most resemblance to a combined MAC channel with feedback and BC channel with feedback (see references in \cite[Ch. 17, Bibliographic Notes]{ElGamalKim:book}, and in particular \cite{willems_FB, elgamal_deg_BC}), though we note that in our two-way model there are no ``free'' feedback links -- any feedback must travel over the same links as the data in the opposite direction, and hence the MAC and BC with feedback results are not directly applicable. 



The second channel model we consider is the two-way Z channel, with 6 messages. The one-way Z channel (with 3 messages, rather than the Z Interference channel with 2 messages) was first studied in  \cite{Vishwanath2003Z}, 
 in which a general outer bound, and a matching inner bound for a special class of degraded Z channels are obtained. 
The capacity region of the one-way deterministic Z channel with invertibility constraints similar in flavor to those in \cite{elgamal_det_IC},   is found in \cite{VCadambe2009Z},   which will be of use here.  

The last channel model considered is the two-way linear deterministic IC in which there are 4 messages and 4 terminals forming ICs in the $\rightarrow$ and $\leftarrow$ directions. The capacity region of the one-way modulo 2 adder  IC is known  \cite{ElGamalKim:book} and is a special example of a more general class of deterministic IC for which capacity is known \cite{elgamal_det_IC}, including the one-way linear deterministic IC \cite{bresler_tse}.  
The work here is also related to one-way ICs with perfect output feedback \cite{sahai2009channel, Changho2010}, with rate-limited feedback \cite{Vahid-IC-rate-FB}, with generalized feedback \cite{tuninetti2010outer}, and interfering feedback \cite{sahai2009channel, Suh:ISIT2012}. In all these channel models only two messages are present and the ``feedback'' links, whether perfect, noisy, or interfering still serve only to further rates in the forward direction. The tradeoff between sending new information versus feedback on each of the links is not addressed.  The only other example of such a 4-message two-way interference channel besides our prior work \cite{zcheng_Allerton, zcheng_CISS, zcheng_ISIT, zcheng_Allerton2012} is in Section  VI of \cite{Suh:ISIT2012, Suh2012}, where an example of a linear deterministic scheme is  provided which shows that, at least for one particular asymmetric linear deterministic two-way IC  in weak  interference in the $\rightarrow$ and strong interference in the $\leftarrow$ direction, that adaptation can significantly improve the capacity region over non-interaction. The general capacity region of the linear deterministic two-way IC (with 4 messages)  remains open in general despite the example in \cite{Suh:ISIT2012, Suh2012} and the progress made here. 
%However, in this work we will show capacity for {\it symmetric} channels with partial adaptation and full adaptation in certain regimes. 
One final word on terminology: we will refer to the 4 message two-way IC as the ``two-way IC'' and the 2 message channel of \cite{Suh2012, Suh:ISIT2012} -- considered in all sections but Section VI --  as the ``two-way interference channel with interfering feedback'' to emphasize that the rates are still flowing in one direction only. % (while \cite{Suh2012} calls this 2 message channel a ``two-way interference channel''). 
Further comparisons and relationships with ICs with/without feedback \cite{Sahai:2009, Changho2010, Suh2012, Suh:ISIT2012, etkin_tse_wang} will be made in Section \ref{IC} and \ref{ICG}.

%The work of Gaussian IC with a deterministic view, IC with feedback, channel output feedback, rate-limited feedback, and interfering feedback are shown in \cite{Changho2010}, \cite{bresler_tse}, \cite{Sahai:2009}, \cite{Vahid-IC-rate-FB}, and \cite{Suh2012} respectively.

%The two-way versions of the binary modulo 2 MAC/BC, Z and IC were first considered in \cite{zcheng_Allerton}, where adaptation was shown to be useless -- time-sharing achieved capacity. Here we extend the limited results on multi-user two-way channels to the more general class of linear deterministic channel models. 
%The results here are also related to the "generalized" degrees of freedom, as shown for example for the interference channel in \cite{bresler_tse, etkin_tse_wang}. %, where we gain insight into when adaptation may be useful/useless. 
%where an outer bound region for the general discrete memoryless model is proposed. The two-way interference channel bears some similarities with interference channels with feedback and generalized feedback \cite{Changho2010, yang:IFCGF, tuninetti2007interference}.
%As we see, little work has emerged on multi-user two-way channel models, except a class of simple modulo 2 adder channel models considered in \cite{zcheng_Allerton}. In this paper, we study the linear deterministic channel models. 

\subsection{Outline}
Channel models are first introduced  in Section \ref{model}.  In Section \ref{MACBC} we consider the two-way MAC/BC channel and show that adaptation is useless for three deterministic channel models: 1) the binary modulo 2 adder channel, 2) a generalization of this which we term the ``deterministic, invertible and cardinality constrained'' model, and finally 3) the linear deterministic channel. 
%In all three channels time-sharing is shown to achieve capacity.
 In Section \ref{Z} we again show that adaptation is useless for the same three deterministic models as in the MAC/BC (but now for the two-way Z channel). %two-way Z channels. %: the 1) binary modulo 2 adder channel, 2) its generalization which we term the linear cardinality constrained model, and the 3) linear deterministic model. 
In Section \ref{IC} we move on to the deterministic IC. For the 1) binary modulo 2 channel we show that adaptation is useless, and show a similar result for its generalization 2) the ``deterministic, invertible and cardinality constrained''  model. For the 3) linear deterministic model we show that adaptation is useless for $2/3<\alpha$ (where $\alpha$ denotes the ratio of cross-over to direct links, as in \cite{etkin_tse_wang}), and show that partial adaptation is useless for the remaining channel conditions. We also obtain the general asymmetric capacity region for the linear deterministic channel model under partial adaptation, which is equal to two parallel ICs in opposite directions.  {In Section \ref{MACBCG} we show that adaptation may only increase the sum-capacity of the two-way Gaussian MAC/BC up to $\frac{1}{2}$ bit per direction. }
Finally, in Section \ref{ICG} we consider the Gaussian two-way IC and show that a non-adaptive scheme achieves within a constant gap (and in one case capacity) of any partially (sometimes fully) adaptive scheme. %, and sometimes a fully adaptive scheme. %\nrd{Change these last sentences once fix up Gaussian cases!}
We conclude in Section \ref{conclusion} with some general observations and intuition as to when adaptation is useful, which may be extracted from these examples of two-way multi-user channel models. 

\section{Models, Definitions and Notations}
\label{model}


We consider three multi-user two-way channels, where all nodes act as both transmitters  (encoders) and receivers (decoders), as  shown in Fig. \ref{fig:channels-journal}, and described by:
%. All nodes act as both transmitters 
%(encoders) and receivers (decoders), and we illustrate two specific channel models and per-channel use input-output equations that we will consider: the binary modulo 2 adder and the linear deterministic channel versions of all models.  
%To summarize, we consider the following three models:




\smallskip
\noindent $\bullet$ {\it the  two-way MAC/BC channel:} transmitters 1 and 3 send independent messages $M_{12}$ and $M_{32}$ to receiver 2, respectively, forming a MAC in the $\rightarrow$ direction. Transmitter 2 sends independent messages $M_{21}$ and $M_{23}$ to receivers 1 and 3, respectively, forming a BC in the $\leftarrow$ direction.

\smallskip
\noindent $\bullet$ {\it the two-way Z  channel:}  transmitters 1 and 4 send messages $M_{12}$ and $M_{43}$  to receivers 2 and 3 respectively. Transmitters 2 and 3 send messages $(M_{21},M_{23})$ and $(M_{32},M_{34})$ to receivers 1,3 and 2,4 respectively. %We thus have two Z channels in opposite directions. %This channel model contains interference, multiple-access, broadcast and two-way features.

\smallskip
\noindent $\bullet$ {\it  the two-way interference channel:} transmitters 1 and 3 send messages $M_{12}$ and $M_{34}$ to receivers 2 and 4, respectively, forming an IC in the $\rightarrow$ direction. Similarly,  transmitters 2 and 4 send messages $M_{21}$ and $M_{43}$ to receivers 1 and 3 respectively, forming another IC in the $\leftarrow$ direction.

For each of these models, let $M_{jk}$ denote the message from node $j$ to node $k$; all messages are independent and uniformly distributed over $\mathcal{M}_{jk} : =\{1,2,\cdots 2^{nR_{jk}}\}$, where the ranges of $j,k$ depend on the channel model (all subsets of $\{1,2,3,4\}$) and $R_{jk}$ is the rate of transmission from node $j$ to node $k$. For example, in the MAC/BC $R_{12}$ is the rate of message $M_{12}$ but $R_{13}$ and $M_{13}$ do not exist. 

 All channels are assumed to be memoryless and at each channel use, described by the input/output relationships in Fig. \ref{fig:channels-journal}. 
Let $X_j$ and $Y_k$ denote the channel input of node $j$ and output at node $k$ used to describe the model (per channel use). Let $X_{j,i} \; (Y_{j,i})$ denote the channel input (output) at node $j$ at channel use $i$, and $X_j^n : = (X_{1,1}, X_{1,2}, \cdots X_{1,n})$. Let $[x]^+ = \max(0,x)$. 
For the binary modulo 2 adder channels the input and output alphabets are $\{0,1\}$, and $\oplus$ denotes modulo 2 addition.
For the linear deterministic models, the channel inputs and outputs are binary vectors, and all addition will be bit-wise and modulo 2. We furthermore let $S$ denote an $N\times N$ lower shift matrix, where $N = \max(n_{jk})$ over the relevant $j,k$ for each channel model, and $n_{jk}$ defines the number of signal bit levels from transmitter $j$ to receiver $k$.  
We will also consider two other types of channel models: the ``deterministic, invertible and cardinality constrained''  deterministic channel models and the Gaussian two-way MAC/BC and interference channels. We will define those channel models in the appropriate sections. 

A node $j$ is said to employ {\it adaptation} or {\it interaction} if  the channel input at time $i$ is a function of the previously received outputs, 
\begin{equation}X_{j,i}=f_j(M_{j{\cal K}}, Y_j^{i-1}),\label{eq:adapt}\end{equation}
 where $f_j$ ($j\in \{1,2,3,4\}$) are deterministic functions, and $M_{j{\cal K}} := \{M_{jk}| k\in {\cal K}\}$ are the (sets of) messages from node $j$ to all the nodes in ${\cal K} \subset \{1,2,3,4\}$ where ${\cal K}$ depends on the channel model, and may be obtained from Fig. \ref{fig:channels-journal}. 
If a node behaves in a {\it non-adaptive} or {\it restricted} fashion then its inputs are functions of its messages only, i.e. $X_{j,i} = f_j(M_{jk})$. If some nodes adapt while the others do not, we refer to this as {\it partial adaptation}, and will specify which nodes adapt. 
Thus, unless otherwise noted, at each time step $0\leq i \leq n$, for $n$ the blocklength,  encoding functions are specified by the mappings $X_{j,i}(M_{j{\cal K}}, Y_j^{i-1})$. Receiver $k$ uses a decoding function $g_k:\mathcal{Y}_k^n \times \mathcal{M}_{k{\cal I}} \rightarrow\mathcal{\widehat{M}}_{{\cal J}k}$ to obtain estimates of all transmitted messages destined to received $k$, $\mathcal{\widehat{M}}_{{\cal J}k} := \{\widehat{M}_{jk}| j\in {\cal J}, {\cal J}\subset \{1,2,3,4\}\}$ depending on the model,  given knowledge of its own message(s) $M_{k{\cal I}}$ for suitable ${\cal I}\subset\{1,2,3,4\}$, which again depends on model. The capacity region of each channel model is the closure of the set of rate tuples for which there exist encoding and decoding functions (of the appropriate rates) which simultaneously drive the probability that any of the estimated messages is not equal to the true message, to zero as $n\rightarrow \infty$. 




 \begin{figure*}
\begin{center}
\includegraphics[width=18cm]{channels-journal-new}
\caption{Three multi-user two-way channel models and two of the classes under consideration.} 
\label{fig:channels-journal}
\end{center}
\end{figure*}


%The messages $M_{jk}$ of rate $R_{jk}$ are uniformly distributed in $\{1,2,\cdots 2^{nR_{jk}}\}$ for $j,k$ in the appropriate sets depending on the channel model. 



%
%%\begin{figure}
%\begin{figure*}
%\includegraphics[width=18cm]{channels}
%\vspace{-0.3cm}
%\caption{Three two-way linear deterministic channel models under consideration.}
%\label{fig:model}
%\end{figure*}
%%\end{figure}



%For the modulo 2 adder channel models, the binary channel inputs and outputs of user $j\in \{1,2,3,4\}$ at discrete time $i$ are $X_{j,i}$ and $Y_{j,i}$, which lie in the alphabets $\mathcal{X}_j\in \{0,1\}$ and ${\cal Y}_j\in \{0,1\}$ respectively. We use ``$\oplus$'' to denote modulo 2 addition.


%For the linear deterministic models, the channel inputs and outputs are binary vectors, and all addition will be bit-wise and modulo 2. We furthermore let $S$ denote an $N\times N$ lower shift matrix, where $N$ will be defined for each channel model.  In addition, we define $n_{jk}=\lfloor \log h_{jk}^2P_j\rfloor$ to indicate the number of signal bit levels from transmitter $j$ to receiver $k$, where $h_{jk}$ is the channel gains and $P_j$ denotes the power of the transmitter $j$. Receiver $k$ uses a decoding function $g_k:\mathcal{Y}_k^n\rightarrow\mathcal{\widehat{M}}_{jk}$ to obtain an estimate $\widehat{M}_{jk}$ of the transmitted message $M_{jk}$. The capacity region is the supremum over all rate tuples which simultaneously drive the probability that any of the estimated messages is not equal to the true message, to zero as $n\rightarrow \infty$. 
% Standard definitions for achievable rate regions and capacity regions are used \cite{Cover:2006}. We give the capacity regions of all three binary symmetric channel models in Fig. \ref{fig:model}. 
%We now proceed to define the channel model for each different channel and obtain its capacity region.


\section{Two-way MAC/BC}
\label{MACBC}
We first consider the 3 user, full-duplex two-way MAC/BCnetwork as shown in Fig. \ref{fig:channels-journal}(a).
% which forms a multiple access channel in one direction, and a broadcast channel in the other: a Multiple Access/Broadcast Channel (MAC/BC). 
As an introductory example, we first show that adaptation is useless for the modulo 2 adder MAC/BC and a slight generalization thereof, and capacity may be achieved via time-sharing.
%. We then proceed to a slight generalization of the modulo 2 adder channel: a class of deterministic channels for which we may show that adaptation is useless and c
%Capacity is achieved via time-sharing as in the modulo 2 adder channel. 
Finally, we  consider the linear deterministic two-way MAC/BC and show that, once again, adaptation is useless. 
 
\subsection{An Introductory Example: Modulo 2 Adder MAC/BC}

In the  two-way modulo 2 adder MAC/BC, we emphasize that all three users may employ full adaptation -- i.e. all channel inputs at time $i$ may be a function of previously received channel outputs at that node. There are no additional orthogonal, or free,  ``feedback'' links.  The capacity region may be stated as follows.

%and at each channel use is  described by
%\begin{align*}
%&Y_1=X_1\oplus X_2\\
%&Y_2=X_1\oplus X_2\oplus X_3\\
%&Y_3=X_2\oplus X_3.
%\end{align*}

%If all nodes employ adaptation, then at channel use $i$
%\begin{align*}
%& X_{1,i}=f_1(M_{12},Y_1^{i-1})\\
%& X_{2,i}=f_2(M_{21},M_{23},Y_2^{i-1})\\
%& X_{3,i}=f_3(M_{32},Y_3^{i-1}). 
%\end{align*}
%In this case, the capacity region may be stated as follows.
\begin{theorem}
\label{thm:3user}
The capacity region of the two-way modulo 2 adder MAC/BC channel is the set of non-negative rate tuples ($R_{12},R_{32},R_{21},R_{23}$) such that
\begin{align}
& R_{12}+R_{32}\leq 1\label{3user1}\\
& R_{23}+R_{21}\leq 1.\label{3user2}
\end{align}
\end{theorem}
\begin{proof}
The outer bound follows from the cut-set bound. The inner bound follows by time-sharing as in Fig. \ref{fig:TS} {\it without adaptation}: $\alpha$ time-shares between channel inputs $X_1$ and $X_3$ for the MAC channel in the $\rightarrow$ direction, while $\beta$ time-shares between the messages $M_{21}$ and $M_{23}$ 
%(encoded as $X_2(M_{21})$ and $X_2(M_{23})$ respectively)
 in the BC in the $\leftarrow$ direction. Both directions ignore the received signals and use i.i.d. Bernoulli(1/2) codebooks.  
\end{proof}

%The outer bound may be derived in three ways 1) the cut-set bound, 2) from Fano's inequality and first principles by properly taking into account the ability of the nodes to adapt to received outputs, and 3) by providing Tx 1 and 3 with both $M_{21}, M_{23}$ and perfect channel output feedback $Y_{2,i-1}$ to be used at time $i$, the $\rightarrow$ direction may be shown to fall into the class of discrete memoryless channels with feedback for which 

%The proof may be found in Appendix \ref{MACBC:mod2}. The converse may follow by the cut-set bound, but we include an alternative derivation in which we see the role of adaptation and how it may be incorporated into the converse; similar steps are made in the more general models considered later on. 

{\begin{remark}
We note that the outer bound may alternatively be derived from either 1) Fano's inequality and first principles, taking into account the ability of the nodes to adapt (we provide one such example for the linear deterministic MAC/BC channel in Theorem \ref{thm:mac/bc} for completeness), or 2)  yet another alternative is to provide Tx 1 and 3 with both $M_{21}, M_{23}$ and perfect channel output feedback $Y_{2,i-1}$  and to provide Tx 2 with channel output feedback $Y_{1,i-1}$ and $Y_{3,i-1}$ at time $i$. Then, one can mimic the outer bound for a class of MACs with FB for 
the $\rightarrow$ direction as derived by Willems \cite{willems_FB} and the physically degraded BC with feedback of \cite{elgamal_deg_BC} for the $\leftarrow$ BC direction (which goes through without a problem for the modulo 2 adder and linear deterministic models). 
Willems' class of channels is one for which (in our notation), at least one of $H(X_1|Y_2,X_3,X_2)$ or $H(X_3|Y_2,X_1,X_2)$ is zero for all input distributions. We note that we provide Tx 1 and Tx 3 with $M_{21}, M_{23}$ in addition to the output feedback in order to be able to construct the inputs $X_{2,i}$, so that $X_2$ would be placed in the conditioning of the bounds of \cite{willems_FB}. 
%and remove these, rendering our channel no more capable than Willems' channel with perfect output feedback. 
 Note also that while the capacity of Willems' class of discrete memoryless channels with feedback is expressed in terms of an auxiliary random variable $U$ which is the result of the feedback and its ability to correlate channel inputs. In general, this would result in a larger region than the MAC channel without feedback. However, for our binary modulo 2 channel law, even with conditioning on $X_2$, and the fact that $X_1,X_2,X_3$ may all be correlated, these evaluate to the same region; adaptation is useless. 
\end{remark}
}


%\begin{proof}
%Please refer to the Appendix.
%\end{proof}
%
\begin{remark} The capacity region of Theorem \ref{thm:3user} is the same as that of a modulo 2 adder MAC and a modulo 2 adder BC channel in parallel, which do {\it not} interact. That is, the capacity of a one-way modulo 2 adder MAC is $R_{12}+R_{32}\leq 1$, while that of a one-way modulo 2 adder BC (which is actually just a BC with $Y_1=X_2=Y_3$ is $R_{23}+R_{21}\leq 1$. No adaptation is needed to achieve these regions. In fact, 
% Without adaptation, the encoding functions would become
%\begin{align*}
%& X_{1,i}=f_1(M_{12})\\
%& X_{2,i}=f_2(M_{21},M_{23})\\
%& X_{3,i}=f_3(M_{32})
%\end{align*}
%This channel model thus resembles a MAC channel simultaneously transmitting with a BC channel with restricted nodes. In this case, we are still able to achieve the capacity region of Theorem \ref{thm:3user} using time-sharing. As such, we see that allowing nodes to adapt does not increase the capacity region of this channel model, and the capacity region is that of a modulo 2 adder MAC and a modulo 2 adder BC channel in parallel. 
%%Now we can consider this channel as a combination of two classic one-way channel models: multiple-access binary symmetric channel and broadcast binary symmetric channel, whose capacity region can be easily seen as exactly the same as the region in Theorem \ref{thm:3user}. This indicates the adaptation does not effect the capacity region of the 3-user two-way binary symmetric channel.
%\end{remark}
%\begin{remark}
we notice that capacity is achieved by time-sharing amongst the data traveling in the same ``direction'' (i.e. between nodes 1 and 3, and between messages $M_{21}$ and $M_{23}$) but not between the two directions themselves. That is, transmission may take place simultaneously between the two directions, as is the case in the point-to-point modulo 2 adder and Gaussian channel models, where no time-sharing is needed between the two transmission directions $\rightarrow$ and $\leftarrow$.
\end{remark}
%First we define the Gaussian two-way MAC/BC channel as:
%\begin{align*}
%&Y_1=h_{11}X_1+ h_{21}X_2+Z_1\\
%&Y_2=h_{12}X_1+ h_{22}X_2+ h_{32}X_3+Z_2\\
%&Y_3=h_{23}X_2+ h_{33}X_3+Z_3
%\end{align*}
%where $X_j, Y_j$ and $Z_j, j\in \{1,2,3\}$ are channel inputs, outputs and Gaussian noises respectively. $h_{jk}, j,k\in \{1,2,3\}$ are channel gains from transmitter $i$ to receiver $k$. Without loss of generality, we normalize signal power and noise power to 1, i.e., $P_j=1, Z_j\sim \mathcal{N} (0,1), j\in\{1,2,3\}$.
\subsection{A more general model for deterministic MAC/BC}
Adaptation is useless for the simple modulo 2 adder MAC/BC channel and capacity is achieved using time-sharing in each direction. We ask whether there exists a larger class of channels for which this holds. We answer this positively by considering the ``deterministic, invertible and alphabet restricted'' class of two-way MAC/BCs  with:
\begin{align*}
&Y_1=F_1(X_1,X_2)\\
&Y_2=F_2(X_1,X_2,X_3)\\
&Y_3=F_3(X_2,X_3)
\end{align*}
where $F_m(),m\in \{1,2,3\}$ are deterministic functions which also satisfy
%. We assume  the following class of ``deterministic, invertible and alphabet restricted'' class of MAC/BC channels:
\begin{itemize}
\item {\bf P1:} $|{\cal X}_1| = |{\cal X}_2|= |{\cal X}_3| = |{\cal X}| = |{\cal Y}_1|= |{\cal Y}_2| = |{\cal Y}_3| = |{\cal Y}| = \kappa$ for known $\kappa \in {\mathbb N}^+$. 
%\item Let $f(x_1,x_3)$ be a function such that $\exists x_3^*$ such that $x_1$ and $y_2  =  f(x_1,x_3^*)$ are in 1-to-1 correspondence, and such that $\exists x_1^*$ such that $x_3$ and $y_2 = f(x_1^*,x_3)$ are in 1-to-1 correspondence.
\item {\bf P2:} Given $X_1$, $Y_1$ is invertible, i.e. $\exists$ a function $G_1$ s.t. $X_2=G_1(X_1,Y_1)$. Similarly, we assume $\exists G_{21}, G_{23}, G_3$: $X_1 = G_{21}(X_2,X_3,Y_2)$, $X_3 = G_{23}(X_1,X_3,Y_2)$, and $X_2 = G_3(X_3,Y_3)$. These conditions exclude two-way channels such as the binary multiplier channel.
\item {\bf P3:}  $\exists x_3^*$ such that given $X_3 = x_3^*$, $X_1,X_2$ both uniform on their alphabets implies both $Y_1$ and $Y_2$ uniform on their alphabets. Similarly, $\exists x_1^*$ such that given $X_1 = x_1^*$, $X_3,X_2$ both uniform on their alphabets implies $Y_2, Y_3$ uniform on their alphabets. This ensures we can achieve the full $\log(\kappa)$, and is true only for channels with a high degree of symmetry. 
\end{itemize}
Under these conditions (which we only claim are sufficient and not necessarily necessary), the capacity region of the deterministic MAC/BC is given in the following Theorem.
%If these conditions are met, %then clearly
%%\begin{align*}
%%R_{12}+R_{32}\leq \max H(Y_2)\leq \log (k)
%%\end{align*}
%we claim that the capacity region of this class of deterministic MAC/BC is the following:
\begin{theorem}
The capacity region of the two-way ``deterministic, invertible and alphabet restricted''  MAC/BC satisfying the conditions {\bf P1}, {\bf P2}, {\bf P3} is the set of non-negative $(R_{12}, R_{32}, R_{21}, R_{43})$ satisfying: 
\begin{align}
%&R_{12}\leq H(X_1)\\
%&R_{32}\leq H(X_3)\\
&R_{12}+R_{32}\leq \log\kappa \label{general1}\\
&R_{21}+R_{23}\leq \log\kappa, \label{general2}
\end{align}
which may be achieved via time-sharing (in each direction). 
\label{thm:generalMABC}
\end{theorem}
\begin{proof}
The outer bound follows directly from the cut set, or may be directly derived from Fano's inequality or as a result of MAC and BC channels with feedback, as in Remark 1. %Achievability is guaranteed by the ``deterministic, invertible and alphabet restricted'' conditions. 
{The restriction on the alphabet sizes condition {\bf P1} prohibits ``coherent gain'' - like phenomena in the outer bound, where correlation between user inputs may be beneficial, as in for example the Gaussian MAC channel with feedback.  
Essentially, this guarantees that the outer bound of Willems for the MAC with feedback \cite{willems_FB} results in the same region as the MAC without feedback. Condition ${\bf P2}$ also guarantees that Willems' outer bound is applicable, though for the outer bound again, the cut-set bound is sufficient. }

Our achievability scheme consists of time-sharing between user 1 and user 3 in the $\rightarrow$ (MAC) direction, while simultaneously time-sharing between sending data to user 1 and 3 in the $\leftarrow$ (BC) direction, as in 
%\nrd{NOT SURE ABOUT THIS PART: The first sum-rate bound can be achieved using time-sharing as follows.
%Let $X_2$ always be selected to be uniform over its $\kappa$ possible symbols. 
% For $\alpha\in [0,1]$ for $n\alpha$ channel uses set $X_3 = x_3^*$ 
%(hence $R_{32}=0$), in which case selecting $X_1$ to be uniform over its $\kappa$ symbols will achieve $H(Y_2) = \log\kappa$, by property \eqref{P3} (noting that $X_2$ is also uniform). For the remaining 
%$n(1-\alpha)$ channel uses set $X_1 = x_1^*$ (hence $R_{12} = 0$) and select $X_3$ to be uniform over its $\kappa$ symbols, again 
%achieving the outer bound. The sum-rate bound for the BC direction \eqref{general2} is achieved via time-sharing using conditions {\bf P2} and {\bf P3}. 
%
%
 Fig. \ref{fig:TS}. There, we see two time-sharing coefficients $0\leq \alpha, \beta\leq 1$, where $\alpha$ time-shares in the $\rightarrow$ direction and $\beta$ in the $\leftarrow$ direction. Let us consider the rates achieved in time slot (1), of duration $\alpha$ (WLOG we have taken $\alpha<\beta$). Node 1 encodes $M_{12}$ into $X_1$ uniformly distributed over the $\log(\kappa)$ input symbols; node 2 encodes $M_{21}$ into $X_2$ uniformly distributed over $\log(\kappa)$ input symbols and node 3 fixes $X_3=x_3^*$ (rate 0). We claim this scheme achieves the rates $R_{12} = R_{21} =  \alpha \log(\kappa)$, $R_{23} = R_{32} = 0$. Consider $R_{12}$: node 2 receives $Y_2=F_2(X_1,X_2,x_3^*)$. Since node 2 knows $X_2$ and knows that $X_3=x_3^*$,  by {\bf P2},  it may construct $X_1 = G_{21}(X_2,x_3^*,Y_2)$ to decode $M_{12}$. By {\bf P3} this may be done at full rate $\alpha \log(\kappa)$. Similar arguments for time slots (2) and (3) demonstrate that the rates in \eqref{general1}-\eqref{general2} are achievable. 


\begin{figure}[h]
\begin{center}
\includegraphics[width=3in]{TS}
\caption{Time-sharing based achievability for the proof of Theorem \ref{thm:generalMABC}.} 
\label{fig:TS}
\end{center}
\end{figure}


\end{proof}

One example, besides the binary modulo 2 adder channel, is the channel with input alphabets $\{0,1,\cdots \kappa-1\}$ for some $\kappa$ and $Y_1 = X_1+X_2 \mod \kappa$, $Y_2 = X_1+X_2+X_3 \mod \kappa$, and $Y_3 = X_2+X_3\mod \kappa$. %; we see that adaptation is useless and that the capacity region may be achieved by time-sharing in the forward direction, in parallel with time-sharing in the backwards direction.
%Other examples?}






\begin{remark}
We comment on restricting the cardinality of the input (which we stress, may not be necessary). This restriction was brought about by simply considering the two-way MAC/BC 
 binary adder channel (not modulo), with inputs ${\cal X}_1 = {\cal X}_3=\{0,1\} $ and outputs $Y_2= X_1+X_3$ with alphabet $\{0,1,2\}$ in the MAC direction. %\textcolor{red}{Zhiyu -- you checked this channel right and concluded that joint input distributions do enlarge the one way capacity region right? If so, please type this example up if it falls into this category of channels.} 
In this channel model, it is easy to derive inner and  outer bounds both of the form $R_{12}+R_{32}\leq H(X_1+X_3)$.  In general, one would hope, like Shannon did for the point-to-point two-way channel \cite{Shannon:1961}, to derive multi-user inner and outer bounds of the same {\it form}. However, even if one is able to do so (which may be too much to hope for in general, but may be reasonable for certain classes of deterministic models), we are left with the distributions over which these bounds are taken.  That is, back to our example, to claim that adaptation is useless, the inner bound should be taken over independent input distributions $p(x_1)p(x_3)$, while the outer bound, in general allowing for adaptation,  is taken over joint distributions $p(x_1,x_3)$ (unless one restricts the set of input distributions perhaps via dependence-balance-bound-like techniques \cite{Hekstra:1989}, an open problem). Inner and outer bounds taken over these different sets of distributions do {\it not} match for the binary adder channel with ternary output. As such, we restricted the channels to those for which a form of cooperation (or adaptation) between users cannot possibly help -- which is the case for the modulo adder channels, and as we will see, the similar, in terms of properties, linear deterministic channels. 

%In general, the main problem in two-way and feedback-type problems  is that, when we allow for full adaptation, for the non-adaptive inner bounds to match the new adaptive outer bounds, two things must occur:
%\begin{itemize}
%\item The outer bounds must be re-derived without the assumption of independent inputs, as all inputs may be functions of previously received outputs plus the associated messages at the nodes. It would then be nice if the {\it form} of the outer bounds matches the {\it form} of the non-adaptive inner bounds.
%\item Even if the {\it form} of the outer bounds may be shown to match the form of the inner bounds, in general the outer bound is taken over the union of all joint input distributions (unless we use something like the dependence-balance condition to restrict the class of input distributions, a direction which we do not pursue here). The most immediate distributions for which the outer bounds would be valid would be the fully general/joint distributions. The non-adaptive inner bounds are taken over all product (or independent) input distributions. So, the evaluation of the outer bounds for the product and joint distributions must match. We will try to find a general class of channels for which this holds in the following.
%\end{itemize}


%
%Numerically, we may see that inner and outer bounds taken over these Therefore, the numerical results for the inner bound is 1.0397, while for the outer bound is 1.0985. We can see that the inner and outer bound do not match even if they have the same expressions. 
\end{remark}

\subsection{Linear Deterministic MAC/BC}
The two-way linear deterministic MAC/BC channel is defined by the input/output equations as in Fig. \ref{fig:channels-journal}(a).
%\begin{align*}
%&Y_1=S^{N-n_{11}}X_1+ S^{N-n_{21}}X_2\\
%&Y_2=S^{N-n_{12}}X_1+ S^{N-n_{22}}X_2+ S^{N-n_{32}}X_3\\
%&Y_3=S^{N-n_{23}}X_2+ S^{N-n_{33}}X_3,
%\end{align*}
%where  $n_{jk}$ denotes the number of signal bit levels from transmitter $i$ to receiver $j$,  $N=\max (n_{jk})$ for $ j,k\in \{1,2,3\}$. %, and $S$ is the $N\times N$ shift matrix. The inputs $X_1,X_2$ and outputs $Y_1,Y_2,Y_3$ are bit-vectors, and addition is bit-wise modulo 2 addition, as in the work of Tse et. al \cite{}.  
We recall that all nodes are permitted to adapt, so that at channel use $i$, 
$ X_{1,i}=f_1(M_{12},Y_1^{i-1})$, $X_{2,i}=f_2(M_{21},M_{23}, Y_2^{i-1})$, 
and $ X_{3,i}=f_3(M_{32},Y_3^{i-1})$.
%\begin{align*}
%& X_{1,i}=f_1(M_{12},Y_1^{i-1})\\
%& X_{2,i}=f_2(M_{21},M_{23}, Y_2^{i-1})\\
%& X_{3,i}=f_3(M_{32},Y_3^{i-1}).
%\end{align*}
The capacity region may be stated as follows:

\begin{theorem}
\label{thm:mac/bc}
The capacity region of the two-way linear deterministic MAC/BC is the set of non-negative rate tuples $(R_{12},R_{32},R_{21},R_{23})$ such that
\begin{equation}  \mbox{MAC } \rightarrow \left\{ \begin{array}{l}
 R_{12}\leq n_{12}, \;\; R_{32}\leq n_{32},\\
 R_{12}+R_{32}\leq \max (n_{12},n_{32})
 \end{array}\right. \label{1}
\end{equation}
\begin{equation}  \mbox{BC } \leftarrow \left\{ \begin{array}{l}
 R_{21}\leq n_{21}, \;\;  R_{23}\leq n_{23}\\
R_{21}+R_{23}\leq \max (n_{21},n_{23}).
 \end{array}\right. \label{2}
\end{equation}%\begin{align}
%& R_{12}\leq n_{12}\label{1}, \;\; R_{32}\leq n_{32}, \;\; R_{21}\leq n_{21}, \;\;  R_{23}\leq n_{23}\\
%& R_{12}+R_{32}\leq \max (n_{12},n_{32})\label{3}\\
%& R_{21}+R_{23}\leq \max (n_{21},n_{23}).\label{4} 
%\end{align}
\end{theorem}

\begin{proof}
Achievability may be argued via \cite{Avestimehr2009} by mimicking a one-way MAC and one-way BC channel in opposite directions and noting that each user may subtract off its own transmitted signal from its received signal. 
The outer bounds may be obtained by the cut-set, or via an alternative direct proof. We include an example of this alternative (to the cut-set) outer bound proof below, out of interest and to demonstrate how adaptation may be taken into account.
%but that we derive it in an alternative way nonetheless as the technique illustrates how adaptation may be taken into account directly. 
%and leads to more general outer bounds (for two-way MAC/BC channels), which are omitted.  
%\noindent {\it Proof of bound $R_{12}\leq n_{12}$: }
\begin{align*}
& n(R_{12}-\epsilon) \leq I(M_{12};Y_2^n|M_{21},M_{23},M_{32})\\
%&= H(Y_2^n|M_{21},M_{23},M_{32})-H(Y_2^n|M_{12},M_{21},M_{23},M_{32})\\
& \overset{(a)}{\leq} \sum_{i=1}^n [H(Y_{2,i}|Y_2^{i-1},M_{21},M_{23},M_{32},X_2^i)]\\
& \overset{(b)}{=} \sum_{i=1}^n [H(Y_{2,i}|Y_2^{i-1},M_{21},M_{23},M_{32},X_2^i,X_3^i)]\\
& \leq \sum_{i=1}^n [H(S^{N-n_{12}}X_{1,i})]\leq n \, (n_{12}),
\end{align*}
where (a) follows since given $(Y_2^{i-2},M_{21},M_{23})$, we may construct $X_2^i$, which cancels out the ``self-interference'' term $X_{2,i}$ in $Y_{2,i}$. We note that the self-interference term can be always cancelled out in this way in the converse of additive models. Step (b) follows from the fact that given $M_{32}, X_2^i$, we may construct $X_3^i$. The other single rate bounds follow similarly. % have $R_{32}\leq n_{32}$.
%\noindent {\it Proof of bound \eqref{2}:}
%\begin{align*}
%& n(R_{21}-\epsilon)\\
%& \leq I(M_{21};Y_1^n|M_{12})\\
%&= H(Y_1^n|M_{12})-H(Y_1^n|M_{12},M_{21})\\
%& \leq \sum_{i=1}^n [H(Y_{1,i}|Y_1^{i-1},M_{12},X_1^i)]\\
%& \leq \sum_{i=1}^n [H(S^{N-n_{21}}X_{2,i})]\\
%&=nn_{21}
%\end{align*}
%By symmetry, we have $R_{23}\leq n_{23}$.
%\noindent {\it Proof of bound $ R_{12}+R_{32}\leq \max (n_{12},n_{32})$:}
%\noindent {\it Proof of sum-rate bounds: }
\begin{align*}
&n(R_{12}+R_{32}-\epsilon) \leq I(M_{12},M_{32};Y_2^n|M_{21},M_{23})\\
%&= H(Y_2^n|M_{21},M_{23})-H(Y_2^n|M_{12},M_{32},M_{21},M_{23})\\
& \leq \sum_{i=1}^n [H(Y_{2,i}|Y_2^{i-1},M_{21},M_{23},X_2^i)]\\
& \leq \sum_{i=1}^n [H(S^{N-n_{12}}X_{1,i}+S^{N-n_{32}}X_{3,i})]\leq n (\max (n_{12},n_{32})).
\end{align*}
We may analogously obtain the other sum-rate bound.
%We also have
%\noindent {\it Proof of bound $R_{21}+R_{23}\leq \max(n_{21},n_{23})$:}
%\begin{align*}
%& n(R_{21}+R_{23}-\epsilon) \leq I(M_{21},M_{23};Y_1^n,Y_3^n|M_{12},M_{32})\\
%& \leq H(Y_1^n,Y_3^n|M_{12},M_{32})\\
%&\leq \sum_{i=1}^n [H(Y_{1,i},Y_{3,i}|M_{12},M_{32},Y_1^{i-1},Y_3^{i-1},X_1^i,X_3^i)]\\
%& \leq \sum_{i=1}^n [H(S^{N-n_{21}}X_{2,i},S^{N-n_{23}}X_{2,i})]\leq n\max (n_{21},n_{23}).
%\end{align*}
%We are able to achieve this outer bound without adaptation by following the lines in \cite{Avestimehr2009} for one-way linear deterministic MAC and BC.
\end{proof}

\begin{remark}
Without adaptation, the channel would correspond to a MAC channel simultaneously transmitting with a BC channel with restricted nodes. 
This coincides with our outer bound with adaptation, which may furthermore be achieved in one channel use: adaptation is useless, and  the capacity region is a four dimensional region that is equivalent to the capacity region of the linear deterministic MAC and the linear deterministic BC in opposite directions.
\end{remark}

 
\section{Two-way Z Channel}
\label{Z}
We now consider the 4 user, full-duplex network as shown in Fig. \ref{fig:channels-journal}(b). The 6 message network, resembles a cascade of three two-way channels, in the shape of a Z (in each direction). Again, we first introduce the modulo 2 adder model and its generalization, and then the linear deterministic model, showing that adaptation is useless through the re-derivation of adaptive outer bounds. %for all channels.
%,  and show that adaptation is useless, mention a generalization to a slightly larger class of deterministic channels, and then show the same is true for the linear deterministic two-way Z channel.
 \subsection{An Introductory Example: Modulo 2 Adder Two-way Z Channel}
 
The two-way modulo 2 adder Z channel is discrete and memoryless, and all four users may employ full adaptation.
% and at each channel use is described by
%begin{align*}
%&Y_1=X_1\oplus X_2\\
%&Y_2=X_1\oplus X_2\oplus X_3\\
%&Y_3=X_2\oplus X_3\oplus X_4\\
%&Y_4=X_3\oplus X_4.
%\end{align*}
%If all nodes employ adaptation, then at channel use $i$
%\begin{align*}
%& X_{1,i}=f_1(M_{12},Y_1^{i-1})\\
%& X_{2,i}=f_2(M_{21},M_{23},Y_2^{i-1})\\
%& X_{3,i}=f_3(M_{32},M_{34},Y_3^{i-1})\\
%& X_{4,i}=f_4(M_{43},Y_4^{i-1}).
%\end{align*}
The capacity region of this channel is stated as follows: 
\begin{theorem}
\label{mod2thm:z}
The capacity region of the two-way modulo 2 adder Z  channel is the set of non-negative rate tuples ($R_{12},R_{21},R_{23},R_{32},R_{34},R_{43}$) such that
\begin{align}
& R_{12}+R_{32}+R_{34}\leq 1\label{z1}\\
& R_{21}+R_{23}+R_{43}\leq 1\label{z2}
\end{align}
\end{theorem}
The proof is found in Appendix \ref{mod2Z} and is not a direct consequence of the cut-set outer bound. 
\begin{remark}
\label{remark:Ztype}
We note that the proof of the sum-rate outer bound of the $Z$ channel in Theorems \ref{mod2thm:z}, \ref{thm:z}, and the sum-rate bounds of the two-way IC in Theorems \ref{mod2thm:intf}, \ref{thm:ICG}, \ref{thm:intf}, \ref{thm:outer-full} all follow the same general idea of giving an asymmetric genie to one receiver, as initially done in \cite{elgamal_det_IC} for the one-way IC, and quite similar to the $Z$ channel outer bound in \cite{VCadambe2009Z},  and in particular \cite{sahai2009channel, Changho2010} for the one-way IC with feedback (IC with FB). That is, in the $\rightarrow$ direction, we provide one of the receivers with the message of the non-desired message in the $\rightarrow$ direction (as in the IC with FB) as well as all messages of the $\leftarrow$ direction (particular to the two-way channels, as no $\leftarrow$ messages in one-way channels), and the desired signal received at the other receiver of the $\rightarrow$ direction (similar to the genies given in the IC with FB). The additional messages and receiver output (relative to one-way models) are needed to create various inputs, as may be done with less side-information in one-way models. \end{remark}

%\textcolor{blue}{I think this remark is correct, and to be specific, the sum-rate bounds refer to the triple sum-rate bounds.}
\begin{remark} We again notice that since time-sharing achieves the above region, adaptation does not enlarge the capacity region.  We again see that the messages in the $\rightarrow$ and the $\leftarrow$ directions may be simultaneously communicated, but that the messages within one direction must be time-shared. 
\end{remark}
%We first define the Gaussian two-way Z channel as:
%\begin{align*}
%&Y_1=h_{11}X_1+ h_{21}X_2+Z_1\\
%&Y_2=h_{12}X_1+ h_{22}X_2+ h_{32}X_3+Z_2\\
%&Y_3=h_{23}X_2+ h_{33}X_3+h_{43}X_4+Z_3\\
%&Y_4=h_{34}X_3+ h_{44}X_4+Z_4
%\end{align*}
%
%where $X_j, Y_j$ and $Z_j, j\in \{1,2,3,4\}$ are channel inputs, outputs and Gaussian noises respectively. $h_{jk}, j,k\in \{1,2,3,4\}$ are channel gains from transmitter $i$ to receiver $k$. Without loss of generality, we normalize signal power and noise power to 1, i.e., $P_j=1, Z_j\sim \mathcal{N} (0,1), j\in\{1,2,3,4\}$.

\subsection{A More General Model for Two-way Z Channel}
%\nrd{Considering leaving this out (there are so many conditions!) and just having a comment like the following: 
Similar to the more general ``deterministic, invertible and restricted'' class of two-way MAC/BC channels where it was shown that non-adaptive time-sharing achieves capacity, we may extend the two-way Z modulo 2 adder model to a more general class of two-way Z channels.  The converse follows along similar lines as for the modulo 2 adder channel. In terms of achieving the outer bounds  $R_{12}+R_{32}+R_{34}\leq \log\kappa$ and $R_{21}+R_{23}+R_{43}\leq \log \kappa$, one sufficient condition involves restricting the input and output alphabet sizes to be equal (eliminating some of the potential benefits of adaptation via user cooperation), as well as several symmetry constraints akin to extensions of {\bf P2} and {\bf P3}. Again, one example of such a channel model is the modulo $\kappa$ channel. We omit the full statement as it follows in a straightforward and analogous fashion to Theorems \ref{thm:generalMABC} and \ref{mod2thm:z}.

%As in the MAC/BC model, the modulo 2 two-way Z channel capacity is achieved via time-sharing, and adaptation does not increase the capacity. We ask whether this is true for a more general class of deterministic channels. 

%Consider a class of deterministic two-way Z channel, which are described by:
%\begin{align*}
%&Y_1=F_1(X_1,X_2)\\
%&Y_2=F_2(X_1,X_2,X_3)\\
%&Y_3=F_3(X_2,X_3,X_4)\\
%&Y_4=F_4(X_3,X_4)
%\end{align*}
%where $F_m(),m\in \{1,2,3,4\}$ are deterministic functions. Consider the following class of channels, which we term the ``deterministic, invertible and restricted'' class of two-way Z channels:

%\begin{itemize}
%\item {\bf P1Z:} Restrict $|{\cal X}_1| = |{\cal X}_2|= |{\cal X}_3| = |{\cal X}_3| = |{\cal Y}_1|= |{\cal Y}_2| = |{\cal Y}_3| =  |{\cal Y}_4| = \kappa$ for known $\kappa \in {\mathbb N}^+$. 
%%\item Let $f(x_1,x_3)$ be a function such that $\exists x_3^*$ such that $x_1$ and $y_2  =  f(x_1,x_3^*)$ are in 1-to-1 correspondence, and such that $\exists x_1^*$ such that $x_3$ and $y_2 = f(x_1^*,x_3)$ are in 1-to-1 correspondence.
%\item {\bf P2Z:} Given $X_1$, $Y_1$ is invertible, i.e. $\exists$ a function $G_{12}$ s.t. $X_2=G_1(X_1,Y_1)$. Similarly, we assume $\exists G_{21}, G_{23}, G_{32}, G_{34}, G_{43}$: $X_1 = G_{21}(X_2,X_3,Y_2)$, $X_3 = G_{23}(X_1,X_3,Y_2)$, $X_2 = G_{32}(X_3,X_4,Y_3)$, $X_4 = G_{34}(X_2,X_3,Y_3)$, $X_3 = G_{43}(X_4,Y_4)$. These conditions excludes two-way channels such as the binary multiplier channel along any of the two-way links in the network.
%\item {\bf P3Z:}  to ensure that we may attain the outer bound through time-sharing, we impose the following symmetric constraints: 
%\begin{itemize}
%\item $\exists x_3^*, x_4^*:$ $X_3 = x_3^*, X_4=x_4^*$ and $X_1,X_2$ uniform $\Rightarrow$ $Y_2,Y_3$ uniform
%\item $\exists x_3^*, x_4^*:$ $X_3 = x_3^*, X_4=x_4^*$ and $X_1,X_2$ uniform $\Rightarrow$ $Y_2,Y_4$ uniform
%\item $\exists x_3^*, x_2^*:$ $X_3 = x_3^*, X_2=x_2^*$ and $X_1,X_4$ uniform $\Rightarrow$ $Y_2,Y_3$ uniform
%\item $\exists x_1^*, x_2^*:$ $X_1 = x_1^*, X_2=x_2^*$ and $X_3,X_4$ uniform $\Rightarrow$ $Y_2,Y_3$ uniform
%\item $\exists x_1^*, x_2^*:$ $X_1 = x_1^*, X_2=x_2^*$ and $X_3,X_4$ uniform $\Rightarrow$ $Y_4,Y_3$ uniform
%\item $\exists x_1^*, x_4^*:$ $X_1 = x_1^*, X_4=x_4^*$ and $X_2,X_3$ uniform $\Rightarrow$ $Y_1,Y_4$ uniform
%\item $\exists x_1^*, x_4^*:$ $X_1 = x_1^*, X_4=x_4^*$ and $X_2,X_3$ uniform $\Rightarrow$ $Y_3,Y_4$ uniform
%\item $\exists x_1^*, x_4^*:$ $X_1 = x_1^*, X_4=x_4^*$ and $X_2,X_3$ uniform $\Rightarrow$ $Y_1,Y_2$ uniform
%\item $\exists x_1^*, x_4^*:$ $X_1 = x_1^*, X_4=x_4^*$ and $X_2,X_3$ uniform $\Rightarrow$ $Y_3,Y_2$ uniform
%\end{itemize}
%%$\exists x_3^*$ such that given $X_3 = x_3^*$, $X_1,X_2$ both uniform on their alphabets implies both $Y_1$ and $Y_2$ uniform on their alphabets. Similarly, $\exists x_1^*$ such that given $X_1 = x_1^*$, $X_3,X_2$ both uniform on their alphabets implies $Y_2, Y_3$ uniform on their alphabets. This ensures we can achieve the full $\log(\kappa)$. 
%\end{itemize}
%Under these conditions, the capacity region of the deterministic MAC/BC is given in the following Theorem.

%
%\begin{itemize}
%\item Restrict $|{\cal X}_1| = |{\cal X}_2|= |{\cal X}_3| = |{\cal X}_4|= |{\cal X}| = |{\cal Y}_1|= |{\cal Y}_2| = |{\cal Y}_3| = |{\cal Y}_4| =|{\cal Y}| = \kappa$. 
%\item Let $f_2(x_1,x_3)$ be a function such that $\exists x_3^*$ such that $x_1$ and $y_2  =  f_2(x_1,x_3^*)$ are in 1-to-1 correspondence, and such that $\exists x_1^*$ such that $x_3$ and $y_2 = f_2(x_1^*,x_3)$ are in 1-to-1 correspondence. Similar argument can be made for $f_3()$.\nrd{Not sure about this restriction}
%\end{itemize}
%If these conditions are met, %then clearly
%%\begin{align*}
%%R_{12}+R_{32}+R_{34}\leq \max H(Y_2)\leq \log (k)
%%\end{align*}
%we claim that the capacity region of this class of deterministic two-way Z channel is the following:
%\begin{align*}
%&R_{12}+R_{32}+R_{34}\leq \log\kappa\\
%&R_{21}+R_{23}+R_{43}\leq \log\kappa
%\end{align*}
%We only consider the first sum-rate bound due to symmetry. \nrd{(Again, not sure about this part:) It can be achieved using time-sharing as follows. For $\alpha\in [0,1]$ for $n\alpha$ channel uses set $X_3 = x_3^*$ 
%(hence $R_{32}=R_{34}=0$), in which case selecting $x_1$ to be uniform over its $\kappa$ symbols will achieve $H(Y_2) = \log\kappa$. For the remaining 
%$n(1-\alpha)$ channel uses set $X_1 = x_1^*$ (hence $R_{12} = 0$) and select $x_3$ to be uniform over its $\kappa$ symbols, again 
%achieving the outer bound, even if the input distributions are not jointly chosen. }The converse: 
%\begin{align*}
%& n(R_{12}+R_{32}+R_{34}-\epsilon)\\
%& \leq I(M_{12};Y_2^n|M_{21},M_{23},M_{43})\\
%&+I(M_{32},M_{34};Y_4^n,Y_2^n|M_{43},M_{12},M_{21},M_{23})\\
%&\leq I(M_{12};Y_2^n|M_{21},M_{23},M_{43})+I(M_{32},M_{34};Y_2^n|M_{43},M_{12}, M_{21},M_{23})+I(M_{32},M_{34};Y_4^n|M_{43},M_{12},M_{21},M_{23},Y_2^n)\\
%&\leq H(Y_2^n|M_{21},M_{23},M_{43})-H(Y_2^n|M_{12},M_{21},M_{23},M_{43})+H(Y_2^n|M_{12},M_{21},M_{23},M_{43})\\
%&+H(Y_4^n|M_{43},M_{12},M_{21},M_{23},Y_2^n)\\
%&\overset{(a)}{\leq} \sum_{i=1}^n [H(Y_{2,i})+H(Y_{4,i}|M_{12},M_{21},M_{23},M_{43},Y_4^{i-1},X_4^i,Y_2^n,X_2^n,X_1^i,X_3^i)]\\
%%& \overset{(d)}{=} \sum_{i=1}^n [H(Y_{2,i})+H(X_{3,i}\oplus X_{4,i}|M_{12},M_{21},M_{23},M_{43},Y_4^{i-1},X_4^i,X_3^{i-1},Y_2^n,X_2^n)]\\
%%& \overset{(e)}{=} \sum_{i=1}^n [H(Y_{2,i})+H(X_{3,i}|M_{12},M_{21},M_{23},M_{43},Y_4^{i-1},X_4^i,X_3^{i-1},X_1^n \oplus X_2^n\oplus X_3^n,X_2^n,X_1^n)]\\
%&=\sum_{i=1}^n [H(Y_{2,i})]\\
%&\leq n\log\kappa
%\end{align*}
%In (a),  $X_1^i$ is constructed by $(M_{12},X_2^n)$, $X_3^i$ is decoded from $Y_2^n$ because of invertibility condition, i.e. $X_3=G(X_1,X_2,Y_2)$. 
\subsection{Linear Deterministic Two-way Z Channel}

The linear deterministic two-way Z channel is defined by the input / output equations in Fig. \ref{fig:channels-journal}(b).
%, where we recall that all nodes may employ adaptation. In this case, t
The capacity region is again that of two parallel Z channels in opposite directions; adaptation is useless.
%\begin{align*}
%&Y_1=S^{N-n_{11}}X_1+ S^{N-n_{21}}X_2\\
%&Y_2=S^{N-n_{12}}X_1+ S^{N-n_{22}}X_2+ S^{N-n_{32}}X_3\\
%&Y_3=S^{N-n_{23}}X_2+ S^{N-n_{33}}X_3+ S^{N-n_{43}}X_4\\
%&Y_4=S^{N-n_{34}}X_3+S^{N-n_{44}}X_4,
%\end{align*}
%where $N=\max (n_{jk}), j,k\in \{1,2,3,4\}$. %, and $S$ is the $N\times N$ shift matrix.
%If all nodes employ adaptation, then at channel use $i$, 
%$ X_{1,i}=f_1(M_{12},Y_1^{i-1})$, $X_{2,i}=f_2(M_{21},M_{23}, Y_2^{i-1})$,
%$X_{3,i}=f_3(M_{32},M_{34},Y_3^{i-1})$,  $X_{4,i}=f_4(M_{43},Y_4^{i-1})$. 
%\begin{align*}
%& X_{1,i}=f_1(M_{12},Y_1^{i-1})\\
%& X_{2,i}=f_2(M_{21},M_{23}, Y_2^{i-1})\\
%& X_{3,i}=f_3(M_{32},M_{34},Y_3^{i-1})\\
%& X_{4,i}=f_4(M_{43},Y_4^{i-1})
%\end{align*}
%The capacity region of this channel is stated as follow:
\begin{theorem} 
\label{thm:z}
The capacity region of the two-way linear deterministic Z channel is the set of all rate-tuples ($R_{12},R_{21},R_{23},R_{32},R_{34},R_{43}$) which satisfy the following:

\begin{equation*}  \mbox{Z } \rightarrow \left\{ \begin{array}{l}
 R_{12}\leq n_{12},  \;\; R_{32}\leq n_{32}, \;\; R_{34}\leq n_{34}\\
 R_{12}+R_{32}\leq \max (n_{12},n_{32})\\
   R_{32}+R_{34}\leq \max (n_{32},n_{34})\\
   R_{12}+R_{32}+R_{34}\leq \max (n_{12},n_{32})+[n_{34}-n_{32}]^+
 \end{array}\right.
\end{equation*}
\begin{equation*}  \mbox{Z } \leftarrow \left\{ \begin{array}{l}
  R_{43}\leq n_{43}, \;\; R_{23}\leq n_{23}, \;\; R_{21}\leq n_{21}\\
R_{43}+R_{23}\leq \max (n_{43},n_{23})\\
 R_{23}+R_{21}\leq \max (n_{23},n_{21})\\
   R_{43}+R_{23}+R_{21}\leq \max (n_{43},n_{23})+[n_{21}-n_{23}]^+. \end{array}\right.
\end{equation*}

%\begin{align}
%& R_{12}\leq n_{12}\label{z1},  \;\; R_{32}\leq n_{32}, \;\; R_{34}\leq n_{34}\\
%& R_{43}\leq n_{43}, \;\; R_{23}\leq n_{23}, \;\; R_{21}\leq n_{21}\\
%& R_{12}+R_{32}\leq \max (n_{12},n_{32})\label{z4}\\
%& R_{32}+R_{34}\leq \max (n_{32},n_{34})\label{z5}\\
%& R_{43}+R_{23}\leq \max (n_{43},n_{23})\\
%& R_{23}+R_{21}\leq \max (n_{23},n_{21})\\
%& R_{12}+R_{32}+R_{34}\leq \max (n_{12},n_{32})+[n_{34}-n_{32}]^+\label{z6}\\ 
%& R_{43}+R_{23}+R_{21}\leq \max (n_{43},n_{23})+[n_{21}-n_{23}]^+
%\end{align}

%\begin{align}
%& R_{12}\leq n_{12}\label{z1},  \;\; R_{32}\leq n_{32}, \;\; R_{34}\leq n_{34}\\
%& R_{43}\leq n_{43}, \;\; R_{23}\leq n_{23}, \;\; R_{21}\leq n_{21}\\
%& R_{12}+R_{32}\leq \max (n_{12},n_{32})\label{z4}\\
%& R_{32}+R_{34}\leq \max (n_{32},n_{34})\label{z5}\\
%& R_{43}+R_{23}\leq \max (n_{43},n_{23})\\
%& R_{23}+R_{21}\leq \max (n_{23},n_{21})\\
%& R_{12}+R_{32}+R_{34}\leq \max (n_{12},n_{32})+[n_{34}-n_{32}]^+\label{z6}\\ 
%& R_{43}+R_{23}+R_{21}\leq \max (n_{43},n_{23})+[n_{21}-n_{23}]^+
%\end{align}
\end{theorem}
%Before we proceed to prove the theorem, we present the capacity of the one-way deterministic Z channel first.
%
%\begin{lemma}
%\label{lemmaz}
%\cite{VCadambe2009Z} 
% The capacity region of the deterministic Z channel is the set of all rate-tuples ($R_{12},R_{32},R_{34}$) which satisfy, for some product probability distribution $p_{X_1}(x_1)p_{X_3}(x_3)$, the following:
%\begin{align*}
%& R_{12}\leq H(Y_2|U)\\
%& R_{32}\leq H(U)\\
%& R_{34}\leq H(Y_4)\\
%& R_{32}+R_{12}\leq H(Y_2)\\
%& R_{32}+R_{34}\leq H(Y_4,U)\\
%& R_{12}+R_{32}+R_{34}\leq H(Y_2)+H(Y_4|U)
%\end{align*}
%where $U=g(X_3)$, $g$ is a deterministic function, and $H(Y_2|X_1,X_2)=H(U)$. 
%\end{lemma}

\begin{proof}
We first note that the capacity of a class of deterministic Z channels is shown in \cite[Th. 3.1]{VCadambe2009Z}. To show achievability of the above, we use the achievability scheme of \cite[Th. 3.1]{VCadambe2009Z} in each $\rightarrow$ and $\leftarrow$ direction with non-adaptive nodes.
% (adaptive may mimic non-adaptive). 
Due to the additive nature of the channel, each receiver may cancel or subtract out its own ``self-interference'' term $S^{N-n_{jj}}X_j$ from its received signal. By making the appropriate correspondences,  the above is achievable and equivalent to two one-way Z channels. 
%For the achievable scheme of two-way linear deterministic Z channel, we assume there's no adaptation at all users, which indicates the two-way Z channel can be regard as two one-way Z channels.
% Since the linear deterministic channel is a special case of the general deterministic model, and \eqref{z1},\eqref{z2},\eqref{z3},\eqref{z4},\eqref{z5} and \eqref{z6} in Theorem \ref{thm:z} corresponds to the six inequalities in Lemma \ref{lemmaz}, we can apply the same achievable scheme by following the lines in \cite{VCadambe2009Z}.

For the converse,  note that all but the triple-rate bounds may be obtained by the cut-set bound, or independently by giving the appropriate side-information or genie to the receivers (as illustrated in previous models). 
The non-cut-set triple rate bound may be obtained as follows:% but that they are left as they illustrate the impact of adaptation and how these bounds made be generalized.  By symmetry, and due to space constraints, we only show the sum-rate bounds for one direction of $Z$ channels.
%\begin{align*}
%&n(R_{12}+R_{32}-\epsilon) \leq I(M_{12},M_{32};Y_2^n|M_{21},M_{23})\\
%%&= H(Y_2^n|M_{21},M_{23})-H(Y_2^n|M_{12},M_{32},M_{21},M_{23})\\
%& \leq \sum_{i=1}^n [H(Y_{2,i}|Y_2^{i-1},M_{21},M_{23},X_2^i)]\\
%& \leq \sum_{i=1}^n [H(S^{N-n_{12}}X_{1,i}+S^{N-n_{32}}X_{3,i})]\leq n\, \max (n_{12},n_{32}).
%\end{align*}
%We also have
%%\noindent {\it Proof of bound $R_{32}+R_{34}\leq \max(n_{32},n_{34})$:} 
%\begin{align*}
%& n(R_{32}+R_{34}-\epsilon) \leq I(M_{32},M_{34};Y_2^n,Y_4^n|M_{21},M_{23},M_{43},M_{12})\\
%%& \leq H(Y_2^n,Y_4^n|M_{21},M_{23},M_{43},M_{12})\\
%&\leq \sum_{i=1}^n [H(Y_{2,i},Y_{4,i}|M_{21},M_{23},M_{43},M_{12},Y_2^{i-1},Y_4^{i-1},X_2^i,X_4^i)]\\
%&\overset{(a)}{\leq} \sum_{i=1}^n [H(Y_{2,i},Y_{4,i}|M_{21},M_{23},M_{43},M_{12},Y_2^{i-1},Y_4^{i-1},X_2^i,X_4^i,X_1^i)]\\
%& \leq \sum_{i=1}^n [H(S^{N-n_{32}}X_{3,i},S^{N-n_{34}}X_{3,i})] \leq n\, \max (n_{32},n_{34}),
%\end{align*}
%where (a) follows from a Markov chain that given $M_{12},X_2^i$, we have $X_1^i$.
%\smallskip
%\noindent {\it Proof of triple sum-rate bound:} 
\begin{align*}
& n(R_{12}+R_{32}+R_{34}-\epsilon) \leq I(M_{12};Y_2^n|M_{21},M_{23},M_{43}) +I(M_{32},M_{34};Y_2^n, Y_4^n|M_{43},M_{12},M_{21},M_{23}) \\
%&+I(M_{32},M_{34};Y_2^n|M_{43},M_{12},M_{21},M_{23})+I(M_{32},M_{34};Y_4^n|M_{43},M_{12},M_{21},M_{23},Y_2^n)\\
%&\leq I(M_{12};Y_2^n|M_{21},M_{23},M_{43})+I(M_{32},M_{34};Y_2^n|M_{43},M_{12},\\
&\leq H(Y_2^n|M_{21},M_{23},M_{43})+H(Y_4^n|M_{43},M_{12},M_{21},M_{23},Y_2^n)\\
%-H(Y_2^n|M_{12},M_{21},M_{23},M_{43})\\
%&+H(Y_2^n|M_{12},M_{21},M_{23},M_{43})+H(Y_4^n|M_{43},M_{12},M_{21},M_{23},Y_2^n)\\
&\leq \sum_{i=1}^n [H(Y_{2,i}|Y_2^{i-1},M_{21},M_{23},X_2^i)+H(Y_{4,i}|M_{12},M_{21},M_{23},M_{43},Y_4^{i-1},X_4^i,Y_2^n, X_2^n)]
%& \leq \sum_{i=1}^n [H(Y_{2,i}|X_2^i)\\
%& \ \ \ +H(S^{N-n_{34}}X_{3,i}|M_{12},M_{21},M_{23},M_{43},Y_4^{i-1},X_4^i,Y_2^n,X_2^n)]\\
\end{align*}
\begin{align*}
& \overset{(a)}{\leq} \sum_{i=1}^n [H(S^{N-n_{12}}X_{1,i}+S^{N-n_{32}}X_{3,i})\\
& \ \ +H(S^{N-n_{34}}X_{3,i}|M_{12},M_{21},M_{23},M_{43},Y_4^{i-1},X_4^i,S^{N-n_{12}}X_{1,i}+S^{N-n_{22}}X_{2,i}+S^{N-n_{32}}X_{3,i},X_2^n,X_1^n)]\\
& \leq \sum_{i=1}^n [H(S^{N-n_{12}}X_{1,i}+S^{N-n_{32}}X_{3,i}) +H(S^{N-n_{34}}X_{3,i}|S^{N-n_{32}}X_{3,i})]\\
&\leq n(\max (n_{12},n_{32})+[n_{34}-n_{32}]^+).
\end{align*}
In (a), $X_1^n$ in the second entropy term follows since given, $M_{12}$ and $X_2^n$, we may construct $X_1^n$.
\end{proof}

\begin{remark}
Again, we are always able to achieve the desired rates in Theorem \ref{thm:z} in only one channel use, therefore adaptation is useless.   The capacity region of this channel, a 6 dimensional region, is exactly equivalent to the capacity region of the two one-way linear deterministic Z channels. %As such, we see that allowing nodes to adapt does not increase the capacity region of this channel model. 
%In addition, the region in  Fig \ref{fig:hsnr} can be paraphrased to the capacity of linear deterministic interference channel given in  \cite{bresler_tse}, which is a special case of the class of more general deterministic channels whose capacity is given in \cite{elgamal_det_IC}.
 \end{remark}
 %\nrd{Where is a nice place to show where we cancel out self-interference? Want to make a remark about that. Gaussian point-to-point.}
 
\section{Deterministic Two-way Interference Channels}
\label{IC}
The last deterministic multi-user two-way network we consider is a 4 user, 4 message, full-duplex network as shown in Fig. \ref{fig:channels-journal}(c).
 This channel model merges elements of two-way, feedback,  and  interference, and forms two parallel interference channels in the $\rightarrow$ and $\leftarrow$ directions. Again, we first introduce the modulo 2 adder model of this channel and show that adaptation is useless, generalizing this to a slightly larger class of symmetric channels. This generalization is not as straightforward as for the  MAC/BC and Z channels, and hence is discussed in somewhat more depth. 
 Finally, for the symmetric linear deterministic two-way interference channel, we show that full adaptation is useless when the interference is very strong, strong, and in some of the weak regimes, while in all other regimes we show that partial adaptation is useless (i.e. if only 2 of the nodes adapt, might as well have none of the nodes adapt). 
 %In addition, we propose a more general model of deterministic two-way interference channel that adaptation is useless.
\subsection{An Introductory Example: Modulo 2 Adder Two-way IC}
We are again motivated by the two-way, modulo 2 adder IC, perhaps the simplest example of a two-way IC channel in which adaptation is useless, and capacity is achieved through time-sharing.
%The two-way modulo 2 adder interference channel is discrete and memoryless, with all four users employing full adaptation.
% and at each channel use is described by\begin{align*}
%&Y_1=X_1\oplus X_2\oplus X_4\\
%&Y_2=X_1\oplus X_2\oplus X_3\\
%&Y_3=X_2\oplus X_3\oplus X_4\\
%&Y_4=X_1\oplus X_3\oplus X_4.
%\end{align*}
%If all nodes employ adaptation, then at channel use $i$
%\begin{align*}
%& X_{1,i}=f_1(M_{12},Y_1^{i-1})\\
%& X_{2,i}=f_2(M_{21},Y_2^{i-1})\\
%& X_{3,i}=f_3(M_{34},Y_3^{i-1})\\
%& X_{4,i}=f_4(M_{43},Y_4^{i-1}).
%\end{align*}
%The capacity region of this channel is stated as follow: 
\begin{theorem}
\label{mod2thm:intf}
The capacity region of the two-way modulo 2 adder interference channel is the set of non-negative rate tules ($R_{12},R_{21},R_{34},R_{43}$) such that
\begin{align}
& R_{12}+R_{34}\leq 1\label{intf1}\\
& R_{21}+R_{43}\leq 1.\label{intf2}
\end{align}
\end{theorem}
\begin{proof}
We may achieve this region using two time-sharing random variables; one between nodes 1 and 3, and a second between nodes 2 and 4. The converse follows by the cut-set bound, or may alternatively be derived as done in the next subsection for a more general class of channels.  
%For the converse,
%
%\noindent{\it Proof of bound \eqref{intf1}:}
%\begin{align*}
%&n(R_{12}+R_{34}-\epsilon)\\
%&\overset{(a)}{\leq} I(M_{12};Y_2^n|M_{21},M_{43})+I(M_{34};Y_4^n,Y_2^n|M_{12},M_{21},M_{43})\\
%&=I(M_{12};Y_2^n|M_{21},M_{43})+I(M_{34};Y_2^n|M_{21},M_{12},M_{43})+I(M_{34};Y_4^n|M_{21},M_{12},M_{43},Y_2^n)\\
%&\overset{(b)}{\leq} I(M_{12};Y_2^n|M_{21},M_{43})+I(M_{34};Y_2^n|M_{21},M_{12},M_{43})+H(Y_4^n|M_{21},M_{12},M_{43},Y_2^n)\\
%&\overset{(c)}{=}I(M_{12};Y_2^n|M_{21},M_{43})+I(M_{34};Y_2^n|M_{21},M_{12},M_{43})\\
%&+\sum_{i=1}^n[H(X_{1,i}\oplus X_{3,i}\oplus X_{4,i}|M_{21},M_{12},M_{43},Y_4^{i-1},X_4^i,Y_2^n,X_2^n)]\\
%&\overset{(d)}{=}I(M_{12};Y_2^n|M_{21},M_{43})+I(M_{34};Y_2^n|M_{21},M_{12},M_{43})\\
%&+\sum_{i=1}^n[H(X_{1,i}\oplus X_{3,i}|M_{21},M_{12},M_{43},Y_4^{i-1},X_4^i,X_1^n\oplus X_3^n,X_2^n)]\\
%&=I(M_{12};Y_2^n|M_{21},M_{43})+I(M_{34};Y_2^n|M_{21},M_{12},M_{43})\\
%&{\leq} \sum_{i-1}^n [H(Y_{2,i})-H(Y_{2,i}|Y_2^{i-1},M_{12},M_{21},M_{43})+H(Y_{2,i}|Y_2^{i-1},M_{12},M_{21},M_{43})]\\
%&=\sum_{i=1}^n [H(Y_{2,i})]\leq n
%\end{align*}
%
%where (a) follows from Fano's inequality and the introduction of a genie $Y_2^n$ in the second mutual information term. We drop a negative entropy term in inequality (b). In (c), we construct $X_4^i=f_4(M_{43},Y_4^{i-1})$ and $X_2^n=f_2(M_{21},Y_2^{n-1})$. In (d), we cancel $X_{4,i}$ in the entropy term since we know $X_4^i$. In addition, $X_1^n\oplus X_3^n$ is decoded from $Y_2^n=X_1^n\oplus X_2^n\oplus X_3^n$ since $X_2^n$ is known. The bound \eqref{intf2} follows by symmetry.
\end{proof}


%\begin{remark} The capacity region of the two-way modulo 2 adder interference channel with full adaptation 
%is the same as the combination of two one-way modulo 2 adder interference channels. Capacity is achieved using time sharing and nodes need not adapt. Thus, adaptation is again useless in this scenario. 
%\end{remark}
%We first define the Gaussian two-way interference channel as:
%\begin{align*}
%&Y_1=h_{11}X_1+ h_{21}X_2+h_{41}X_4+Z_1\\
%&Y_2=h_{12}X_1+ h_{22}X_2+ h_{32}X_3+Z_2\\
%&Y_3=h_{23}X_2+ h_{33}X_3+h_{43}X_4+Z_3\\
%&Y_4=h_{14}X_1+h_{34}X_3+ h_{44}X_4+Z_4
%\end{align*}
%
%where $X_j, Y_j$ and $Z_j, j\in \{1,2,3,4\}$ are channel inputs, outputs and Gaussian noises respectively. $h_{jk}, j,k\in \{1,2,3,4\}$ are channel gains from transmitter $i$ to receiver $k$. Without loss of generality, we normalize signal power and noise power to 1, i.e., $P_j=1, Z_j\sim \mathcal{N} (0,1), j\in\{1,2,3,4\}$.


\subsection{Comments on a more general class of two-way deterministic ICs}
%\nrd{Natasha: to check again! Here don't need the strange uniform implies uniform -- is this only because there is no self-interference? Make a comment!}
We ask whether the above two-way modulo 2 IC results may be extended to a more general class of deterministic ICs in which adaptation is useless and capacity is achieved through time-sharing. In both the MAC/BC and Z channel models we were able to accomplish this by imposing certain cardinality, invertibility and symmetry constraints. One example of a channel in this class is the modulo-$\kappa$ (for some $\kappa$) channel. We now extend results to the two-way IC, but note that we must make two additional restrictions: 1) we do not consider ``self-interference'' (which we did in the previous two models), and 2) we impose symmetry of the outputs (common output in each direction). 
Both of these conditions are sufficient for obtaining sum-rate outer bounds equal to $\log\kappa$ in each direction (where $\kappa$ is the input/output alphabet size); whether they are necessary remains open. 

Consider a class of deterministic two-way interference channels without self-interference,
% which we would call it two-way IC with backup receivers, in the sense of the two receivers in one side are exactly the same, acting as the backup to each other, which may model some practical communications scenario that have backup receivers to enhance the system reliability. This channel model are 
described by:
\begin{align*}
&Y_1=F_\rightarrow(X_2,X_4) = Y_3  \mbox{ (there is no self-interference, symmetric channel)} \\
&Y_2=F_\leftarrow(X_1,X_3) = Y_4  \mbox{ (there is no self-interference, symmetric channel)}
\end{align*}
where $F_\rightarrow, F_\leftarrow$ are deterministic functions. Further restrict the class of channels to those with:
\begin{itemize}
\item {\bf P1IC:} $|{\cal X}_1| = |{\cal X}_2|= |{\cal X}_3| = |{\cal X}_4| = |{\cal Y}_1|= |{\cal Y}_2| = |{\cal Y}_3| =  |{\cal Y}_4| = \kappa$ for known $\kappa \in {\mathbb N}^+$. 
%\item Let $f(x_1,x_3)$ be a function such that $\exists x_3^*$ such that $x_1$ and $y_2  =  f(x_1,x_3^*)$ are in 1-to-1 correspondence, and such that $\exists x_1^*$ such that $x_3$ and $y_2 = f(x_1^*,x_3)$ are in 1-to-1 correspondence.
\item {\bf P2IC:} ``Invertibility'' constraints reminiscent of Costa and El Gamal \cite{elgamal_det_IC}.  In the notation of \cite{elgamal_det_IC}, we assume $f_1 = f_2 = F_\rightarrow$ (and similarly, in the reverse direction we have $f_1=f_2=F_\leftarrow$), and that $g_1 = g_2$ are the identity functions, i.e. $g_1(X_1) = X_1$ and $g_2(X_3) = X_3$ (and similarly for the reverse direction). 
Then we require that, given $X_1$, $Y_2$ is invertible, i.e. $\exists$ a function $G_{2}$ s.t. $X_3=G_2(X_1,Y_2)$. Similarly, we assume $\exists G_{1},G_3, G_4$: $X_4 = G_1(X_2,Y_1)$, $X_2 = G_{3}(X_4,Y_3)$, and $X_1 = G_{4}(X_3,Y_4)$. 
%These conditions excludes two-way channels such as the binary multiplier channel along any of the two-way links in the network.
\item {\bf P3IC:}  to ensure that we may attain the outer bound through time-sharing, we impose that $F_\rightarrow$ is a function such that $\exists x_3^*$ such that $X_1$ and $Y_2 = Y_4 =  F_\rightarrow(X_1,X_3=x_3^*)$ are in 1-to-1 correspondence, and $\exists  x_1^*$ such that $X_3$ and $Y_2 = Y_4 =  F_\rightarrow(X_1=x_1^*,X_3)$ are in 1-to-1 correspondence,  (and similarly for $F_\leftarrow$). %and such that $\exists x_1^*$ such that $x_3$ and $y_2 = y_4 = f(x_1^*,x_3)$ are in 1-to-1 correspondence.
\end{itemize}

% which satisfy the ``invertibility'' constraints of Costa and El Gamal \cite{elgamal_det_IC}. In the notation of \cite{elgamal_det_IC}, we assume $f_1 = f_2 = f_F$ (and similarly, in the reverse direction we have $f_1=f_2=f_R$), and that $g_1 = g_2$ are the identity functions, i.e. $g_1(X_1) = X_1$ and $g_2(X_3) = X_3$ (and similarly for the reverse direction). 

%Let us consider one direction only, i.e. the $\rightarrow$ direction in which messages $M_{12}$ (at node 1 destined for node 2) and $M_{34}$ (at node 3 destined for node 4) wish to be communicated to nodes 2 and 4 respectively. The other direction will follow analogously by symmetry. 
% We assume, as in Costa and El Gamal, that  (using our indices)
% \begin{align}
% H(Y_2|X_1)  = H(X_3), \;\; H(Y_4|X_3) = H(X_1).\label{ce1}
% \end{align}
%Since in our model $Y_2=Y_4$, we see that we furthermore may conclude that
%\begin{align}
%H(Y_4|X_1) = H(X_3), \;\; H(Y_2|X_3) = H(X_1).\label{ce2}
%\end{align}
%We ask whether constraining ourselves to this class of channels is enough to claim that ``adaptation does not help''? 

%\textcolor{green}{Here's a simple counter example to show that the condition of the same size of alphabet of the inputs and outputs is necessary:}

%\textcolor{green}{Consider the following two-way MAC/BC binary adder channel in MAC direction: $Y_2=X_1+X_3$, where $X_1,X_3$ are the channel inputs with alphabet $\mathcal{X}=[0,1]$, and $Y_2$ is the channel outputs with alphabet $\mathcal{Y}=[0,1,2]$. It is easy to get an inner bound and an outer bound both in the forms of $R_{12}+R_{32}\leq H(X_1+X_3)$, however, the inner bound is taking over independent input distribution $p(x_1)p(x_3)$, while the outer bound is taking over joint input distribution $p(x_1,x_3)$ because the two inputs are correlated. Therefore, the numerical results for the inner bound is 1.0397, while for the outer bound is 1.0985. We can see that the inner and outer bound do not match even if they have the same expressions. My intuition about why different sizes of the inputs and outputs do matter is whenever the mapping function can generate a new element or remove an element in the alphabet, then the correlation between the inputs will take effect, i.e. enlarge the one-way non-adaptive capacity.}

For this class of channels, the capacity is given by the following:
\begin{theorem}
The capacity region of the two-way ``deterministic, invertible and alphabet restricted''  IC satisfying the conditions {\bf P1IC}, {\bf P2IC}, {\bf P3IC} is the set of non-negative rates $(R_{12}, R_{34}, R_{21}, R_{43})$ satisfying: 
\begin{align}
%&R_{12}\leq H(X_1)\\
%&R_{32}\leq H(X_3)\\
&R_{12}+R_{34}\leq \log\kappa \label{general1IC}\\
&R_{21}+R_{43}\leq \log\kappa, \label{general2IC}
\end{align}
which may be achieved via time-sharing (in each direction). 
\label{thm:ICG}
\end{theorem}
\begin{proof}
%While the converse would be a consequence of the cut-set bound, we provide another direct alternative proof as it provides additional insight on how one may handle adaptation, and why we need the above conditions. 
Let us consider only the $\rightarrow$ direction for now. Under the above restrictions, the capacity region of the class of deterministic (one-way) interference channels in \cite{elgamal_det_IC} may be simplified to 
\begin{equation} R_{12}+R_{34} \leq \log\kappa  \label{eq:sum_in} \end{equation}
which may be achieved by time-sharing between the inputs $X_1$ uniform over the $\kappa$ input symbols, while $X_3=x_3^*$ and vice versa. 
That the rates \eqref{eq:sum_in} are achievable may alternatively be directly verified. 
%\begin{align}
%R_{12}&\leq H(X_1) \label{eq:sin_in}\\
%R_{34}&\leq H(X_3) \\ 
%R_{12}+R_{34}&\leq H(Y_2) = H(Y_4) \label{eq:sum_in}
%\end{align}
%taken over all input distributions $p(x_1)p(x_3)$. 

%We try to match the {\it form} of these bounds for our outer bounds. First consider single-rate bound:%\textcolor{red}{What about the single-rate bounds -- Zhiyu can you show those?} 
%%\textcolor{green}{Sure:}
%\begin{align*}
%&n(R_{12}-\epsilon)\\
%&\leq I(M_{12};Y_2^n,Y_3^n|M_{21},M_{43},M_{34})\\
%&\leq \sum_{i=1}^n [H(Y_{2,i},Y_{3,i}|Y_2^{i-1},Y_3^{i-1},M_{21},X_{2,i},M_{34},X_{3,i},M_{43},Y_4^{i-1},X_{4,i})]\\
%&\leq \sum_{i=1}^n [H(X_{1,i})]\\
%&= nH(X_1)
%\end{align*}
We find the matching outer bound:
\begin{align*}
&n(R_{12}+R_{34}-\epsilon)\\
&\leq I(M_{12};Y_2^n|M_{21},M_{43})+I(M_{34};Y_4^n,Y_2^n|M_{12},M_{21},M_{43})\\
&= I(M_{12};Y_2^n|M_{21},M_{43})+I(M_{34};Y_2^n|M_{21},M_{12},M_{43})+I(M_{34};Y_4^n|M_{21},M_{12},M_{43},Y_2^n)\\
&\overset{(a)}{\leq} \sum_{i=1}^n [H(Y_{2,i}|Y_2^{i-1},M_{21},M_{43})-H(Y_{2,i}|Y_2^{i-1},M_{12},M_{21},M_{43})+H(Y_{2,i}|Y_2^{i-1},M_{12},M_{21},M_{43})]\\
&\leq \sum_{i=1}^n [H(Y_{2.i})]\\
&= n\max_{p(x_1,x_3)} H(Y_2) \leq n \log \kappa,
\end{align*}
where in (a) we dropped two negative entropy terms, and were able to replace $Y_{4,i}$ by $Y_{2,i}$, allowing us to cancel the 2nd and 3rd terms. This is the central reason why we have restricted $Y_2=Y_4$ and $Y_1=Y_3$, whether one may somehow cancel these terms when this is not the case is open.  Restricting the alphabet size as in {\bf P1IC} yields the final inequality. %Notice however that is this symmetry is  not always necessary, as we will see in the linear deterministic two-way IC in the next section.}
%\nrd{Where do we need "no-self interference"? I though it was in (a), but is there another proof for the IC? Zhiyu: $Y_2=Y_4$ already implies no self-interference.}
%Notice that this bound is of the same form as \eqref{eq:sum_in}, but taken over joint distributions (due to adaptation) rather than a product input distribution. Thus, it seems we need to show the following to achieve capacity:
%\begin{itemize}
%\item Either we need to show outer bounds like $R_{12}\leq H(X_1)$, $R_{34}\leq H(X_3)$ or we need to show that these are redundant, and
%\item We need to restrict the class of channels to those for which 
%\begin{equation}\max_{p(x_1)p(x_3)} H(Y_2) \mbox{ (achievable) } = \max_{p(x_1,x_3)} H(Y_2) \mbox{ (outer bound)} \label{eq:eq} \end{equation}
%\end{itemize}

\end{proof}

%Here is one such sufficient class of channels -- it need not be necessary;  a more general class of channels may exist. 
%We want to place a restriction on the function $f()$ which will guarantee \eqref{eq:eq}. One restriction, beyond the ones already made is the following:
%\begin{itemize}
%\item Restrict $|{\cal X}_1| = |{\cal X}_3| = |{\cal X}| = |{\cal Y}_2| = |{\cal Y}_4| = |{\cal Y}| = k$. 
%\item Let $f(x_1,x_3)$ be a function such that $\exists x_3^*$ such that $x_1$ and $y_2 = y_4 =  f(x_1,x_3^*)$ are in 1-to-1 correspondence, and such that $\exists x_1^*$ such that $x_3$ and $y_2 = y_4 = f(x_1^*,x_3)$ are in 1-to-1 correspondence.
%\end{itemize}
%The first condition restricts the sizes of alphabets of inputs and outputs to be the same. Because intuitively, if the mapping function generates a new element or removes an element in the alphabet, which indicates the correlation between the inputs takes effect, i.e. enlarge the one-way non-adaptive capacity. The second condition assures one-to-one mapping functions so that satisfying \eqref{ce1} and \eqref{ce2}.
%If these conditions are met, then clearly
%\begin{align}
%R_{12}+R_{34}&\leq \max H(Y_2) \\
%&\leq \log(k)
%\end{align}
%and we may furthermore achieve this using time-sharing as follows. For $\alpha\in [0,1]$ for $n\alpha$ channel uses set $X_3 = x_3^*$ 
%(hence $R_{34}=0$), in which case selecting $x_1$ to be uniform over its $k$ symbols will achieve $H(Y_2) = k$. For the remaining 
%$n(1-\alpha)$ channel uses set $X_1 = x_1^*$ (hence $R_{12} = 0$) and select $x_3$ to be uniform over its $k$ symbols, again 
%achieving the outer bound, even if the input distributions are not jointly chosen. 

\begin{remark}
We have proposed a slightly more general model for deterministic two-way ICs in which adaptation is useless. However, it should be pointed out that our conditions are sufficient but by no means necessary.  For instance, consider a binary multiplier two-way interference channel described by $Y_1=Y_3=X_2X_4$ and $Y_2=Y_4=X_1X_3$, with all inputs and outputs binary. It is not difficult to show that adaptation is useless for this model and the capacity of this channel is equivalent to the capacity of two one-way binary multiplier interference channels in parallel,  the same capacity region as in Theorem \ref{mod2thm:intf}. In addition, we will show in Section \ref{ICG} that adaptation is also useless for the Gaussian two-way interference channel with partial adaptation when the two-way interference is very strong; this channel is not in the class of channels considered above either. 
\end{remark}
%In addition, let $\mathcal{Y}$ denote the alphabet of channel outputs $Y_k, k\in (1,2,3,4)$ and $\mathcal{X}$ denote the alphabet of channel inputs $X_j, j\in (1,2,3,4)$, we assume $|\mathcal{X}|=|\mathcal{Y}|=\kappa$, where $\kappa$ is a constant number. We will show that the capacity of this channel model is the following:
%

%\textcolor{red}{Find some nice examples of channels for which this applies, i.e. show that previous stuff falls into this category, or not? I think the binary adder channel does, but not sure about the linear deterministic high SNR channel. In particular, I think a non-linear channel would be interesting. The binary multiplier channel is out if we use El Gamal + Costa's requirements.}


%For the linear deterministic channel, we use $n_{jk}, j,k\in\{1,2,3,4\}$ to denote the number of signal bit levels from transmitter $i$ to receiver $j$. These values correspond to channel gains in dB scale: $n_{jk}=\lfloor \log |h_{jk}|^2\rfloor$. Then we have
\subsection{Linear Deterministic Two-way IC}
The two-way linear deterministic interference channel is defined by the input / output equations in Fig. \ref{fig:channels-journal}(c). 
In this section we will be considering the general linear deterministic IC, as well as the ``symmetric'' linear deterministic IC for which $p:=n_{12}=n_{21}=n_{34}=n_{43}$,  $q:=n_{14}=n_{41}=n_{23}=n_{32}$, and $\alpha: = q/p$. This will allow us to compare the symmetric, normalized sum capacity of various one and two-way interference channels, defined as $C_{sym}(\alpha) : = \frac{R_{12}+R_{34}}{2}$.
%\[ 
%\mbox{Symmetric linear deterministic sum-capacity = } C_{sym} := \begin{array}{ll}
%\frac{R_{12}+R_{34}}{2} & \mbox{ For one-way interference channels} \\
%\frac{R_{12}+R_{34}+R_{21}+R_{43}}{4} & \mbox{ For two-way interference channels} \\
%\end{array}
%\]

%\begin{align*}
%&Y_1=S^{N-n_{11}}X_1+ S^{N-n_{21}}X_2+S^{N-n_{41}}X_4\\
%&Y_2=S^{N-n_{12}}X_1+ S^{N-n_{22}}X_2+ S^{N-n_{32}}X_3\\
%&Y_3=S^{N-n_{23}}X_2+ S^{N-n_{33}}X_3+ S^{N-n_{43}}X_4\\
%&Y_4=S^{N-n_{14}}X_1+S^{N-n_{34}}X_3+S^{N-n_{44}}X_4,
%\end{align*}
%where $N=\max (n_{jk}), j,k\in \{1,2,3,4\}$. %, and $S$ is the $N\times N$ shift matrix.
Recall our definition of partial adaptation (nodes 1 and 3 are fixed or ``restricted'') of Section \ref{model}:
\begin{align}
& X_{1,i}=f_1(M_{12}), \;\; X_{2,i}=f_2(M_{21},Y_2^{i-1}) \label{eq:partial1}\\
& X_{3,i}=f_3(M_{34}), \;\; X_{4,i}=f_4(M_{43},Y_4^{i-1}) \label{eq:partial2}
\end{align}
%\begin{align}
%& X_{1,i}=f_1(M_{12})\label{eq:p1}\\
%& X_{2,i}=f_2(M_{21},Y_2^{i-1})\\
%& X_{3,i}=f_3(M_{34})\\
%& X_{4,i}=f_4(M_{43},Y_4^{i-1})\label{eq:p2}
%\end{align}
%The  full adaptation case is the subject of ongoing work. 
We first prove a  Lemma regarding partial adaptation, which is key in showing that partial adaptation is useless, and that the inability of {\it certain} nodes to adapt essentially ``blocks'' the ability of adaptation to help at all. 
\begin{lemma}
Under partial adaptation conditions \eqref{eq:partial1} -- \eqref{eq:partial2}, for some deterministic functions $f_5$ and $f_6$, 
\begin{align}
X_{2,i}& = f_5(M_{12}, M_{21}, M_{34}) \perp M_{43}, \;\; \forall i \label{eq:X2}\\
X_{4,i}& = f_6(M_{43}, M_{34}, M_{12}) \perp M_{21}, \;\; \forall i \label{eq:X4}
\end{align}
where $\perp$ denotes independence.
\label{lemma:partial}
\end{lemma}
\begin{proof}
Note that $X_{2,i} = f_2(M_{21}, Y_2^{i-1})$ and $Y_2^{i-1} = S^{N - n_{12}}X_{1}^{i-1} + S^{N-n_{22}} X_2^{i-1}+S^{N-n_{32}}X_3^{i-1}$. Since $X_1^{i-1}$ and $X_3^{i-1}$ are functions only of $M_{12}$ and $M_{34}$ respectively, we may conclude that  there exists a function $f^*$ such that $X_{2,i} = f^*(M_{21}, M_{12}, M_{34}, X_2^{i-1})$. Iterating this argument, and noting that $X_{2,1}$ is only a function of $M_{21}$, we obtain the theorem. The result for $X_{4,i}$ follows by a similar argument.
\end{proof}

\medskip
%In this case, the capacity region can be stated as follow:
\begin{theorem}
\label{thm:intf}
The capacity region of the two-way linear deterministic interference channel under partial adaptation constraints is the set of ($R_{12},R_{21},R_{34},R_{43}$) which satisfy the following:
%(A) and (B) in Fig. \ref{fig:ICcap}.
%\begin{equation}  \mbox{IC } \rightarrow \left\{ \begin{array}{l}
 %R_{12}\leq n_{12},   \;\; R_{34}\leq n_{34}\\
 %R_{12}+R_{34}\leq \max (n_{12},n_{32})+[n_{34}-n_{32}]^+\\
  %R_{12}+R_{34}\leq \max (n_{34},n_{14})+[n_{12}-n_{14}]^+\\
   %R_{12}+R_{34}\leq \max ([n_{12}-n_{14}]^+,n_{32})+\max ([n_{34}-n_{32}]^+,n_{14})\\
   %2R_{12}+R_{34}\leq \max (n_{12},n_{32})+[n_{12}-n_{14}]^++\max ([n_{34}-n_{32}]^+,n_{14})\\
  % R_{12}+2R_{34}\leq \max (n_{34},n_{14})+[n_{34}-n_{32}]^++\max ([n_{12}-n_{14}]^+,n_{32})\\
 %\end{array}\right. \label{IC1}
%\end{equation}
%\begin{equation}  \mbox{IC } \leftarrow \left\{ \begin{array}{l}
  %R_{21}\leq n_{21},  \;\; R_{43}\leq n_{43}\\
 %R_{21}+R_{43}\leq \max (n_{21},n_{41})+[n_{43}-n_{41}]^+\\
  %R_{21}+R_{43}\leq \max (n_{43},n_{23})+[n_{21}-n_{23}]^+\\
   %R_{21}+R_{43}\leq \max ([n_{21}-n_{23}]^+,n_{41})+\max ([n_{43}-n_{41}]^+,n_{23})\\
  % 2R_{21}+R_{43}\leq \max (n_{21},n_{41})+[n_{21}-n_{23}]^++\max ([n_{43}-n_{41}]^+,n_{23})\\
%   R_{21}+2R_{43}\leq \max (n_{43},n_{23})+[n_{43}-n_{41}]^++\max ([n_{21}-n_{23}]^+,n_{41}). \end{array}\right. \label{IC2}
%\end{equation}

\begin{align}  
 &R_{12}\leq n_{12},   \;\; R_{34}\leq n_{34}\tag{IC$\rightarrow$ a}\\
 &R_{12}+R_{34}\leq \max (n_{12},n_{32})+[n_{34}-n_{32}]^+\tag{IC$\rightarrow$ b}\\
  &R_{12}+R_{34}\leq \max (n_{34},n_{14})+[n_{12}-n_{14}]^+\tag{IC$\rightarrow$ c}\\
   &R_{12}+R_{34}\leq \max ([n_{12}-n_{14}]^+,n_{32})+\max ([n_{34}-n_{32}]^+,n_{14})\tag{IC$\rightarrow$ d}\\
   &2R_{12}+R_{34}\leq \max (n_{12},n_{32})+[n_{12}-n_{14}]^++\max ([n_{34}-n_{32}]^+,n_{14})\tag{IC$\rightarrow$ e}\\
   &R_{12}+2R_{34}\leq \max (n_{34},n_{14})+[n_{34}-n_{32}]^++\max ([n_{12}-n_{14}]^+,n_{32})\tag{IC$\rightarrow$ f}
\end{align}
\begin{align}  
 & R_{21}\leq n_{21},  \;\; R_{43}\leq n_{43}\tag{IC$\leftarrow$ a}\\
 &R_{21}+R_{43}\leq \max (n_{21},n_{41})+[n_{43}-n_{41}]^+\tag{IC$\leftarrow$ b}\\
  &R_{21}+R_{43}\leq \max (n_{43},n_{23})+[n_{21}-n_{23}]^+\tag{IC$\leftarrow$ c}\\
   &R_{21}+R_{43}\leq \max ([n_{21}-n_{23}]^+,n_{41})+\max ([n_{43}-n_{41}]^+,n_{23})\tag{IC$\leftarrow$ d}\\
   &2R_{21}+R_{43}\leq \max (n_{21},n_{41})+[n_{21}-n_{23}]^++\max ([n_{43}-n_{41}]^+,n_{23})\tag{IC$\leftarrow$ e}\\
   &R_{21}+2R_{43}\leq \max (n_{43},n_{23})+[n_{43}-n_{41}]^++\max ([n_{21}-n_{23}]^+,n_{41}). \tag{IC$\leftarrow$ f}
\end{align}
%\begin{figure*}
%{\small \begin{align}
%& R_{12}\leq n_{12}\label{intf1}, \;\; R_{34}\leq n_{34}, \\
%& R_{12}+R_{34}\leq \max (n_{12},n_{32})+[n_{34}-n_{32}]^+\nonumber\\%\label{intf5}\\ 
%& R_{12}+R_{34}\leq \max (n_{34},n_{14})+[n_{12}-n_{14}]^+\nonumber\\%\label{intf7}\\ 
%& R_{12}+R_{34}\leq \max ([n_{12}-n_{14}]^+,n_{32})+\max ([n_{34}-n_{32}]^+,n_{14})\nonumber\\%\label{intf9}\\
%& 2R_{12}+R_{34}\leq \max (n_{12},n_{32})+[n_{12}-n_{14}]^++\max ([n_{34}-n_{32}]^+,n_{14})\nonumber\\%\label{intf11}\\
%& R_{12}+2R_{34}\leq \max (n_{34},n_{14})+[n_{34}-n_{32}]^++\max ([n_{12}-n_{14}]^+,n_{32})\nonumber\\%\label{intf13}\\
%& \\
%& R_{21}\leq n_{21}, \;\; R_{43}\leq n_{43}\\
%& R_{21}+R_{43}\leq \max (n_{21},n_{41})+[n_{43}-n_{41}]^+\label{intf6}\\
%& R_{21}+R_{43}\leq \max (n_{43},n_{23})+[n_{21}-n_{23}]^+\label{intf8}\\
%& R_{21}+R_{43}\leq \max ([n_{21}-n_{23}]^+,n_{41})+\max ([n_{43}-n_{41}]^+,n_{23})\label{intf10}\\
%& 2R_{21}+R_{43}\leq \max (n_{21},n_{41})+[n_{21}-n_{23}]^++\max ([n_{43}-n_{41}]^+,n_{23})\label{intf12}\\
%& R_{21}+2R_{43}\leq \max (n_{43},n_{23})+[n_{43}-n_{41}]^++\max ([n_{21}-n_{23}]^+,n_{41})\label{intf14}
%\end{align}}
%\end{figure*}

%\begin{figure*}
%\begin{figure}
%\includegraphics[width=19cm]{ICcap}
%\vspace{-0.9cm}
%\caption{Capacity region of two-way linear deterministic interference channel with partial adaptation.}
%\label{fig:ICcap}
%\end{figure}
%\end{figure*}

%{\small \begin{align}
%& R_{12}\leq n_{12}\label{intf1}, \;\; R_{21}\leq n_{21}, \;\; R_{34}\leq n_{34}, \;\; R_{43}\leq n_{43}\\
%& R_{12}+R_{34}\leq \max (n_{12},n_{32})+[n_{34}-n_{32}]^+\label{intf5}\\ 
%& R_{21}+R_{43}\leq \max (n_{21},n_{41})+[n_{43}-n_{41}]^+\label{intf6}\\
%& R_{12}+R_{34}\leq \max (n_{34},n_{14})+[n_{12}-n_{14}]^+\label{intf7}\\ 
%& R_{21}+R_{43}\leq \max (n_{43},n_{23})+[n_{21}-n_{23}]^+\label{intf8}\\
%& R_{12}+R_{34}\leq \max ([n_{12}-n_{14}]^+,n_{32})+\max ([n_{34}-n_{32}]^+,n_{14})\label{intf9}\\
%& R_{21}+R_{43}\leq \max ([n_{21}-n_{23}]^+,n_{41})+\max ([n_{43}-n_{41}]^+,n_{23})\label{intf10}\\
%& 2R_{12}+R_{34}\leq \max (n_{12},n_{32})+[n_{12}-n_{14}]^++\max ([n_{34}-n_{32}]^+,n_{14})\label{intf11}\\
%& 2R_{21}+R_{43}\leq \max (n_{21},n_{41})+[n_{21}-n_{23}]^++\max ([n_{43}-n_{41}]^+,n_{23})\label{intf12}\\
%& R_{12}+2R_{34}\leq \max (n_{34},n_{14})+[n_{34}-n_{32}]^++\max ([n_{12}-n_{14}]^+,n_{32})\label{intf13}\\
%& R_{21}+2R_{43}\leq \max (n_{43},n_{23})+[n_{43}-n_{41}]^++\max ([n_{21}-n_{23}]^+,n_{41})\label{intf14}
%\end{align}}
\end{theorem}

\begin{proof}
For achievability, note that self-interference may be removed at each receiver due to this channel model's linearity, in which case the physical channel model reduces to two one-way IC in opposite directions. We may thus apply the well-known Han-Kobayashi scheme \cite{H+K} in each direction, ignoring the ability of the nodes to adapt, achieving the expression in (IC$\rightarrow$) and (IC$\leftarrow$).%\nrd{ZHIYU: We put equations in Fig. 3 to make it fit for ISIT, but in this journal should write it out, like in other Theorems 3 and 5 (same style to be consistent). Then we can also number the equations and refer to them in the proof in the next few pages! Please change.}

Now we prove the converse. Single-rates follow as in \eqref{1}, and using Lemma \ref{lemma:partial} (where we use partial adaptation). For the sum-rate (IC$\rightarrow$ b): %\nrd{which one? reference right equation \#}
%Due to space constraints we only focus on the sum-rates; single-rates follow as in \eqref{1}, and using Lemma \ref{block}. 
%\noindent {\it Proof of bound \eqref{intf1}:} 
%First we consider single-rate bound:
%This bound is derived with full adaptation, and if we consider in symmetric case, the bound coincides with the line in W curve when $\alpha>2$ (very strong interference), thus we obtain the capacity and conclude that adaptation is useless when the interference is very strong.  
%
%In (a), we drop $S^{N-n_{32}}X_{3,i}$ since $X_{3,i}$ is only a function of $M_{34}$. By symmetry, we have $R_{34}\leq n_{34}$.
% 
%\noindent {\it Proof of bound \eqref{intf2}:} 
%\begin{align*}
%& n(R_{21}-\epsilon)\\
%& \leq I(M_{21};Y_1^n|M_{12},M_{34},M_{43})\\
%& \leq H(Y_1^n|M_{12},M_{34},M_{43})\\
%& = \sum_{i=1}^n [H(Y_{1,i}|Y_1^{i-1},M_{12},M_{34},M_{43})]\\
%& = \sum_{i=1}^n [H(S^{N-n_{41}}X_{4,i}+S^{N-n_{21}}X_{2,i}|Y_1^{i-1},M_{12},M_{34},M_{43})]\\
%& \overset{(a)}{=} \sum_{i=1}^n [H(S^{N-n_{21}}X_{2,i}|Y_1^{i-1},M_{12},M_{34},M_{43})]\\
%& \leq \sum_{i=1}^n [H(S^{N-n_{21}}X_{2,i})]\\
%& = nn_{21}
%\end{align*}
%
%In (a), $X_{4,i}$ is a function of $(M_{43},Y_4^{i-1})$, and $Y_4^{i-1}=S^{N-n_{14}}X_1^{i-1}+S^{N-n_{34}}X_3^{i-1}+S^{N-n_{44}}X_4^{i-1}$, where $X_1^{i-1}$ and $X_3^{i-1}$ are functions of $M_{12}$ and $M_{34}$ respectively due to the definition of partial adaptation. By iterating the same argument, it's easy to see $X_{4,i}$ is a function of $(M_{12},M_{34},M_{43})$. By symmetry, we have $R_{43}\leq n_{43}$.
%\begin{remark} The single rate bounds are tight for the two-way linear deterministic symmetric interference channel when $ \alpha\geq 2, \alpha=m/n, n=n_{12}=n_{21}=n_{34}=n_{43}, m=n_{14}=n_{41}=n_{23}=n_{32}$. In other words, partial adaptation is useless when the two-way interference is very strong ($ \alpha\geq 2$).
%\end{remark}
%\noindent {\it Proof of first bound on $R_{12}+R_{34}$:} 
\begin{align*}
&n(R_{12}+R_{34}-\epsilon)\\
&\overset{(a)}{\leq} I(M_{12};Y_2^n|M_{21},M_{43})+I(M_{34};Y_4^n,Y_2^n|M_{12},M_{21},M_{43})\\
%&=I(M_{12};Y_2^n|M_{21},M_{43})+I(M_{34};Y_2^n|M_{21},M_{12},M_{43})\\
%&+I(M_{34};Y_4^n|M_{21},M_{12},M_{43},Y_2^n)\\
&\leq I(M_{12};Y_2^n|M_{21},M_{43})+I(M_{34};Y_2^n|M_{21},M_{12},M_{43})+H(Y_4^n|M_{21},M_{12},M_{43},Y_2^n)\\
%&=I(M_{12};Y_2^n|M_{21},M_{43})+I(M_{34};Y_2^n|M_{21},M_{12},M_{43})\\
%&+\sum_{i=1}^n[H(S^{N-n_{12}}X_{1,i}+S^{N-n_{34}}X_{3,i}+S^{N-n_{44}}X_{4,i}|\\
%& \ \ M_{21},M_{12},M_{43},Y_4^{i-1},X_4^i,Y_2^n,X_2^n)]\\
&\overset{(b)}{=}I(M_{12};Y_2^n|M_{21},M_{43})+I(M_{34};Y_2^n|M_{21},M_{12},M_{43})\\
&+\sum_{i=1}^n[H(S^{N-n_{34}}X_{3,i}|M_{21},M_{12},M_{43},Y_4^{i-1},X_4^i,Y_2^n,X_2^n,X_1^i)]\\
&\leq \sum_{i-1}^n [H(Y_{2,i}|Y_2^{i-1},M_{21},X_2^i)-H(Y_{2,i}|Y_2^{i-1},M_{12},M_{21},M_{43})+H(Y_{2,i}|Y_2^{i-1},M_{12},M_{21},M_{43})\\
&+H(S^{N-n_{34}}X_{3,i}|M_{21},M_{12},M_{43},Y_4^{i-1},X_4^i,S^{N-n_{12}}X_1^n+S^{N-n_{22}}X_2^n+S^{N-n_{32}}X_3^n,X_2^n,X_1^i)]\\
&\leq \sum_{i=1}^n [H(S^{N-n_{12}}X_{1,i}+S^{N-n_{32}}X_{3,i})+H(S^{N-n_{34}}X_{3,i}|S^{N-n_{32}}X_{3,i})]\\
&\leq n(\max (n_{12},n_{32})+[n_{34}-n_{32}]^+)\\
&\overset{(c)}{=} n(\max (p,q)+[p-q]^+).
\end{align*}

We introduce the genie $Y_2^n$ in the second mutual information term in (a), i.e. we provide asymmetric side information to only one receiver. In (b), we add $X_1^i$ in the entropy term because of the iterated argument that, given $M_{12},X_2^n,X_4^i$, we can construct $X_1^i$. For (c), we assumed a symmetric channel. 

%To obtain the bound $R_{12}+R_{34}\leq \max(n_{34},n_{14})+[n_{12}-n_{14}]^+$ we mimic the above proof starting with $n(R_{12}+R_{34}-\epsilon)\leq I(M_{12};Y_2^n, Y_4^n|M_{21},M_{43},M_{34})+I(M_{34};Y_4^n|M_{21},M_{43})$.


%consider the symmetric case with $p=n_{12}=n_{21}=n_{34}=n_{43}, q=n_{14}=n_{41}=n_{23}=n_{32}$. 

\begin{remark} 
%This bound is tight for the two-way linear deterministic symmetric interference channel when $2/3\leq \alpha\leq 2, \alpha=m/n$. In other words, adaptation is useless when the two-way interference is strong ($1\leq \alpha\leq 2$) and weak in some interval ($2/3\leq \alpha\leq 1$).% 
Note that we do {\bf not} need partial adaptation in this bound, and so these conclusions actually hold for {\bf full adaptation.} This implies that for the symmetric channel, full adaptation is useless when two-way interference is strong ($1\leq \alpha\leq 2, \alpha=q/p$) and weak in some interval ($2/3\leq \alpha\leq 1, \alpha=q/p$) where this outer bound may be achieved. Interestingly, when $2/3\leq \alpha\leq 2$, the ``V'' curve is also the capacity for the linear deterministic symmetric interference channel with  feedback \cite{Changho2010}.
%Thus, by symmetry, we can prove bound \eqref{intf6}. 
If we add another asymmetric genie $Y_4^n$ in the first term in (a), then we obtain the second sum-rate bound (IC$\rightarrow$ c). %Fig. \ref{fig:ICcap} (A). \nrd{change}%Again, bound \eqref{intf8} follows from symmetry. 
%\textcolor{red}{ND: I DON'T GET THIS REASONING? Intuitively, full adaptation is useless in this bound is because we an asymmetric genie is provided to only one receiver, 
%which results in the nodes can not full cooperate with each other.}
\end{remark}
 
It may further be shown that for symmetric channels,  adaptation is also useless when two-way interference is very strong ($\alpha>2, \alpha=q/p$). To show this, we re-derive the single-rate bounds this time {\it not} assuming partial adaptation (allowing for full adaptation), and using symmetry in the last step:%all we need to do is deriving the single-rate bounds alternatively, with full adaptation:
\begin{align*}
 n(R_{12}-\epsilon)
& \leq I(M_{12};Y_2^n,Y_3^n|M_{21},M_{34})\\
& \leq H(Y_2^n,Y_3^n|M_{21},M_{34})\\
& = \sum_{i=1}^n [H(Y_{2,i},Y_{3,i}|Y_2^{i-1},Y_3^{i-1},M_{21},M_{34},X_2^i,X_3^i)]\\
& = \sum_{i=1}^n [H(S^{N-n_{12}}X_{1,i},S^{N-n_{43}}X_{4,i}|Y_2^{i-1},Y_3^{i-1},M_{21},M_{34},X_2^i,X_3^i)]\\
& \leq \sum_{i=1}^n [H(S^{N-n_{12}}X_{1,i},S^{N-n_{43}}X_{4,i})]\\
& = n\max (n_{12},n_{43})\\
&=np
\end{align*}
%\begin{figure}
%\begin{center}
%\includegraphics[width=3.0in]{intfchain}
%\caption{The Markov chain for the sum-rate bound for two-way IC.} 
%\label{fig:intfchain}
%\end{center}
%\end{figure}
Under very strong interference constraints,  this is also known to be achievable.  Thus, we have obtained the capacity  for the  symmetric linear deterministic two-way IC when $\alpha\geq 2/3$, where we see that full adaptation is useless.  We will comment more on this in Remark \ref{curves}, and in Fig. \ref{fig:curves}.

{We now continue with the sum-rate outer bound (IC$\rightarrow$ d), which uses a similar genie to that in  Costa and El Gamal's \cite{elgamal_det_IC} capacity result for a class of deterministic ICs, i.e. gives to one receiver the interference created at the {\it other} receiver by the desired message. The same type of genie (though this time noisy) is used in the new outer bound for the Gaussian one-way interference channel by Etkin, Tse and Wang \cite{etkin_tse_wang}. The main difference is that we also provide the transmitters in the $\rightarrow$ direction the messages in the $\leftarrow$ direction, or $M_{21}$ and $M_{43}$, in order to be able to create $X_2$ and $X_4$ and remove these from the entropy terms, obtaining only entropies of combinations of the variables in the $\rightarrow$ direction (of $X_1, X_3$) for the sum-rate bound for the $\rightarrow$ direction.}

%\nrd{which one? use numbers when change}
 \begin{align*}
&n(R_{12}+R_{34}-\epsilon)\leq I(M_{12};Y_2^n,S^{N-n_{14}}X_1^n,M_{21},M_{43})+I(M_{34};Y_4^n,S^{N-n_{32}}X_3^n,M_{21},M_{43})\\
%&\overset{(a)}{=} H(Y_2^n,S^{N-n_{14}}X_1^n|M_{21},M_{43})\\
%& -H(Y_2^n,S^{N-n_{14}}X_1^n|M_{12},M_{21},M_{43})\\
%&+ H(Y_4^n,S^{N-n_{32}}X_3^n|M_{21},M_{43})\\
%& -H(Y_4^n,S^{N-n_{32}}X_3^n|M_{34},M_{21},M_{43})\\
%&\overset{(d)}{=} H(Y_2^n,S^{N-n_{14}}X_1^n|M_{21},M_{43})-H(Y_2^n,S^{N-n_{14}}X_1^n|M_{12},M_{21},M_{43})\\
%&+ H(Y_4^n,S^{N-n_{32}}X_3^n|M_{21},M_{43})-H(Y_4^n,S^{N-n_{32}}X_3^n|M_{34},M_{21},M_{43})\\
& \overset{(d)}{=} H(Y_2^n|S^{N-n_{14}}X_1^n,M_{43},M_{21})+H(S^{N-n_{14}}X_1^n|M_{43},M_{21})-H(Y_2^n,S^{N-n_{14}}X_1^n|M_{12},M_{21},M_{43})\\
& +H(Y_4^n|S^{N-n_{32}}X_3^n,M_{43},M_{21})+H(S^{N-n_{32}}X_3^n|M_{43},M_{21})-H(Y_4^n,S^{N-n_{32}}X_3^n|M_{34},M_{21},M_{43})\\
& \overset{(e)}{=} H(Y_2^n|S^{N-n_{14}}X_1^n,M_{43},M_{21})+H(Y_4^n|S^{N-n_{32}}X_3^n,M_{43},M_{21}) \\
& + \sum_{i=1}^n H(S^{N-n_{14}}X_{1,i}|S^{N-n_{14}}X_{1}^{i-1},M_{43},M_{21}, M_{34})-H(Y_{2,i}|Y_2^{i-1}, M_{12},M_{21},M_{43}, X_2^i, X_1^i)\\
& + \sum_{i=1}^n H(S^{N-n_{32}}X_{3,i}|S^{N-n_{32}}X_{3}^{i-1}, M_{43},M_{21}, M_{12})-H(Y_{4,i}|Y_4^{i-1}, M_{34},M_{21},M_{43}, X_4^i, X_3^i)\\
%& \overset{(f)}{=} H(Y_2^n|S^{N-n_{14}}X_1^n,M_{43},M_{21})+H(Y_4^n|S^{N-n_{32}}X_3^n,M_{43},M_{21}) \\
%& + \sum_{i=1}^n H(S^{N-n_{14}}X_{1,i}|S^{N-n_{14}}X_{1}^{i-1},M_{43},M_{21}, M_{34})-H(S^{N-n_{32}}X_{3,i}|Y_2^{i-1}, S^{N-n_{14}}X_{1}^{i-1}, M_{12},M_{21},M_{43},  X_2^i, X_1^i)\\
%& + \sum_{i=1}^n H(S^{N-n_{32}}X_{3,i}|S^{N-n_{32}}X_{3}^{i-1}, M_{43},M_{21}, M_{12})-H(S^{N-n_{14}}X_{1,i}|Y_4^{i-1}, S^{N-n_{32}}X_3^{i-1}, M_{34},M_{21},M_{43}, X_4^i, X_3^i)\\
& \overset{(f)}{=} H(Y_2^n|S^{N-n_{14}}X_1^n,M_{43},M_{21})+H(Y_4^n|S^{N-n_{32}}X_3^n,M_{43},M_{21}) \\
& + \sum_{i=1}^n H(S^{N-n_{14}}X_{1,i}|S^{N-n_{14}}X_{1}^{i-1},M_{43},M_{21}, M_{34}, X_3^i, X_4^i)-H(S^{N-n_{32}}X_{3,i}|S^{N-n_{32}}X_3^{i-1}, M_{12},M_{21},M_{43},  X_2^i, X_1^i)\\
& + \sum_{i=1}^n H(S^{N-n_{32}}X_{3,i}|S^{N-n_{32}}X_{3}^{i-1}, M_{43},M_{21}, M_{12}, X_1^i, X_2^i)-H(S^{N-n_{14}}X_{1,i}|S^{N-n_{14}}X_1^{i-1},  M_{34},M_{21},M_{43}, X_4^i, X_3^i)\\
%& \overset{(e)}{=} H(Y_2^n|S^{N-n_{14}}X_1^n,M_{43},M_{21})+H(S^{N-n_{14}}X_1^n|M_{43},M_{21},M_{34})-H(S^{N-n_{32}}X_3^n|M_{12},M_{43},M_{21})\\
%& +H(Y_4^n|S^{N-n_{32}}X_3^n,M_{43},M_{21})+H(S^{N-n_{32}}X_3^n|M_{43},M_{21},M_{12})-H(S^{N-n_{14}}X_1^n|M_{43},M_{21},M_{34})\\
& = H(Y_2^n|S^{N-n_{14}}X_1^n,M_{43},M_{21})+H(Y_4^n|S^{N-n_{32}}X_3^n,M_{43},M_{21})\\
%& =\sum_{i=1}^n [H(Y_{2,i}|Y_2^{i-1},S^{N-n_{14}}X_1^n,M_{43},M_{21},X_{2,i})\\
%& +H(Y_{4,i}|Y_4^{i-1},S^{N-n_{32}}X_3^n,M_{43},M_{21},X_{4,i})]\\
& \leq \sum_{i=1}^n [H(S^{N-n_{12}}X_{1,i}+S^{N-n_{32}}X_{3,i}|S^{N-n_{14}}X_{1,i})+H(S^{N-n_{14}}X_{1,i}+S^{N-n_{34}}X_{3,i}|S^{N-n_{32}}X_{3,i})]\\
& \leq n(\max ([n_{12}-n_{14}]^+,n_{32})+\max ([n_{34}-n_{32}]^+,n_{14})),
\end{align*}
where (d) follows from the independence of the messages, (e)  by the chain rule of entropy and the fact that we can create $X_1^i$ given $M_{12}$, and we can create $X_2^i$ given $M_{21}, Y_2^{i-1}$ (similarly for $X_3^i, X_4^i$). We have also added the independent messages $M_{34}$ and $M_{12}$ to the 3rd and 5th term conditioning. For (f), we have expanded the $Y_{2,i}$ and $Y_{4,i}$ in the entropy terms of the 4th and 6th terms and removed the contributions from the conditioning. 
In the 3rd term, we can create $X_3^i$ from $M_{34}$ and $X_4^i$ from $S^{N-n_{14}}X_1^{i-1}$, $X_3^i$ and $M_{43}$ (similarly for the 5th term creating $X_1^i$ and $X_2^i$). 
%the 3rd and 6th terms cancel, as do the 4th and 5th, due to the fact that $X_{1,i}$ and $X_{3,i}$ are functions only of their own message, and reducing knowledge of $Y_4^{i-1}$ to $S^{N-n_{14}}X_1^{i-1}$ given knowledge of $X_3^i$ and $X_4^i$.  Finally, $Y_4^{i-1},X_4^i$ can be constructed in the conditioning of the 3rd term (and similarly for the 4th and 5th terms). In the conditioning of the 4th and 6th terms, we have expanded out $Y_2^{i-1}$ and $Y_4^{i-1}$ and canceled the known terms which are functions of $X_1^i$ and $X_3^i$ respectively. 
%where (d) follows from the independence of the messages and the fact that we can create $X_1^i$ given $M_{12}$, and we can create $X_2^i$ given $M_{21}, Y_2^{i-1}$ (similarly for $X_3^i, X_4^i$). For (e), 
%the 3rd and 6th terms cancel, as do the 4th and 5th, due to the fact that $X_{1,i}$ and $X_{3,i}$ are functions only of their own message, and reducing knowledge of $Y_4^{i-1}$ to $S^{N-n_{14}}X_1^{i-1}$ given knowledge of $X_3^i$ and $X_4^i$.  
%Finally, $Y_4^{i-1},X_4^i$ can be constructed in the conditioning of the 3rd term (and similarly for the 4th and 5th terms). 
%the 2nd and 5th terms follow since $X_1$ and $X_3$ are functions only of $M_{12}$ and $M_{34}$. 
%The 3rd and 6th terms follow from the same reason, together with Lemma \ref{lemma:partial}.
%$H(S^{N-n_{14}}X_1^n|M_{43},M_{21})=H(S^{N-n_{14}}X_1^n|M_{43},M_{21},M_{34})$ and $H(Y_2^n,S^{N-n_{14}}X_1^n|M_{12},M_{21},M_{43})=H(S^{N-n_{32}}X_3^n|M_{12},M_{43},M_{21})$ are because (1),$X_1^n$ is only a function of $M_{12}$; (2), $X_{2,i}=f_2(M_{21},Y_2^{i-1})$; (3), $Y_2^{i-1}$ is a function of $(X_3^{i-1}, M_{12},M_{21})$. By symmetry, we have $H(S^{N-n_{32}}X_3^n|M_{43},M_{21})=H(S^{N-n_{32}}X_3^n|M_{43},M_{21},M_{12})$ and $H(Y_4^n,S^{N-n_{32}}X_3^n|M_{34},M_{21},M_{43})=H(S^{N-n_{14}}X_1^n|M_{43},M_{21},M_{34})$.

The sum-rate bound in the opposite direction (which we must consider given the fact that under partial adaptation, not everything is symmetric):
%\noindent {\it Proof of bound \eqref{intf10}:} 
\begin{align*}
&n(R_{21}+R_{43}-\epsilon) \leq I(M_{21};Y_1^n,S^{N-n_{23}}X_2^n,M_{12},M_{34})+I(M_{43};Y_3^n,S^{N-n_{41}}X_4^n,M_{12},M_{34})\\
%&\overset{(a)}{=} H(Y_1^n,S^{N-n_{23}}X_2^n|M_{12},M_{34})\\
%& -H(Y_1^n,S^{N-n_{23}}X_2^n|M_{12},M_{34},M_{21})\\
%& +H(Y_3^n,S^{N-n_{41}}X_4^n|M_{12},M_{34})\\
%& -H(Y_3^n,S^{N-n_{41}}X_4^n|M_{12},M_{34},M_{43})\\
& \overset{(f)}{=}H(Y_1^n|S^{N-n_{23}}X_2^n,M_{12},M_{34})+H(S^{N-n_{23}}X_2^n|M_{12},M_{34})-H(Y_1^n,S^{N-n_{23}}X_2^n|M_{12},M_{34},M_{21})\\
& +H(Y_3^n|S^{N-n_{41}}X_4^n,M_{12},M_{34})+H(S^{N-n_{41}}X_4^n|M_{12},M_{34})-H(Y_3^n,S^{N-n_{41}}X_4^n|M_{12},M_{34},M_{43})\\
& \overset{(g)}{=}H(Y_1^n|S^{N-n_{23}}X_2^n,M_{12},M_{34})+H(S^{N-n_{23}}X_2^n|M_{12},M_{34},M_{43})-H(S^{N-n_{41}}X_4^n|M_{12},M_{34},M_{21})\\
& +H(Y_3^n|S^{N-n_{41}}X_4^n,M_{12},M_{34})+H(S^{N-n_{41}}X_4^n|M_{12},M_{34},M_{21})-H(S^{N-n_{23}}X_2^n|M_{12},M_{34},M_{43})\\
%& =H(Y_1^n|S^{N-n_{23}}X_2^n,M_{12},M_{34})+H(Y_3^n|S^{N-n_{41}}X_4^n,M_{12},M_{34})\\
%& =\sum_{i=1}^n [H(Y_{1,i}|Y_1^{i-1},S^{N-n_{23}}X_2^n,M_{12},M_{34},X_{1,i})\\
%& +H(Y_{3,i}|Y_3^{i-1},S^{N-n_{41}}X_4^n,M_{12},M_{34},X_{3,i})]\\
& \leq \sum_{i=1}^n [H(S^{N-n_{21}}X_{2,i}+S^{N-n_{41}}X_{4,i}|S^{N-n_{23}}X_{2,i})+H(S^{N-n_{43}}X_{4,i}+S^{N-n_{23}}X_{2,i}|S^{N-n_{41}}X_{4,i})]\\
& \leq n(\max ([n_{21}-n_{23}]^+,n_{41})+\max ([n_{43}-n_{41}]^+,n_{23})),
\end{align*}

where (f) follows from the independence of the messages. Equation (g) follows since $X_1$ and $X_3$ are functions only of $M_{12}$ and $M_{34}$ and from Lemma \ref{lemma:partial}.
%$H(S^{N-n_{23}}X_2^n|M_{12},M_{34})=H(S^{N-n_{23}}X_2^n|M_{12},M_{34},M_{43})$ and $H(Y_1^n,S^{N-n_{23}}X_2^n|M_{12},M_{34},M_{21})=H(S^{N-n_{41}}X_4^n|M_{12},M_{34},M_{21})$ are because (1),$X_2^n$ is a function of $(M_{12},M_{21},M_{34})$ and independent of $M_{43}$; (2), $X_1^n$ is only a function of $M_{12}$. By symmetry, we have $H(S^{N-n_{41}}X_4^n|M_{12},M_{34})=H(S^{N-n_{41}}X_4^n|M_{12},M_{34},M_{21})$ and $H(Y_1^n,S^{N-n_{23}}X_2^n|M_{12},M_{34},M_{21})=H(S^{N-n_{23}}X_2^n|M_{12},M_{34},M_{43})$.

\begin{remark} We needed partial adaptation (Lemma \ref{lemma:partial}) in the proof of the previous two bounds (IC$\rightarrow$ d) and (IC$\leftarrow$ d). % \nrd{Cite bounds by number when change.} 
 In the above, nodes 1 and 3 were restricted. 
 By symmetry, we may obtain the same result if nodes 2 and 4 were restricted.
 
 Finally, %\nrd{THIS ONE ALSO HAD AN ISSUE BUT OK IF MULTI-LETTER.... JUST LEFT OUT!}% Essentially, this shows that restriction at a subset of nodes results in inhibiting cooperation at all, as the alternative path (say of $M_{12}$) of a message, which could be relayed by the other nodes, is blocked. 
% \textcolor{red}{ND: DON'T GET THIS. We add symmetric genies at both receivers which results in the nodes can cooperate with each other. However, partial adaptation blocks this cooperation, that's why the partial adaptation cannot increase the capacity region.}
% \textcolor{green}{ND: DO WE REALLY NEED THIS STATEMENTS? AREN'T THEY OBVIOUS IS WE GET THE CAPACITY REGION?  These bounds are tight for the two-way linear deterministic symmetric interference channel when $0\leq \alpha\leq 2/3, \alpha=m/n$. In other words, partial adaptation is useless when the two-way interference is very weak ($0\leq \alpha\leq 1/2$) and weak in some interval ($1/2\leq \alpha\leq 2/3$).}
\end{remark}
%\noindent {\it Proof of bound \eqref{intf11}:} 
\begin{align*}
& n(2R_{12}+R_{34}-\epsilon)\\
& \leq I(M_{12};Y_2^n|M_{21},M_{43})+I(M_{12};Y_2^n,Y_4^n|M_{21},M_{43},M_{34})+I(M_{34};Y_4^n,S^{N-n_{32}}X_3^n|M_{21},M_{43})\\
&=H(Y_2^n|M_{21},M_{43})-H(Y_2^n|M_{21},M_{43},M_{12})+H(Y_4^n|M_{21},M_{43},M_{34})\\
& +H(Y_2^n|M_{21},M_{43},M_{34},Y_4^n)+H(Y_4^n,S^{N-n_{32}}X_3^n|M_{21},M_{43})-H(Y_4^n,S^{N-n_{32}}X_3^n|M_{34},M_{21},M_{43})\\
%\end{align*}
%\begin{align*}
& \overset{(h)}{=}H(Y_2^n|M_{21},M_{43})-H(Y_2^n|M_{21},M_{43},M_{12})+H(S^{N-n_{32}}X_3^n|M_{43},M_{21},M_{12})\\
& +H(Y_4^n|S^{N-n_{32}}X_3^n,M_{43},M_{21})+H(Y_4^n|M_{21},M_{43},M_{34})\\
& -H(Y_4^n,S^{N-n_{32}}X_3^n|M_{34},M_{21},M_{43})+H(Y_2^n|M_{21},M_{43},M_{34},Y_4^n)\\
%& =H(Y_2^n|M_{21},M_{43})+H(Y_4^n|S^{N-n_{32}}X_3^n,M_{43},M_{21})\\
%& +H(Y_2^n|M_{21},M_{43},M_{34},Y_4^n)\\
& \overset{(i)}{\leq} \sum_{i=1}^n [H(S^{N-n_{12}}X_{1,i}+S^{N-n_{32}}X_{3,i})+H(S^{N-n_{14}}X_{1,i}+S^{N-n{34}}X_{3,i}|S^{N-n_{32}}X_{3,i})+H(S^{N-n_{12}}X_{1,i}|S^{N-n_{14}}X_{1,i})]\\
& =n(\max (n_{12},n_{32})+\max ([n_{34}-n_{32}]^+,n_{14})+[n_{12}-n_{14}]^+),
\end{align*}
%where (h) follows from the independence of the messages, and 
where  (h) follows from the definition of partial adaptation and Lemma \ref{lemma:partial} (skipping a transition to multi-letter for brevity), and (i) by canceling the 2nd and 3rd terms, as well as the 5th and the 6th terms.
%,$-H(Y_2^n|M_{21},M_{43},M_{12})+H(S^{N-n_{32}}X_3^n|M_{43},M_{21},M_{12})=0$ and  $+H(Y_4^n|M_{21},M_{43},M_{34})-H(Y_4^n,S^{N-n_{32}}X_3^n|M_{34},M_{21},M_{43})=0$ are because Lemma \ref{block}. Similarly, we can prove bound \eqref{intf12}. By symmetry, bounds \eqref{intf13} and \eqref{intf14} are obtained.
We may similarly prove the other bounds of this form (IC$\rightarrow$ f), (IC$\leftarrow$ e) and (IC$\leftarrow$ f).\end{proof}

We again see that, under partial adaptation constraints, adaptation is useless and we obtain the capacity region of two one-way ICs. Essentially, {\it partial} adaptation prevented messages being relayed by other messages (which was also impossible in the MAC/BC and Z channels). For example,  under full adaptation, message $M_{12}$ may be relayed from Tx1 to Rx 2 through nodes 3 and 4. This path is ``blocked'' by the partial adaptation assumption,  as node 3 could not adapt to carry $M_{12}$. However, it should be pointed out that this is not necessary in general: full adaptation in the two-way modulo 2 adder IC is useless as we showed in the previous subsection, but the path is not blocked.
%Specifically, if we consider symmetric case, our result is two well-known "W" curves \cite{etkin_tse_wang} in opposite directions.  
%This may be an important observation in larger networks, where we may have mixtures of nodes which may/may not adapt. In essence, this points at the very rough intuition that we need all nodes along the various relaying paths to adapt in 
%If we consider the symmetric interference channel, then bound \eqref{intf5}=\eqref{intf7}, \eqref{intf6}=\eqref{intf8}, and bounds \eqref{intf11}-\eqref{intf14} are redundant. In this case, our results are reduced to the well-known "W" curve in \cite{etkin_tse_wang}.
 

\subsection{Symmetric rate comparison with other interference channel models} 
For symmetric deterministic linear ICs, we may compare the symmetric sum-capacity $C_{sym}$ of various one-way and two-way models. 
Recalling $\alpha : = q/p$, we plot $C_{sym}$ as a function of $\alpha$ for the IC \cite{etkin_tse_wang}, the IC with noiseless output feedback \cite{Changho2010}, the IC with rate-limited feedback \cite{Vahid-IC-rate-FB} (for a fixed value of $\beta = 0.125$ in the notation of \cite{Vahid-IC-rate-FB}), and the two-way IC with full adaptation considered here (for $\alpha\geq 2/3$ only). Several observations may be made: the two-way IC with {\it partial} adaptation behaves like  two one-way interference channels operating in parallel over the forward and backwards link, as seen by the coinciding lines for the one-way and two-way IC with partial adaptation. This tells us that allowing partial adaptation is useless -- i.e. may as well not adapt. Interestingly, the same holds true even for full adaptation for $\alpha>2/3$. This was also concluded for the linear deterministic one-way interference channel with interfering feedback links in \cite{sahai2009channel}; what is interesting is that we can just as well squeeze in extra information messages in the feedback link (in the two-way interference channel model) rather than use the backwards links for feedback.  The symmetric sum-capacity for the {\it fully} adaptive two-way IC remains open for $\alpha<2/3$; it is solved for partial adaptation.

Recently, the work in \cite{Suh:ISIT2012, Suh2012} has considered a one-way interference channel with interfering feedback links (again forming an interference channel), a generalization of some of the deterministic interference channels with feedback considered in \cite{sahai2009channel}, where the feedback link spends fraction $\lambda$ of its time sending feedback, and uses the remaining $(1-\lambda)$ for other things (such as for example sending independent backwards messages, though adaptation as in \eqref{eq:adapt} is not considered). This is quite different from our model which integrates sending feedback and messages over all links, allows for adaptation, and does not force this separation.  While the symmetric sum-capacity for this two-message interference channel with interfering feedback links is obtained in \cite{Suh:ISIT2012, Suh2012} in our notation for $\alpha\geq 2/3$, it is a function of this parameter $\lambda$ and is thus not plotted here. 
\begin{figure*}
%\begin{figure}
\centerline{\includegraphics[width=12cm]{curves}}
\vspace{-3.9cm}
\caption{$C_{sym}$ for various linear deterministic ICs as a function of $\alpha:= \frac{q}{p}$; $q$ interfering link strength, $p$ direct link strength.}
\label{fig:curves}
%\end{figure}
\end{figure*}
%\noindent {\it Proof of bound \eqref{intf9}:} 
\label{curves}


\section{Gaussian Two-way Multiple-Access Broadcast Channel}
\label{MACBCG}
%\nrd{ENTIRELY NEW SECTION - Zhiyu please check *very* carefully!}

All previous channel models considered were deterministic. We now ask whether we may obtain insight into whether adaptation is useless / useful in certain noisy channels. We do so by considering the Gaussian two-way MAC/BC in this section, and the Gaussian two-way IC in the next.

We demonstrate that adaptation in the real Gaussian two-way MAC/BC can only improve the sum-capacity up to $1/2$ bit per direction. We show this by comparing non-adaptive inner bounds for this channel to outer bounds to the two-way Gaussian MAC/BC. Our outer bound for the $\rightarrow$ MAC direction is derived directly; the outer bound for the $\leftarrow$ BC direction 
follows by enhancing the BC channel by giving Tx 2 perfect output feedback and rendering the channel degraded, at which point the converse of \cite[Thm.2]{elgamal_deg_BC} follows.  

\subsection{Channel model}
At each channel use, the Gaussian two-way MAC/BC is described by the input/output relationships
\begin{align*}
&Y_1=X_2+Z_1\\
&Y_2=X_1+X_3+Z_2\\
&Y_3=X_2+Z_3,
\end{align*}
%\textcolor{blue}{This i/o relationships imply all the channel gains are 1, which seem to me that the outer bound is derived in the DoF sense, but the constant gap result is based on the argument of "regardless of various channel gains". But I think this is MAC/BC, there's no distinction between SNR and INR, so should be ok.}
subject to power constraints $E[|X_j|^2]\leq P_j, j\in\{1,2,3\}$, and independent, identically distributed complex Gaussian noise $Z_j\sim \mathcal{CN}(0, N_j)$ at all nodes  $j\in (1,2,3)$, which may be done without loss of generality (notice both the arbitrary power and noise constraints). WLOG assume that $N_3\geq N_1$. 
 Note that we have removed the ``self-interference'' terms such as $X_1$ in the expression of $Y_1$ (for example) in contrast to the deterministic models considered. This is for ease of exposition, to make the parallels with the MAC and BC channels more direct. Note that in a Gaussian model, these ``self-interference'' terms can always be subtracted at a given node in any case. To contrast, in the Gaussian interference channel next, we will NOT eliminate the self-interference terms from the channel model, to demonstrate how they may be handled directly. 
%  In this Gaussian model, it is clear that since node $1$ knows $X_1$ we may equivalently remove this self-interference term due to the additive nature of the channel and hence including it is unnecessary. However,  we leave it in our expressions to emphasize precisely this fact. In other channels such as the two-way binary multiplier channel, where $Y=X_1 X_2$ one cannot ``undo'' ones' own channel, which is one source of difficulty for this elusive two-way channel. In all converses, the fact that we can cancel or subtract out a node's ``self-interference'' is shown explicitly. This is one of the reasons two-way channels of this form, as seen for example in the Gaussian two-way channel as well \cite{Han:1984}, are easier to deal with, which we hope to emphasize. %\textcolor{blue}{This is also similar to the Gaussian point-to-point two-way channel, in which the author keeps the self-interference in his channel model but it can be subtracted out in the proof.}
Recall that the inputs of the Gaussian two-way MAC/BC are fully adaptive, i.e. 
\begin{align}
& X_{1,i}=f_1(M_{12},Y_1^{i-1}), \;\; X_{2,i}=f_2(M_{21}, M_{23}, Y_2^{i-1}), \;\;  X_{3,i}=f_3(M_{32},Y_3^{i-1}).
\end{align}

\subsection{The limited utility of adaptation in the Gaussian two-way MAC/BC}
We now have the following theorem. 

\medskip
\begin{theorem}
Adaptation in the Gaussian MAC/BC channel may only improve the sum-rate in the $\rightarrow$ and $\leftarrow$ directions by up to $1/2$ bit per direction. 
\label{thm:MACBCG}
\end{theorem}
\begin{proof}
First let us consider the $\rightarrow$ direction. 
For achievability, let the $\rightarrow$ direction use the capacity achieving scheme for the non-adaptive Gaussian multiple access channel, whose sum-rate is  dominated by 
\begin{equation}
R_{12}+R_{32} \leq \frac{1}{2}\log\left(1+\frac{P_1+P_3}{N_2}\right)
\label{eq:MAC}
\end{equation}

For the converse, first consider the MAC direction, and follow steps along the lines of a MAC with feedback as in \cite{Ozarow:MAC-FB, willems_FB}, which are however not immediately applicable:
% first enhance the two-way MAC/BC by providing perfect output feedback in the two directions. That is, at time $i$, for the MAC $\rightarrow$ direction, give Tx 1 and Tx 3 access to $Y_{2,i-1}$ in addition to $M_{21}$ and $M_{23}$.
%Clearly any outer bound to this enhanced channel will form an outer bound to our original two-way Gaussian MAC/BC.
%For the MAC $\rightarrow$ direction, then, at time $i$,  Tx 1 and Tx 3 (the transmitters) have access to the following, since given perfect output feedback $Y_2^{i-1}$ and messages $M_{21}, M_{23}$, they can both create $X_2^i$.
%\begin{align*}
%\mbox{Tx 1: } & M_{12}, M_{21}, M_{23}, Y_1^{i-1} = X_2^{i-1}+Z_1^{i-1}, Y_2^{i-1} = X_1^{i-1}+X_3^{i-1}+Z_2^{i-1}, X_i^{i-1}, X_2^{i}\\
%\mbox{Tx 3: } & M_{32}, M_{21}, M_{23}, Y_3^{i-1} = X_2^{i-1}+Z_3^{i-1}, Y_2^{i-1} = X_1^{i-1}+X_3^{i-1}+Z_2^{i-1}, X_3^{i-1}, X_2^i\\
%\end{align*}
%We now outer bound the above  channel model in a similar way as the Gaussian MAC with feedback; the difference here additional access to $Y_1^{i-1}, Y_3^{i-1}, X_2^i$ at time $i$.  These do not change much conceptually with respect to the Gaussian MAC with feedback, but we include the below derivation for completeness. We consider only the sum-rate, and follow the steps of Ozarow \cite{Ozarow:MAC-FB}:
%\begin{align}
%n(R_{12}+R_{32}) & = H(M_{12}, M_{32}) = H(M_{12}, M_{32}|M_{21}, M_{23}, Z_1^n, Z_3^n) \\
%& = H(M_{12}, M_{32}|M_{21}, M_{23}, Z_1^n, Z_3^n, Y_2^n) + I(M_{12}, M_{32};Y_2^n|M_{21}, M_{23}, Z_1^n, Z_3^n) \\
%&\overset{(a)}{\leq} n\epsilon_n + \sum_{i=1}^n H(Y_{2,i}|Y_2^{i-1}, M_{21}, M_{23}, Z_1^n, Z_3^n) - H(Y_{2,i}|Y_2^{i-1}, M_{12}, M_{32}, M_{21}, M_{23}, Z_1^n, Z_3^n) \\
%& \overset{(b)}{\leq}  n\epsilon_n + \sum_{i=1}^n H(Y_{2,i}|X_{2,i}) - H(Y_{2,i}|X_{1,i}, X_{2,i}, X_{3,i}) \\
%& \overset{(c)}{=}  n\epsilon_n + \sum_{i=1}^n I(X_{1,i}, X_{3,i}; Y_{2,i}|X_{2,i}) \\
%& \overset{(d)}{\leq}  n\epsilon_n + nI(X_1,X_3;Y_2|X_2) \\
%& \overset{(e)}{\leq}  n\epsilon_n + n\left(\frac{1}{2}\log\left(1+P_1+P_3+2\sqrt{P_1 P_3} \right)\right) \label{eq:28}
%\end{align}
%where (a) follow by Fano's inequality, (b) since for the 2nd term, given $M_{21}, M_{23}$ and $Y_2^{i-1}$ one can construct $X_{2,i}$ and then conditioning reduces entropy, and for the 3rd term since given all the terms in the conditioning we may create $X_{1,i}, X_{2,i}$, and $X_{3,i}$ and then use the memoryless property of the channel model, (c) follows by definition, (d) by Jensen's inequality. Finally, (e) follows as we maximize this expression over all possible 
%joint distributions $p(X_1,X_2,X_3)$. Along the lines of \cite{Ozarow:MAC-FB}, it suffices to consider $X_1,X_3$ to be jointly Gaussian conditioned on $X_2$; further outer bounding the expression which would contain correlation coefficients to eliminate them results in \eqref{eq:28}. 
\begin{align}
n(R_{12}+R_{32}) & = H(M_{12}, M_{32}) = H(M_{12}, M_{32}|M_{21}, M_{23}) \\
& = H(M_{12}, M_{32}|M_{21}, M_{23}, Y_2^n) + I(M_{12}, M_{32};Y_2^n|M_{21}, M_{23}) \\
&\overset{(a)}{\leq} n\epsilon_n + \sum_{i=1}^n H(Y_{2,i}|Y_2^{i-1}, M_{21}, M_{23}) - H(Y_{2,i}|Y_2^{i-1}, M_{12}, M_{32}, M_{21}, M_{23}, Y_1^n, Y_3^n) \\
& \overset{(b)}{\leq}  n\epsilon_n + \sum_{i=1}^n H(Y_{2,i}|X_{2,i}) - H(Y_{2,i}|X_{1,i}, X_{2,i}, X_{3,i}) \\
& \overset{(c)}{\leq} n\epsilon_n + \sum_{i=1}^n H(X_{1,i}+X_{3,i}+Z_{2,i}) - H(Z_{2,i}) \\
& \overset{(d)}{\leq} n\epsilon_n + \frac{n}{2}\log\left(1+\frac{P_1+P_3+2\sqrt{P_1P_3}}{N_2}\right)  \label{eq:MACG2}
%& \overset{(c)}{=}  n\epsilon_n + \sum_{i=1}^n I(X_{1,i}, X_{3,i}; Y_{2,i}|X_{2,i})
%& \overset{(d)}{\leq}  n\epsilon_n + nI(X_1,X_3;Y_2|X_2) \\
%& \overset{(e)}{\leq}  n\epsilon_n + n\left(\frac{1}{2}\log\left(1+P_1+P_3+2\sqrt{P_1 P_3} \right)\right) \label{eq:28}
\end{align}
%\textcolor{blue}{It seems to me that after step (b), we can directly proceed in this way: $\leq n\epsilon_n+\sum_{i=1}^nH(X_{1,i}+X_{3,i}+Z_{2,i})-H(Z_{2,i})\leq n\epsilon_n+n/2\log (1+P_1+P_3+2\sqrt{P_1P_3})$}
where (a) follows by Fano's inequality for the first term, by the chain rule of entropy for the 2nd and 3rd terms, and by conditioning reduces entropy in adding $Y_1^n$ and $Y_3^n$ to the 3rd term, (b) since for the 2nd term, given $M_{21}, M_{23}$ and $Y_2^{i-1}$ one can construct $X_{2,i}$ and then conditioning reduces entropy, and for the 3rd term since given all the terms in the conditioning we may create $X_{1,i}, X_{2,i}$, and $X_{3,i}$ and then use the memoryless property of the channel model, (c) follows by conditioning reduces entropy and by the memoryless channel, (d) since it suffices to consider $X_1,X_3$ to be jointly Gaussian and is outer bounded when they are maximally correlated, as adaptation may permit joint $p(x_1,x_3)$. 

%From here we may single-letterize using Jensen's inequality to obtain the following outer bound:
%\begin{align}
%R_{12}+R_{32} &\leq \max_{p(x_1,x_2,x_3)} I(X_1,X_3;Y_2|X_2) \\
%& = \max_{p(x_1,x_2,x_3)} H(X_1+X_3\textcolor{blue}{+Z_2}|X_2) - H(Z_2) \\
%& \leq  \max_{p(x_1,x_2,x_3)} H(X_1+X_3\textcolor{blue}{+Z_2}) - H(Z_2) \\
%&\leq  \frac{1}{2}\log\left(1+\frac{P_1+P_3+2\sqrt{P_1 P_3}}{N_2} \right)\ \label{eq:MACG2}
%\end{align}
%which follows as conditioning reduces entropy, and, 
%along the lines of \cite{Ozarow:MAC-FB}, it suffices to consider $X_1,X_3$ to be jointly Gaussian.
%% conditioned on $X_2$, leading to an expression with several correlation coefficients between $x_1,x_2,x_3$. Further outer bounding this by proper choice of correlation coefficient yields
%%\begin{equation}
%%R_{12}+R_{32} \leq \frac{1}{2}\log\left(1+P_1+P_3+2\sqrt{P_1 P_3} \right)\ \label{eq:MACG2}
%%\end{equation}


% further outer bounding the expression which would contain correlation coefficients to eliminate them results in \eqref{eq:28}. 

Now, taking the difference between the outer bound to the adaptive two-way MAC/BC in the MAC direction in \eqref{eq:MACG2} and the non-adaptive inner bound of \eqref{eq:MAC} yields
\begin{align*}
\eqref{eq:MACG2} - \eqref{eq:MAC} & = \frac{1}{2}\log\left(\frac{N_2+P_1+P_3+2\sqrt{P_1 P_3}}{N_2+P_1+P_3} \right)\\
& = \frac{1}{2}\log\left(1+ \frac{2\sqrt{P_1 P_3}}{N_2+P_1+P_3} \right)\\
& \leq \frac{1}{2}\log\left(1+ \frac{P_1+P_3}{P_1+P_3} \right) = \frac{1}{2},
\end{align*}
where the inequality follows as $(\sqrt{P_1}-\sqrt{P_3})^2 \geq 0$ implies ${2}\sqrt{P_1P_3}\leq P_1 + P_3$, and we have decreased the denominator.

For the $\leftarrow$ direction use the capacity achieving scheme for the non-adaptive single-antenna Gaussian broadcast channel, which yields the rates, for $0\leq \alpha\leq 1$
\begin{equation}
R_{21} \leq \frac{1}{2}\log\left(1+\frac{\alpha P_2}{N_1}\right), \;\; R_{23} \leq \frac{1}{2}\log\left(1+\frac{(1-\alpha)P_2}{N_3+\alpha P_2}\right). \label{eq:BCin}
\end{equation}

For the converse, for the BC $\leftarrow$ direction we enhance the channel as follows:
\begin{itemize}
\item Give Tx 2 perfect output feedback, i.e. access to $Y_{1,i-1}$, $Y_{3,i-1}$ at time $i$ as well as access to $M_{12}, M_{32}$.  Together with feedback, this allows it to create $X_{1}^i, X_{3}^i$. 
\item Render the channel physically degraded by providing Rx 1 with $Y_3^n$. Then Rx 3's output is trivially a physically degraded version of Rx 1's output. This is where we use the fact that, WLOG $N_3\geq N_1$ (if the reverse had been true we would have given $Y_1^n$ to Rx 3 instead). This is crucial in ensuring a constant gap to a non-adaptive scheme. 
\end{itemize}
The converse of \cite[Thm. 2]{elgamal_deg_BC}, which shows that feedback does not change the capacity region of the physically degraded BC,  then follows along all the same steps with the notation correspondences \break (\cite{elgamal_deg_BC} $\leftrightarrow$ this paper) as follows: 
\[ W_1 \leftrightarrow M_{21}, \; W_2 \leftrightarrow M_{23}, \; Y^n \leftrightarrow (Y_1^n, Y_3^n), \; Z^n \leftrightarrow Y_3^n, \; X^n \leftrightarrow X_2^n \]
The key point in proving the converse is \cite[Lemma 3]{elgamal_deg_BC}, which follows in a straightforward manner even given the added adaptation constraint (i.e. $X_{2,i}$ is also a function of $Y_2^{i-1}$ which is not present in the original \cite[Thm. 2]{elgamal_deg_BC}), but we re-state and prove it here in our notation for clarity and completeness.
\begin{lemma}{\it Analogous to Lemma 3 of \cite{elgamal_deg_BC}.} For all $\lambda\geq 0$,
\begin{align*}
n(R_{21}+R_{23}) &\leq I(M_{23};Y_3^n)+\lambda I(M_{21};Y_1^n,Y_3^n|M_{23}) \\
& \leq \sum_{i=1}^n I(U_i;Y_{3,i})+\lambda I(X_{2,i};Y_{1,i},Y_{3,i}|U_i) 
\end{align*}
where $U_i := (M_{23}, Y_1^{i-1}, Y_3^{i-1})$.
\end{lemma}
\begin{proof}
For the first term,
\begin{align}
I(M_{23};Y_3^n) &= \sum_{i=1}^n I(M_{23};Y_{3,i}|Y_3^{i-1}) \\
& = \sum_{i=1}^n H(Y_{3,i}|Y_3^{i-1}) - H(Y_{3,i}|M_{23}, Y_3^{i-1}) \\
&\leq \sum_{i=1}^n H(Y_{3,i}) - H(Y_{3,i}|M_{23}, Y_3^{i-1}, Y_1^{i-1}) \\
& = \sum_{i=1}^n I(Y_{3,i};U_i)
\end{align}
by definition of $U_i : = (M_{23}, Y_3^{i-1}, Y_1^{i-1})$. For the second term,
\begin{align}
I(M_{21};Y_1^n, Y_3^n|M_{23}) & = \sum_{i=1}^n I(M_{21};Y_{1,i}, Y_{3,i}|M_{23}, Y_1^{i-1}, Y_3^{i-1}) \\
& = \sum_{i=1}^n I(M_{21};Y_{1,i}, Y_{3,i}|U_i) \\
&\leq \sum_{i=1}^n I(M_{21}, X_{2,i};Y_{1,i}, Y_{3,i}|U_i) \\
& = \sum_{i=1}^n H(Y_{1,i}, Y_{3,i}|U_i) - H(Y_{1,i}, Y_{3,i}|U_i, M_{21}, X_{2,i})\\
&=\sum_{i=1}^n I(Y_{1,i}, Y_{3,i};X_{2,i}|U_i)
\end{align}
where several steps in the proof of \cite[Lemma 3]{elgamal_deg_BC} are not needed as our channel is trivially degraded. 
\end{proof}
%by giving Tx 2 access to $Y_{1,i-1}$, $Y_{3,i-1}$, $M_{12}$ and $M_{32}$ at time $i$ (perfect output feedback). We further outer bound this as done in \cite{Ozarow:1984} and render the channel physically degraded, as in \cite{elgamal_deg_BC}, by providing Rx 1 with $Y_3^n$ (thus Rx 3's output is trivially a physically degraded version of Rx 1's output). 
%%As done in \cite{Ozarow:1984}, we will thus form an outer bound to this Gaussian BC with feedback and adaptation by rendering it a physically degraded BC with FB, whose capacity is known, see \cite{elgamal_deg_BC}. 
%The only difference with the Gaussian BC with channel output feedback is that Tx 2 not only has access to $Y_1^{i-1}$ and $Y_3^{i-1}$, and $M_{21}, M_{23}$ when constructing $X_{2,i}$, but also has access to $Y_2^{i-1} = X_1^{i-1}+X_3^{i-1}+Z_2^{i-1}$, $M_{12}$ and $M_{32}$. Note that given $M_{12}$ and $Y_1^{i-1}$ Tx 2 may construct $X_{1,i}$, and given $M_{32}$ and $Y_3^{i-1}$ it may construct $X_{3,i}$. Considering the converse of \cite{elgamal_deg_BC}, where it is shown that feedback does not change the capacity region of the physically degraded BC, instead of taking auxiliary random variable $U_i = (M_{23}, Y_1^{i-1}, Y_3^{i-1})$  (of \cite{elgamal_deg_BC} put into our notation), we take $U_i = (M_{23}, M_{12}, M_{32}, Y_1^{i-1}, Y_3^{i-1}, Z_2^n)$. The outer bound then follows through without any changes \nrd{Zhiyu this is something I want you to carefully double check!}, yielding the following outer bound (noting that receiver Rx 1 has access to both $Y_1, Y_3$ which is how we create the degraded channel)
Following the same arguments as in \cite[Thm. 2]{elgamal_deg_BC}, the above Lemma yields an outer bound equivalent to the region in \eqref{eq:BCout}, where we note that in addition to $U_i = (M_{23}, Y_1^{i-1}, Y_3^{i-1})$ to construct $X_{2,i} = f(M_{21}, M_{23}, Y_2^{i-1}, M_{12}, M_{32}, Y_1^{i-1}, Y_3^{i-1}) \equiv f(U_i, M_{21}, Z_2^{i-1}, M_{12}, M_{32})$ we also need $M_{12}, M_{32}, M_{21}, Z_2^n$, but that, given the above definition of the random variable $U_i$, the factorization of the inputs as $p(u)p(x_2|u)$ still holds. Note that with some abuse of notation we have left the channel distribution as $p(y_1, y_3|x_2)p(y_3|y_1,y_3)$ to emphasize that Rx 1 has access to both $Y_1^n, Y_3^n$ (we have forced the channel to be degraded) and thus that Rx 3, with access to $Y_3^n$ only is trivially a degraded version of this. The outer bound for the $\leftarrow$ BC direction is thus given by the set of all non-negative $R_{21}, R_{23}$ such that
\begin{align}
R_{21} &\leq I(X_2;Y_1, Y_3|U), \;\;\;\;\;\; R_{23} \leq I(U;Y_3) \label{eq:BCout}
\end{align}
over all distributions of the form $p(u)p(x_2|u)p(y_1,y_3|x_2)p(y_3|y_1,y_3)$.  Evaluation for the Gaussian channel, as done in \cite{Ozarow:1984}, yields an outer bound of 
\begin{align}
R_{21} \leq \frac{1}{2}\log\left(1+\frac{\alpha P_2}{N_1}\frac{N_1+N_3}{N_3}\right), \;\;\;\; R_{23} \leq \frac{1}{2}\log\left( 1+\frac{(1-\alpha)P_2}{N_3+\alpha P_2} \right) \label{BCoutG}
\end{align}
for $0\leq \alpha \leq 1$.

%\textcolor{blue}{It looks okay but I want to recheck this part today, and I think we need to rigorously prove this converse.}
Taking the difference between the sum of the outer bound to the adaptive two-way MAC/BC in the BC direction in \eqref{BCoutG} and the sum of the  non-adaptive inner bounds of \eqref{eq:BCin} yields
\begin{align*}
\eqref{BCoutG} - \eqref{eq:BCin} & = \frac{1}{2}\log\left(1+\frac{\alpha P_2}{N_1}\frac{N_1+N_3}{N_3}\right) - \frac{1}{2}\log\left(1+\frac{\alpha P_2}{N_1}\right)   \\
& \overset{(a)}{\leq}\frac{1}{2}\log\left(1+\frac{2 \alpha P_2}{N_1}\right) - \frac{1}{2}\log\left(1+\frac{\alpha P_2}{N_1}\right)  \leq  \frac{1}{2}.
\end{align*}
where (a) follows as $\frac{N_1+N_3}{N_3} = 1+\frac{N_1}{N_3} \leq 2$ since $N_1\leq N_3$. 
\end{proof}

\begin{remark}
We note that this result also implies that for the one-way Gaussian MAC with FB and the one-way BC with FB, feedback and adaptation of the nodes can only increase capacity by up to 1/2 bit (sum-rate) per direction. This fact has been partially noted in \cite{Changho2010}.
\end{remark}

%has the following when constructing its next input $X_{2,i}$:
%\begin{align*}
%\mbox{Tx 2: } & M_{21},M_{23}, Y_2^{i-1} = X_1^{i-1}+X_3^{i-1}+Z_2^{i-1}, X_2^{i-1}\\
%& \;\;\;\;\; Y_1^{i-1} = X_2^{i-1}+Z_1^{i-1}, Y_3^{i-1} = X_2^{i-1}+Z_3^{i-1}.
%\end{align*}
%This is almost identical to what a Gaussian broadcast channel with perfect output feedback would have access, except for the additional access to $Y_2^{i-1}$, which 
%\end{proof}
%

%All previous channel models considered were deterministic. We now ask whether we may obtain insight into whether adaptation is useless / useful in certain noisy channels. We do so by considering the Gaussian two-way MAC/BC in this section, and the Gaussian two-way IC in the next.
%
%We demonstrate that adaptation in the real Gaussian two-way MAC/BC can only improve sum-capacity up to $1/2$ bit per direction. We show this by comparing non-adaptive inner bounds for this channel to outer bounds to the two-way Gaussian MAC/BC. Our outer bounds are derived by showing that the Gaussian MAC with perfect output feedback forms an outer bound to our channel model for the $\rightarrow$ direction, while outer bounds to the Gaussian BC with perfect output feedback form outer bounds to our channel model in the $\leftarrow$ direction.
%
%\subsection{Channel model}
%At each channel use, the Gaussian two-way MAC/BC is described by the input/output relationships
%\begin{align*}
%&Y_1=X_1+X_2+Z_1\\
%&Y_2=X_1+X_2+X_3+Z_2\\
%&Y_3=X_2+X_3+Z_3,
%\end{align*}
%subject to power constraints $E[|X_j|^2]\leq P_j=1, j\in\{1,2,3\}$, and independent, identically distributed complex Gaussian noise $Z_j\sim \mathcal{CN}(0, 1)$ at all nodes  $j\in (1,2,3)$, which may be done without loss of generality. 
% Note that we have kept the ``self-interference'' terms such as $X_1$ in the expression of $Y_1$ (for example). In this Gaussian model, it is clear that since node $1$ knows $X_1$ we may equivalently remove this self-interference term due to the additive nature of the channel and hence including it is unnecessary. However,  we leave it in our expressions to emphasize precisely this fact. In other channels such as the two-way binary multiplier channel, where $Y=X_1 X_2$ one cannot ``undo'' ones' own channel, which is one source of difficulty for this elusive two-way channel. In all converses, the fact that we can cancel or subtract out a node's ``self-interference'' is shown explicitly. This is one of the reasons two-way channels of this form, as seen for example in the Gaussian two-way channel as well \cite{Han:1984}, are easier to deal with, which we hope to emphasize. %\textcolor{blue}{This is also similar to the Gaussian point-to-point two-way channel, in which the author keeps the self-interference in his channel model but it can be subtracted out in the proof.}
%
%The inputs of the Gaussian two-way MAC/BC are fully adaptive, i.e. 
%\begin{align}
%& X_{1,i}=f_1(M_{12},Y_1^{i-1}), \;\; X_{2,i}=f_2(M_{21}, M_{23}, Y_2^{i-1}), \;\;  X_{3,i}=f_3(M_{32},Y_3^{i-1}).
%\end{align}
%
%\subsection{The limited utility of adaptation in the Gaussian two-way MAC/BC}
%We now have the following theorem. 
%
%\medskip
%\begin{theorem}
%Adaptation in the Gaussian MAC/BC channel may only improve the sum-rate in the $\rightarrow$ and $\leftarrow$ directions by up to $1/2$ bit per direction. 
%\label{thm:MACBCG}
%\end{theorem}
%\begin{proof}
%First let us consider the $\rightarrow$ direction. 
%For achievability, let the $\rightarrow$ direction use the capacity achieving scheme for the non-adaptive Gaussian multiple access channel. The sum-rate is then dominated by 
%\begin{equation}
%R_{12}+R_{32} \leq \frac{1}{2}\log(1+P_1+P_3)
%\label{eq:MAC}
%\end{equation}
%
%For the converse, first enhance the two-way MAC/BC by providing perfect output feedback in the two directions. That is, at time $i$, for the MAC $\rightarrow$ direction, give Tx 1 and Tx 3 access to $Y_{2,i-1}$ in addition to $M_{21}$ and $M_{23}$.
%Clearly any outer bound to this enhanced channel will form an outer bound to our original two-way Gaussian MAC/BC.
%For the MAC $\rightarrow$ direction, then, at time $i$,  Tx 1 and Tx 3 (the transmitters) have access to the following, since given perfect output feedback $Y_2^{i-1}$ and messages $M_{21}, M_{23}$, they can both create $X_2^i$.
%\begin{align*}
%\mbox{Tx 1: } & M_{12}, M_{21}, M_{23}, Y_1^{i-1} = X_1^{i-1}+X_2^{i-1}+Z_1^{i-1}, Y_2^{i-1} = X_1^{i-1}+X_2^{i-1}+X_3^{i-1}+Z_2^{i-1}, X_i^{i-1}, X_2^{i}\\
%\mbox{Tx 3: } & M_{32}, M_{21}, M_{23}, Y_3^{i-1} = X_3^{i-1}+X_2^{i-1}+Z_3^{i-1}, Y_2^{i-1} = X_1^{i-1}+X_2^{i-1}+X_3^{i-1}+Z_2^{i-1}, X_3^{i-1}, X_2^i\\
%\end{align*}
%We now outer bound the above  channel model in a similar way as the Gaussian MAC with feedback; the difference here is the presence of the additional $X_2$ due to the full-duplex nature of the channel, as well as the additional i.i.d. Gaussian noise terms $Z_1^n, Z_3^n$.  These do not change anything conceptually with respect to the Gaussian MAC with feedback, but we include the below derivation for completeness. We consider only the sum-rate, and follow the steps of Ozarow for the Gaussian MAC with feedback \cite{Ozarow:MAC-FB}, 
%\begin{align}
%n(R_{12}+R_{32}) & = H(M_{12}, M_{32}) = H(M_{12}, M_{32}|M_{21}, M_{23}, Z_1^n, Z_3^n) \\
%& = H(M_{12}, M_{32}|M_{21}|, M_{23}, Z_1^n, Z_3^n, Y_2^n) + I(M_{12}, M_{32};Y_2^n|M_{21}, M_{23}, Z_1^n, Z_3^n) \\
%&\overset{(a)}{\leq} n\epsilon_n + \sum_{i=1}^n H(Y_{2,i}|Y_2^{i-1}, M_{21}, M_{23}, Z_1^n, Z_3^n) - H(Y_{2,i}|Y_2^{i-1}, M_{12}, M_{32}, M_{21}, M_{23}, Z_1^n, Z_3^n) \\
%& \overset{(b)}{\leq}  n\epsilon_n + \sum_{i=1}^n H(Y_{2,i}|X_{2,i}) - H(Y_{2,i}|X_{1,i}, X_{2,i}, X_{3,i}) \\
%& \overset{(c)}{=}  n\epsilon_n + \sum_{i=1}^n I(X_{1,i}, X_{3,i}; Y_{2,i}|X_{2,i}) \\
%& \overset{(d)}{\leq}  n\epsilon_n + nI(X_1,X_3;Y_2|X_2) \\
%& \overset{(e)}{\leq}  n\epsilon_n + n\left(\frac{1}{2}\log\left(1+P_1+P_3+2\sqrt{P_1 P_3} \right)\right) \label{eq:28}
%\end{align}
%where (a) follow by Fano's inequality, (b) since for the 2nd term, given $M_{21}, M_{23}$ and $Y_2^{i-1}$ one can construct $X_{2,i}$ and then conditioning reduces entropy, and for the 3rd term since given all the terms in the conditioning we may create $X_{1,i}, X_{2,i}$, and $X_{3,i}$ and then use the memoryless property of the channel model, (c) follows by definition, (d) by Jensen's inequality. Finally, (e) follows as we maximize this expression over all possible 
%joint distributions $p(X_1,X_2,X_3)$. Along the lines of \cite{Ozarow:MAC-FB}, it suffices to consider $X_1,X_3$ to be jointly Gaussian conditioned on $X_2$; further outer bounding the expression which would contain correlation coefficients to eliminate them results in \eqref{eq:28}. 
%
%Now, taking the difference between the outer bound to the adaptive two-way MAC/BC in the MAC direction in \eqref{eq:28} and the non-adaptive inner bound of \eqref{eq:MAC} yields
%\begin{align*}
%\eqref{eq:28} - \eqref{eq:MAC} & = \frac{1}{2}\log\left(\frac{1+P_1+P_3+2\sqrt{P_1 P_3}}{1+P_1+P_3} \right)\\
%& = \frac{1}{2}\log\left(1+ \frac{2\sqrt{P_1 P_3}}{1+P_1+P_3} \right)\\
%& \leq \frac{1}{2}\log\left(1+ \frac{P_1+P_3}{P_1+P_3} \right) = \frac{1}{2},
%\end{align*}
%where the inequality follows as $(\sqrt{P_1}-\sqrt{P_3})^2 \geq 0$ implies $\sqrt{P_1P_3}\leq P_1 + P_3$, and we have decreased the denominator.
%
%For the $\leftarrow$ direction use the capacity achieving scheme for the non-adaptive single-antenna Gaussian broadcast channel, which yields the rates, for $0\leq \alpha\leq 1$
%\begin{equation}
%R_{21} \leq \frac{1}{2}\log(1+\alpha P_2), \;\; R_{23} \leq \frac{1}{2}\log\left(1+\frac{(1-\alpha)P_2}{1+\alpha P_2}\right).
%\label{eq:BC}
%\end{equation}
%For the converse, for the BC $\leftarrow$ direction we again enhance the channel by giving Tx 2 access to $Y_{1,i-1}$, $Y_{3,i-1}$, $M_{12}$ and $M_{32}$. Thus, Tx 2 has the following when constructing its next input $X_{2,i}$:
%\begin{align*}
%\mbox{Tx 2: } & M_{21},M_{23}, Y_2^{i-1} = X_1^{i-1}+X_2^{i-1}+X_3^{i-1}+Z_2^{i-1}, X_2^{i-1}\\
%& \;\;\;\;\; Y_1^{i-1} = X_1^{i-1}+X_2^{i-1}+Z_1^{i-1}, Y_3^{i-1} = X_3^{i-1}+X_2^{i-1}+Z_3^{i-1}, \\
%\end{align*}
%\end{proof}

\section{Gaussian Two-way Interference Channel}
\label{ICG}

We now consider the Gaussian two-way interference channel, and ask when non-adaptive schemes such as the celebrated Han and Kobayashi \cite{H+K} perform as well, or nearly as well, as adaptive schemes. 

We do not construct any inner bounds which employ adaptation; our focus is on showing when non-adaptive schemes perform ``well''. Rather, we derive an outer bounds for the Gaussian two-way IC under full adaptation (all 4 nodes may adapt) and several under partial adaptation (only 2 of the 4 may adapt) constraints. We then show that non-adaptive schemes sometimes achieve the capacity, or at least to within a constant gap of either the fully or partially adaptive schemes. 
%To study this problem, we derive outer bounds with full adaptation or partial adaptation and compare with the capacity of one-way Gaussian interference channel with strong interference, and Han+Kobayashi region for the one-way Gaussian interference channel with weak interference. 
%We show that full adaptation is useless when the two-way interference is very strong. 
%In strong interference, non-adaptive scheme may achieve to within 1 bit of our new fully adaptive outer bound.
%In weak interference, we show that non-adaptive schemes may achieve to within 2 bits of our new outer bounds assuming full adaptation in certain regimes, and partial adaptation in others. 
We note that while the converses and the steps are new and exploit carefully chosen genies, when we evaluate these by further outer-bounding our outer-bounds, interestingly, we  sometimes re-obtain some of the outer bounds of the interference channel \cite{etkin_tse_wang} {\it or} the interference channel with feedback \cite{Changho2010}. This  in turn is sufficient to achieve capacity to within a constant gap, which we emphasize, sometimes is limited to {\it partial} adaptation for some of the weak interference regimes but this will be explicitly mentioned when it is the case. 

%; in some regimes we assume partial adaptation) regardless of the channel gains. 

\subsection{Channel model, definitions, and partial adaptation lemma}
At each channel use, the Gaussian two-way IC is described by the input/output relationships
\begin{align*}
&Y_1=g_{11}X_1+g_{21}X_2+g_{41}X_4+Z_1\\
&Y_2=g_{12}X_1+g_{22}X_2+g_{32}X_3+Z_2\\
&Y_3=g_{23}X_2+g_{33}X_3+g_{43}X_4+Z_3\\
&Y_4=g_{14}X_1+g_{34}X_3+g_{44}X_4+Z_4,
\end{align*}
where $g_{jk}$, for $j,k\in \{1,2,3,4\}$ are the complex channel gains. We assume the power constraints $E[|X_j|^2]\leq P_j=1, j\in\{1,2,3,4\}$, and independent, identically distributed complex Gaussian noise $Z_j\sim \mathcal{CN}(0, 1)$ at all nodes  $j\in (1,2,3,4)$, which may be done without loss of generality. 
%\nrd{	ZHIYU: we should do everything for complex Gaussian channels rather than real Gaussian channels if possible. What I mean is that the noise of i.i.d. $\sim {\cal CN}(0,1)$ and channel gains are complex, not real. This means the correlation coefficients will be complex and in many spots you'll need to put $|\rho|^2$ instead of $\rho^2$ and complex conjugates. See Changho's paper on the two-way IC with feedback -- there it's complex channels. Capacity of complex AWGN channel is $\log(1+SNR)$ while that of a real channel is $\frac{1}{2}\log(1+SNR)$ so there is a difference. You've evaluated everything as complex, but our model and the intermediate steps are real. This needs to be changed! I did most of it, but please double check I didn't miss anything!}
Furthermore, we define ${\tt SNR}_{12}=|g_{12}|^2, {\tt SNR}_{21}=|g_{21}|^2, {\tt SNR}_{34}=|g_{34}|^2, {\tt SNR}_{43}=|g_{43}|^2$, and ${\tt INR}_{14}=|g_{14}|^2, {\tt INR}_{41}=|g_{41}|^2, {\tt INR}_{23}=|g_{23}|^2, {\tt INR}_{32}=|g_{32}|^2$. Note that we have kept the ``self-interference'' terms such as $g_{11}X_1$ in the expression of $Y_1$ (for example). In this Gaussian model, it is clear that since node $1$ knows $X_1$ we may remove this self-interference term due to the additive nature of the channel. However,  we leave it in our expressions to emphasize precisely this fact. In other channels such as the two-way binary multiplier channel, where $Y=X_1 X_2$ one cannot ``undo'' ones' own channel, which is one source of difficulty for this elusive two-way channel. In all converses, the fact that we can cancel or subtract out a node's ``self-interference'' is shown explicitly. This is one of the reasons two-way channels of this form, as seen for example in the Gaussian two-way channel as well \cite{Han:1984}, are easier to deal with, which we emphasize. %\textcolor{blue}{This is also similar to the Gaussian point-to-point two-way channel, in which the author keeps the self-interference in his channel model but it can be subtracted out in the proof.}

We say that the Gaussian two-way interference channel operates under ``full adaptation'' if we allow
\begin{align}
& X_{1,i}=f_1(M_{12},Y_1^{i-1}), \;\; X_{2,i}=f_2(M_{21},Y_2^{i-1})\\
& X_{3,i}=f_3(M_{34},Y_3^{i-1}), \;\; X_{4,i}=f_4(M_{43},Y_4^{i-1}).
\end{align}
Similarly, it operates under ``partial adaptation'' if we only allow the following:  
\begin{align}
& X_{1,i}=f_1(M_{12})\label{eq:p1}, \;\; X_{2,i}=f_2(M_{21},Y_2^{i-1})\\
& X_{3,i}=f_3(M_{34}), \;\; X_{4,i}=f_4(M_{43},Y_4^{i-1}),\label{eq:p2}
\end{align}
i.e.  nodes 1 and 3 are ``restricted'' \cite{Shannon:1961}.  By symmetry, we may alternatively allow nodes $2$ and $4$ to be restricted and $1,3$ to be fully adaptive; whether allowing $1,2$ or $1,4$ to be restricted and the complement fully adaptive remains an open problem.
%\nrd{Who first used the term ''restricted' - -a citation? Can we allow 1 and 4 to be restricted equivalently, or 1 and 2?} \textcolor{blue}{Shannon first used the term "restricted" in his paper "Two-way communications channels". I think allowing 1 and 4 or 1 and 2 to be restricted would change a lot since in those cases the two-way interference channel may not be considered as $\rightarrow$ and $\leftarrow$ directions. Still thinking about this. }

We are interested in the symmetric capacity (or sum-rate), when all the SNRs equal a given ${\tt SNR}$, and all the INRs equal a given ${\tt INR}$. For full adaptation, due to the symmetry,  we consider the per-user rates $R_{sym} = \frac{R_{12}+R_{34}}{2} = \frac{R_{21}+ R_{43}}{2}$.  In partial adaptation, there is only partial symmetry (nodes 1 and 3 are fixed, while 2 and 4 are not),  and hence we will consider the per user rates $R_{sym\rightarrow} = \frac{R_{12}+R_{34}}{2}$ and $R_{sym\leftarrow} = \frac{R_{21}+R_{43}}{2}$ for the forward and reverse directions respectively. We will derive outer bounds for $R_{sym}$ under full adaptation and $R_{sym\rightarrow}$, $R_{sym\leftarrow}$ under partial adaptation, and show these to be achievable to within constant gaps by non-adaptive schemes. 

We first prove a modified version of Lemma \ref{lemma:partial} relevant in partial adaptation for the Gaussian channel.
\begin{lemma}
Under partial adaptation \eqref{eq:p1} -- \eqref{eq:p2}, for some deterministic functions $f_5$ and $f_6$, 
\begin{align}
X_{2,i}& = f_5(M_{12}, M_{21}, M_{34}, Z_2^{i-1}) \perp M_{43}, \;\; \forall i \label{eq:X2}\\
X_{4,i}& = f_6(M_{43}, M_{34}, M_{12},Z_4^{i-1}) \perp M_{21}, \;\; \forall i \label{eq:X4}
\end{align}
where $\perp$ denotes independence.
\label{lemma:partialG}
\end{lemma}
\begin{proof}
Note that $X_{2,i} = f_2(M_{21}, Y_2^{i-1})$ and $Y_2^{i-1} = g_{12}X_{1}^{i-1} + g_{22}X_2^{i-1}+g_{32}X_3^{i-1}+Z_2^{i-1}$. Since $X_1^{i-1}$ and $X_3^{i-1}$ are functions only of $M_{12}$ and $M_{34}$ respectively, we may conclude that  there exists a function $f^*$ such that $X_{2,i} = f^*(M_{21}, M_{12}, M_{34}, X_2^{i-1},Z_2^{i-1})$. Iterating this argument, and noting that $X_{2,1}$ is only a function of $M_{21}$, we obtain the lemma. The result for $X_{4,i}$ follows similarly. That $X_{2,i}$ is independent of $M_{43}$ follows since $M_{43}$ is independent of all the arguments inside $f^*$. 
\end{proof}


\subsection{Outer bounds}
We now present two outer bounds for the Gaussian two-way IC under full and partial adaptation respectively. These bounds are either within a constant gap, or sufficient to show the capacity, for the different regimes. We derive general outer bounds, imposing symmetry only in the final step.

\begin{theorem} {\it Outer bound: full adaptation.}
%For the Gaussian two-way IC under full adaptation, any rates $(R_{12}, R_{21}, R_{34}, R_{43})$ in the capacity region lie in the region
%\begin{align}
%&R_{12}\leq \log ((2\pi e)^2(1+ SNR_{12})) \label{R_single1}\\
%&R_{21}\leq \log ((2\pi e)^2(1+ SNR_{21}))\label{R_single2}\\
%&R_{34}\leq \log ((2\pi e)^2(1+ SNR_{34}))\label{R_single3}\\
%&R_{43}\leq \log ((2\pi e)^2(1+ SNR_{43})).\label{R_single4}
%\end{align}
%\nrd{Something is fishy here -- what happens at small SNR, i.e. $SNR<\frac{1}{2\pi e}$? This outer bound is negative!? -- }\textcolor{blue}{Maybe we could consider whether $SNR$ is too small or not as two seperate cases and derive two outer bounds (almost the same) for each case? ETW did the similar thing: they consider whether $INR<1$ or $INR>1$ when they paraphrase the HK region in Gaussian case.}
%\begin{align}
%&R_{12}\leq \log (1+SNR_{12})\\
%&R_{21}\leq \log (1+SNR_{21})\\
%&R_{34}\leq \log (1+SNR_{34})\\
%&R_{43}\leq \log (1+SNR_{43}).
%\end{align}
For the Gaussian two-way symmetric IC under full adaptation, any achievable symmetric rate $R_{sym} = \frac{R_{12}+R_{34}}{2} = \frac{R_{21}+R_{43}}{2}$, achievable by each user,  satisfies, %for $\lambda_{jk} \in [-1,1]$:
\begin{align}
R_{sym}%&\leq \frac{1}{2}\log  (SNR(1-\lambda_{12}^2)+INR(1-\lambda_{23}^2)+2\sqrt{SNR\times INR}(\lambda_{13}-\lambda_{12}\lambda_{32})+1) \notag \\
%&+\frac{1}{2}\log \left(1+\frac{SNR(1-\lambda_{34}^2)}{INR(1-\lambda_{34}^2)+1}\right) \label{R_strong} \\
&\leq \frac{1}{2} \log \left(1+{\tt SNR}+{\tt INR}+2\sqrt{{\tt SNR}\times {\tt INR}}\right)+\frac{1}{2}\log \left(1+\frac{{\tt SNR}}{1+{\tt INR}}\right) \label{R_strong_E}
\end{align}
%\nrd{Add with lambdas maximized}
\label{thm:outer-full}
\end{theorem}
\begin{proof}
%We first show one single-rate bound.  
% OLD ONE:
%\begin{align*}
%& n(R_{12}-\epsilon)\\
%&\leq I(M_{12};Y_2^n,Y_3^n|M_{21},M_{34},M_{43},Z_2^{n-1},Z_3^n,Z_4^{n-1})\\
%&\leq H(Y_2^n,Y_3^n|M_{21},M_{34},M_{43},Z_2^{n-1},Z_3^n,Z_4^{n-1})-H(Z_2^n)\\
%& \leq \sum_{i=1}^n [H(g_{12}X_{1,i}+g_{22}X_{2,i}+g_{32}X_{3,i}+Z_{2,i}, g_{23}X_{2,i}+g_{33}X_{3,i}+g_{43}X_{4,i}+Z_{3,i}|M_{21},M_{34},M_{43},Z_3^n,Y_2^{i-1},Y_3^{i-1},\\
%& \ \ \ \ \ \ \ X_2^i,X_3^i,Z_2^{i-1},Z_4^{i-1})-H(Z_{2,i})]\\
%& \overset{(a)}{\leq} \sum_{i=1}^n [H(g_{12}X_{1,i}+Z_{2,i},g_{43}X_{4,i}+Z_{3,i}|M_{21},M_{34},M_{43},Z_3^n,Y_2^{i-1},Y_3^{i-1},X_2^i,X_3^i,Z_2^{i-1},Z_4^{i-1},X_1^{i-1},X_{4,i})-H(Z_{2,i})]\\
%%& \leq \sum_{i=1}^n [H(g_{12}X_{1,i}+Z_{2,i}|X_{2,i},X_{3,i}+Z_{4,i})-H(Z_{2,i})]\\
%&\leq \sum_{i=1}^n [H(g_{12}X_{1,i}+Z_{2,i})-H(Z_{2,i})]
%\end{align*}
%In (a), $X_1^{i-1}$ can be decoded from $Y_2^{i-1}$ since we know $X_2^i,X_3^i,Z_2^{i-1}$, and $X_{4,i}$ can be constructed since we know $X_1^{i-1},X_3^{i-1},Z_4^{i-1},M_{43}$. 
%\begin{align*}
%& n(R_{12}-\epsilon)\\
%&\leq I(M_{12};Y_2^n,Y_3^n,Z_2^n|M_{21},M_{34},M_{43},Z_3^n,Z_4^{n-1})\\
%&\leq H(Y_2^n,Y_3^n,Z_2^n|M_{21},M_{34},M_{43},Z_3^n,Z_4^{n-1})-H(Y_2^n,Y_3^n,Z_2^n|M_{21},M_{34},M_{43},Z_3^n,Z_4^{n-1},M_{12},Y_2^n,Y_3^n)\\
%&=H(Y_2^n,Y_3^n,Z_2^n|M_{21},M_{34},M_{43},Z_3^n,Z_4^{n-1})-H(Z_2^n)\\
%& \leq \sum_{i=1}^n [H(g_{12}X_{1,i}+g_{22}X_{2,i}+g_{32}X_{3,i}+Z_{2,i}, g_{23}X_{2,i}+g_{33}X_{3,i}+g_{43}X_{4,i}+Z_{3,i},Z_{2,i}|M_{21},M_{34},M_{43},Z_3^n,\\
%& \ \ \ \ \ \ \ Y_2^{i-1},Y_3^{i-1}, X_2^i,X_3^i,Z_2^{i-1},Z_4^{n-1})-H(Z_{2,i})]\\
%& \overset{(a)}{\leq} \sum_{i=1}^n [H(g_{12}X_{1,i}+Z_{2,i},g_{43}X_{4,i}+Z_{3,i},Z_{2,i}|M_{21},M_{34},M_{43},Z_3^n,Y_2^{i-1},Y_3^{i-1},X_2^i,X_3^i,Z_2^{i-1},Z_4^{n-1},X_1^{i-1},X_{4,i}) \\
%& \ \ \ \ \ \ \ \ \ \ \ \ \ \ -H(Z_{2,i})]\\
%%& \leq \sum_{i=1}^n [H(g_{12}X_{1,i}+Z_{2,i}|X_{2,i},X_{3,i}+Z_{4,i})-H(Z_{2,i})]\\
%&\leq \sum_{i=1}^n [H(g_{12}X_{1,i}+Z_{2,i},Z_{2,i})-H(Z_{2,i})\\
%&=\sum_{i=1}^n [H(Z_{2,i})+H(g_{12}X_{1,i}+Z_{2,i}|Z_{2,i})-H(Z_{2,i})]\\
%&= \sum_{i=1}^n[H(g_{12}X_{1,i})]
%\end{align*}
%\begin{align*}
%& n(R_{12}-\epsilon)\\
%&\leq I(M_{12};Y_2^n,Y_3^n,Z_2^n|M_{21},M_{34},M_{43},Z_3^n,Z_4^{n-1})\\
%&\leq H(Y_2^n,Y_3^n,Z_2^n|M_{21},M_{34},M_{43},Z_3^n,Z_4^{n-1})\\%-H(Y_2^n,Y_3^n,Z_2^n|M_{21},M_{34},M_{43},Z_3^n,Z_4^{n-1},M_{12},Y_2^n,Y_3^n)\\
%&=H(Y_2^n,Y_3^n,Z_2^n|M_{21},M_{34},M_{43},Z_3^n,Z_4^{n-1})\\%-H(Z_2^n)\\
%& \leq \sum_{i=1}^n [H(g_{12}X_{1,i}+g_{22}X_{2,i}+g_{32}X_{3,i}+Z_{2,i}, g_{23}X_{2,i}+g_{33}X_{3,i}+g_{43}X_{4,i}+Z_{3,i},Z_{2,i}|M_{21},M_{34},M_{43},Z_3^n,\\
%& \ \ \ \ \ \ \ Y_2^{i-1},Y_3^{i-1}, X_2^i,X_3^i,Z_2^{i-1},Z_4^{n-1})\\%-H(Z_{2,i})]\\
%& \overset{(a)}{\leq} \sum_{i=1}^n [H(g_{12}X_{1,i}+Z_{2,i},g_{43}X_{4,i}+Z_{3,i},Z_{2,i}|M_{21},M_{34},M_{43},Z_3^n,Y_2^{i-1},Y_3^{i-1},X_2^i,X_3^i,Z_2^{i-1},Z_4^{n-1},X_1^{i-1},X_{4,i}) \\
%& \ \ \ \ \ \ \ \ \ \ \ \ \ \ -H(Z_{2,i})]\\
%& \leq \sum_{i=1}^n [H(g_{12}X_{1,i}+Z_{2,i}|X_{2,i},X_{3,i}+Z_{4,i})-H(Z_{2,i})]\\
%&\leq \sum_{i=1}^n [H(g_{12}X_{1,i}+Z_{2,i},Z_{2,i})\\%-H(Z_{2,i})\\
%&\leq \sum_{i=1}^n [H(Z_{2,i})+H(g_{12}X_{1,i})]\\%-H(Z_{2,i})]\\
%&\leq \sum_{i=1}^n[ \log((2\pi e)^2 (1+SNR_{12})]
%&= \sum_{i=1}^n[H(g_{12}X_{1,i})]
%\end{align*}


%In (a), $X_1^{i-1}$ can be decoded from $Y_2^{i-1}$ and knowledge of $X_2^i,X_3^i,Z_2^{i-1}$, and $X_{4,i}$ can be constructed from $X_1^{i-1},X_3^{i-1},Z_4^{n-1},M_{43}$. 
%In the last step, we may bound this differential entropy using our power constraint to obtain the final step.
%$R_{12}\leq \log (2\pi e \, SNR_{12})$. %, and the constant gap to the capacity can be bounded as $\log 2\pi e$.
% The other three bounds follow by symmetry. \nrd{What was the single rate bound inspired by? Notice the trick of adding noise as side-information and how it leads to $2\pi e$ term..... and how I tricked added 1+ in order not to have to worry about small SNR and the bound getting megative.} \textcolor{blue}{The single-rate bound is inspired by the linear deterministic model, but Changho said the single rate bound for his channel model (linear deterministic channel with two messages) is based on the standard cutset argument, which is not the case for our channel model I think. }
It is sufficient to consider $R_{12}+R_{34}$ due to symmetry. This bound is inspired by the corresponding sum-rate bound in the linear deterministic model, i.e., we add asymmetric genie $Y_2^n$ at node 4 and this resembles the bounding technique used by Suh and Tse for the interference channel with feedback \cite{Changho2010}. We also note that we could have equivalently provided node 4 with the genie $g_{32}X_3^n+Z_2^n$ instead of $Y_2^n$ (in addition to $M_{12},  M_{21}, M_{43}$ and $Z_1^n$) which more resembles the type of genie seen in ICs and ICs with feedback. We have given $Y_2^n$ as it is then easier to see how node 4 may create $X_2^n$ based on $M_{21}$ and $Y_2^n$, and the bounds work out to the same. 
Notice the genie $Z_1^n$ in the conditioning of both terms as well which is not seen in the feedback bounds \cite{Changho2010}; this is needed in order to, together with the genie $M_{12}, M_{21}, M_{43}, Y_2^n$, be able to create $X_1^n$ at node 4 (essentially, to create $Y_1^n$ to create $X_1^n$). %For the sum-rate bound,  \nrd{what was the sum-rate bound for full adaptation inspired by?} \textcolor{blue}{also the linear deterministic model I think.}
%For the Gaussian two-way interference channel with strong interference, we derive outer bound with full adaptation:
%\begin{theorem}
%The capacity region of Gaussian two-way interference channel with strong interference and partial adaptation is %the set of rate pairs ($R_{12},R_{21},R_{34},R_{43}$), such that
%\begin{align}
%&R_{12}\leq \log (1+SNR_{12})\\
%&R_{21}\leq \log (1+SNR_{21})\\
%&R_{34}\leq \log (1+SNR_{34})\\
%&R_{43}\leq \log (1+SNR_{43})\\
%&R_{12}+R_{34}\leq \min \{\log (1+INR_{32}+SNR_{12}),\log (1+INR_{14}+SNR_{34})\}\\
%&R_{21}+R_{43}\leq \min \{\log (1+INR_{23}+SNR_{21}),\log (1+INR_{41}+SNR_{43})\}\\
%\end{align}
%The capacity region of Gaussian two-way interference channel with strong interference and full adaptation is within 1 bit to the capacity region of one-way Gaussian interference channel in both directions.
%\end{theorem}
\begin{align}
&n(R_{12}+R_{34}-\epsilon) \notag\\
&\leq I(M_{12};Y_2^n|M_{21},M_{43},Z_1^n)+I(M_{34};Y_4^n,Y_2^n|M_{12},M_{21},M_{43},Z_1^n)\notag\\
&\leq I(M_{12};Y_2^n|M_{21},M_{43},Z_1^n)+I(M_{34};Y_2^n|M_{21},M_{12},M_{43},Z_1^n)+H(Y_4^n|M_{21},M_{12},M_{43},Y_2^n,Z_1^n)-H(Z_4^n)\notag\\
&\overset{(a)}{=}I(M_{12};Y_2^n|M_{21},M_{43},Z_1^n)+I(M_{34};Y_2^n|M_{21},M_{12},M_{43},Z_1^n)\notag\\
&+\sum_{i=1}^n[H(g_{34}X_{3,i}+Z_{4,i}|M_{21},M_{12},M_{43},Y_4^{i-1},X_4^i,Y_2^n,X_2^n,Z_1^n,X_1^i)]-H(Z_4^n)\notag\\
&\overset{(b)}{\leq} \sum_{i=1}^n [H(Y_{2,i}|Y_2^{i-1},M_{21},X_{2,i})-H(Y_{2,i}|Y_2^{i-1},M_{12},M_{21},M_{43},Z_1^n)\notag\\
& \ \ +H(Y_{2,i}|Y_2^{i-1},M_{12},M_{21},M_{43},Z_1^n)-H(Z_{2,i})+H(g_{34}X_{3,i}+Z_{4,i}|X_{4,i}, g_{32}X_{3,i}+Z_{2,i},X_1^i,X_2^n)-H(Z_{4,i})]\notag\\
&\overset{(c)}{\leq} \sum_{i=1}^nH(g_{12}X_{1,i}+g_{32}X_{3,i}+Z_{2,i}|X_{2,i})-H(Z_{2,i})+H(g_{34}X_{3,i}+Z_{4,i}|X_{4,i}, g_{32}X_{3,i}+Z_{2,i})-H(Z_{4,i}) \label{(c)}
\end{align}
In step (a), $X_1^i$ in the conditioning of the third term is constructed from ($M_{12},X_2^n,X_4^i,Z_1^n$). In step (b), we used conditioning reduces entropy, the second and the third term cancelled each other and $g_{32}X_{3,i}+Z_{2,i}$ in the conditioning of the fifth term is decoded from $Y_2^n$. In step (c), we only keep the self-interference $X_{4,i}$ and drop the terms  $X_1^i, X_2^n$ in the conditioning of the third term. We could leave these and express the outer bound in terms of correlation coefficients between the inputs (which in general may be correlated due to full adaptation). However, in subsequent steps we will seek to maximize, or outer bound this outer bound to obtain a simple analytical expression, which amounts to setting certain correlation coefficients to $0$, or equivalently, dropping the   terms  $X_1^i, X_2^n$ in the conditioning. 
%due to the simplification of the evaluation and we want to see whether the self-interference in Gaussian two-way IC could make the bound tighter. Interestingly, the evaluation will show that the self-interference is completely redundant (similar for $X_{2,i}$ in the first term). 
Further evaluation yields \eqref{R_strong_E}, for details please refer to Appendix \ref{Gsum}.

%\nrd{could move this to Appendix?}   %, which makes intuitive sense. 
%For $\lambda_{34}$, we take derivative of the last term and make it equal zero, then we have $\lambda_{34}=0$.

% CORRECT, 		AS A BACKUP CHEKC
%\begin{align*}
%&Var(g_{34}X_3+Z_4|g_{32}X_3+Z_2,X_4)\\
%&=SNR+1-\begin{bmatrix} \sqrt{SNR\times INR} & \lambda_{34}\sqrt{SNR} \end{bmatrix} \times \begin{bmatrix} INR+1 & \lambda_{34}\sqrt{INR} \\ \lambda_{34}\sqrt{INR} & 1 \end{bmatrix}^{-1} \times \left[ \begin{array}{c} \sqrt{SNR\times INR} \\ \lambda_{34}\sqrt{SNR} \end{array} \right]\\
%&=1+\frac{SNR(1-\lambda_{34}^2)}{INR(1-\lambda_{34}^2)+1}
%\end{align*}

%\begin{align}
%&R_{sym1}\leq 1/2\log  (SNR(1-\lambda_{12}^2)+INR(1-\lambda_{23}^2)+2\sqrt{SNR\times INR}(\lambda_{13}-\lambda_{12}\lambda_{32})+1) \notag \\
%&+1/2\log (1+\frac{SNR(1-\lambda_{34}^2)}{INR(1-\lambda_{34}^2)+1})\label{R_strong}
%\end{align}
%To maximize $R_{sym1}$, it's obvious to pick $\lambda_{12}=\lambda_{23}=0$ and $\lambda_{13}=1$. For $\lambda_{34}$, we take derivative of the last term and make it equal zero, then we have $\lambda_{34}=0$.



%\begin{align*}
%& n(R_{12}-\epsilon)\\
%&\leq I(M_{12};Y_2^n|M_{21},M_{34})\\
%&=H(Y_2^n|M_{21},M_{34})-H(Z_2^n)\\
%& \overset{(a)}{\leq} \sum_{i=1}^n [H(g_{12}X_{1,i}+g_{22}X_{2,i}+g_{32}X_{3,i}+Z_{2,i}|M_{21},M_{34},Y_2^{i-1},X_{2,i},X_{3,i})-H(Z_{2,i})]\\
%& \leq \sum_{i=1}^n [H(g_{12}X_{1,i}+Z_{2,i}|M_{21},M_{34},Y_2^{i-1},X_{2,i},X_{3,i})-H(Z_{2,i})]\\
%&\leq \sum_{i=1}^n [H(g_{12}X_{1,i}+Z_{2,i})-H(Z_{2,i})]
%\end{align*}
%From here we may conclude that $R_{12}\leq \log (1+SNR_{12})$, using  the fact that Gaussian maximizes entropy. The other three bounds follow by symmetry.
\end{proof}

%OLD REMARK IF 	ADAPTATION IS TOTALLY USELESS
%\begin{remark}
%Interestingly, in the two-way IC under full adaptation, we have single-rate bounds equal to those of the plain one-way interference channel. \nrd{This is in sharp contrast to interference channels with feedback (IC with FB)  \cite{Changho2010}, where the perfect, and in some sense ``free'' (or over orthogonal links) feedback allows for ideal cooperation between sources. For high interference, this means that the symmetric capacity of the IC with FB is unboundedly higher than that of the IC without FB; this is not the case here as ``feedback'' must travel over at least one constraining ``direct'' link in the path from say user $1$ to user $2$,  i.e. $1\rightarrow 2$ is constrained, but so is the path $1\rightarrow 4 \rightarrow 3 \rightarrow 2$ (an alternative path allowed for by full adaptation) due to our symmetric assumption that the $4 \rightarrow 3$ link is just as constrained as the $1\rightarrow 2$ link at high INR.}
%\end{remark}
%\begin{remark}
%{\it Single-rate bounds:} Interestingly, in the two-way IC under full adaptation, we have single-rate bounds similar to those of the plain one-way interference channel. That is, for very high interference (which we will describe precisely in the next sub-section), one can achieve $R_{12} = \log(1+SNR_{12})$ (and similarly for all other single-rates), which is within $\log(2\pi e)^2$ bits (per user) of this outer bound. This means that in the high interference regime, adaptation can only lead to a constant number of bits gain over a non-adaptive scheme, which is in sharp contrast  to interference channels with feedback (IC with FB)  \cite{Changho2010}, where the perfect, and in some sense ``free'' (or over orthogonal links) feedback allows for ideal cooperation between sources. For high interference, this means that the symmetric capacity of the IC with FB is unboundedly higher than that of the IC without FB; this is not the case here as ``feedback'' must travel over at least one constraining ``direct'' link in the path from say user $1$ to user $2$,  i.e. $1\rightarrow 2$ is constrained, but so is the path $1\rightarrow 4 \rightarrow 3 \rightarrow 2$ (an alternative path allowed for by full adaptation) due to our symmetric assumption that the $4 \rightarrow 3$ link is just as constrained as the $1\rightarrow 2$ link at high INR.
%\end{remark}

%\nrd{HERE -- complex or real channels and how will affect the logs?}

\begin{remark} 
{\it Sum-rate bound:} Note that the final, evaluated symmetric, normalized sum-rate bound in \eqref{R_strong_E} has the same form as the IC with perfect output feedback outer bound \cite[upper bound on (7)]{Changho2010}, though they are arrived at using different genies (though similar in many senses as mentioned above). In both channel models, inputs may be arbitrarily correlated as no additional arguments for restricting the input distributions have been made, leading to similar bounding techniques using correlation coefficients. 

% \nrd{	ZHIYU: I did not agree or still did not understand your comment as to why they're the same. Any other thoughts on this? I tried seeing if we could reconstruct $Y_2^n$ at Tx1 using the genies we gave, but it does not seem that way, which is odd.}
%\textcolor{blue}{I think in deriving our bound and Changho's bound, both of us allow for arbitrary correlation between the transmitters, that's why after optimizing over $\lambda$s (although different optimizing procedures because of different $\lambda$s), we got the same form. }
%This is because the evaluation renders the correlation coefficients $\lambda_{12}=\lambda_{23}=\lambda_{34}=0$, which means the the self-interference $X_2$ and $X_4$ (in forward direction) are redundant, and in that case our channel is similar to two parallel interference channels with feedback in the sense of converse.%\nrd{NOT SURE WHAT YOU MEAN WITH THIS SENTENCE?} 
\end{remark}

\bigskip

\begin{theorem} {\it Outer bound: partial adaptation.}
%\nrd{Also add new Zhiyu result for single rate bounds for partial adaptation which leads to capacity at very strong interference.}
% \nrd{ned a forward and a bakcwards direction separate only when evaluate with lambda -- 2 different final expressions!}
For the Gaussian two-way IC under partial adaptation \eqref{eq:p1} -- \eqref{eq:p2}, in addition to the bounds in Theorem \ref{thm:outer-full}, we may also conclude that any achievable rates ($R_{12},R_{21},R_{34},R_{43}$), and $R_{sym\rightarrow} = \frac{R_{12}+R_{34}}{2}$ and $R_{sym\leftarrow} = \frac{R_{21}+R_{43}}{2}$ must satisfy, %for $\lambda_{ij} \in [-1,1]$ 
\begin{align}
&R_{12}\leq \log (1+{\tt SNR}_{12})\label{partial_single_rate1}\\
&R_{21}\leq \log (1+{\tt SNR}_{21})\\
&R_{34}\leq \log (1+{\tt SNR}_{34})\\
&R_{43}\leq \log (1+{\tt SNR}_{43})\label{partial_singe-rate4}
\end{align}
\begin{align}
R_{sym\rightarrow} %\overset{\lambda_{13}=0}{\leq} & \log 2\pi e\left( 1+SNR+INR-\frac{SNR\times INR(1-\lambda_{12}^2)}{1+INR-\lambda_{12}^2INR}
% \right. \notag \\
%&\left. -\frac{\lambda_{23}^2INR^2+\lambda_{23}^2INR+\lambda_{12}^2SNR+2\lambda_{12}\lambda_{23}\sqrt{SNR\times INR}}{1+INR-\lambda_{12}^2INR}
 %\right)  \label{R_weak} \\
 & \leq \log \left( 1+{\tt INR}+{\tt SNR}-\frac{{\tt INR}\times {\tt SNR}}{1+{\tt INR}}\right) \label{R_weak_E} 
 \end{align}
 \begin{equation}   R_{sym\leftarrow} \leq \left\{ \begin{array}{l}
% \log \left(1+{\tt INR}+{\tt SNR}+\frac{2{\tt SNR}}{{\tt INR}}-\frac{\left(\frac{\sqrt{{\tt SNR}\times {\tt INR}}}{{\tt INR}}+\sqrt{{\tt SNR}\times {\tt INR}}\right)^2}{1+{\tt INR}}\right), \ \ \mbox{if} \ {\tt SNR}\leq {\tt INR}^3\\
\log \left(1+{\tt INR}+\frac{{\tt SNR}}{{\tt INR}}\right), \ \ \mbox{if} \ {\tt SNR}\leq {\tt INR}^3\\
%\log \left( 1+{\tt INR}+{\tt SNR}+2\sqrt{{\tt SNR}\times {\tt INR}}-\frac{\left({\tt INR}+\sqrt{{\tt SNR}\times {\tt INR}}\right)^2}{1+{\tt INR}}\right), \ \ \mbox{if} \ {\tt SNR}>{\tt INR}^3
\log \left(1+\frac{(\sqrt{{\tt SNR}}+\sqrt{{\tt INR}})^2}{1+{\tt INR}}\right), \ \ \mbox{if} \ {\tt SNR}>{\tt INR}^3
 \end{array}\right. \label{R_weak_BE}
\end{equation}
% R_{sym\leftarrow} %\overset{\lambda_{24}\neq 0}{\leq} &  \log 2\pi e\left( 1+INR+SNR+2\lambda_{24}\sqrt{SNR\times INR}
 %\right. \notag\\
 %&\left.
 %-\frac{[INR(\lambda_{24}-\lambda_{12}\lambda_{14})+\sqrt{SNR\times INR}(1-\lambda_{12}^2)](\lambda_{24}INR+\sqrt{SNR\times INR})}{1+INR-\lambda_{12}^2INR}
 %\right. \notag \\
 %&\left.
 %-\frac{[\lambda_{14}\sqrt{INR}(1+INR)+\lambda_{12}(\sqrt{SNR}-\lambda_{24}INR^{3/2})](\lambda_{14}\sqrt{INR}+\lambda_{12}\sqrt{SNR})}{1+INR-\lambda_{12}^2INR}
 %\right) \label{R_weak_B}\\
 %& \leq \left[\log \left(1+INR+SNR+\frac{2SNR}{INR}-\frac{(\frac{\sqrt{SNR\times INR}}{INR}+\sqrt{SNR\times INR})^2}{1+INR}\right),
 %\right. \notag\\
 %&\left.
 %\log \left( 1+INR+SNR+2\sqrt{SNR\times INR}-\frac{(INR+\sqrt{SNR\times INR})^2}{1+INR}\right)\right] \label{R_weak_BE}
%\end{align}
\label{thm:outer-partial}
\end{theorem}
\begin{proof}
For the single-rate bounds, it is sufficient to show the first two due to symmetry. Notice that we must treat the $\rightarrow$ and $\leftarrow$ directions separately however due to the asymmetry of the partial adaptation problem definition.
\begin{align*}
n(R_{12}-\epsilon)&\leq I(M_{12};Y_2^n|M_{21},M_{34})\\
&\leq H(Y_2^n|M_{21},M_{34})-H(Y_2^n|M_{21},M_{34},M_{12},X_1^n,X_2^n,X_3^n)\\
&\overset{(a)}{\leq} \sum_{i=1}^n [H(Y_{2,i}|Y_2^{i-1},M_{21},X_{2,i},M_{34},X_{3,i})-H(Z_{2,i})]\\
&\leq \sum_{i=1}^n [H(g_{12}X_{1,i}+Z_{2,i})-H(Z_{2,i})]\\
&\leq \sum_{i=1}^n [\log (1+{\tt SNR}_{12})] \\
%\end{align*}
%\begin{align*}
 n(R_{21}-\epsilon)&\leq I(M_{21};Y_1^n|M_{12},M_{43},M_{34},Z_4^{n-1})\\
& \leq H(Y_1^n|M_{12},M_{34},M_{43},Z_4^{n-1})- H(Y_1^n|M_{12},M_{34},M_{43},Z_4^{n-1},M_{21},X_1^n,X_2^n,X_4^n)\\
&\overset{(b)}{\leq} \sum_{i=1}^n [H(Y_{1,i}|M_{12},M_{34},M_{43},Z_4^{n-1},Y_1^{i-1},X_{1,i},X_{4,i})-H(Z_{1,i})]\\
&\leq \sum_{i=1}^n [H(g_{21}X_{2,i}+Z_{1,i})-H(Z_{1,i})]\\
&\leq \sum_{i=1}^n [\log (1+{\tt SNR}_{21})]
\end{align*}
where (a) follows from the definition of partial adaptation and (b) follows by the same reason, as well as Lemma \ref{lemma:partialG}.
%\nrd{FOR THE FINAL EDITS WILL PLAY AROUND WITH THE SPACING SO DON'T HAVE LARGE WHITE SPACES; FOR NOW LEAVE AND THIS IS THE LAST THING TO DO.}

%\nrd{ZHIYU: can you please add which bound resembles what of previous papers for this whole section? I want to give credit to what we based our bounds on. Please include your changes in a different color.}
%
%\textcolor{blue}{Z-type bound \eqref{R_strong_E} is similar to Suh and Tse's bound \cite{Changho2010} , and \eqref{R_weak_E} and \eqref{R_weak_BE} are ETW-type bounds \cite{etkin_tse_wang} . The technique of deriving single-rate bounds seems standard, I just derived them by myself.}

Next, we consider the sum-rate bounds \eqref{R_weak_E} and \eqref{R_weak_BE}, which are inspired by the techniques used by Etkin, Tse and Wang for the interference channel \cite{etkin_tse_wang}.
For the $\rightarrow$ direction of the symmetric rate, 
%Now we derive a new outer bound to bound the second term in \ref{HK} if it is active, and for this new outer bound, we consider partial adaptation only.
\begin{align}
&n(R_{12}+R_{34}-\epsilon)\leq I(M_{12};Y_2^n,g_{14}X_1^n+Z_4^n,M_{21},M_{43})+I(M_{34};Y_4^n,g_{32}X_3^n+Z_2^n,M_{21},M_{43})\notag \\
& \overset{(a)}{=} H(Y_2^n|g_{14}X_1^n+Z_4^n,M_{43},M_{21})+H(g_{14}X_1^n+Z_4^n|M_{43},M_{21})-H(Y_2^n,g_{14}X_1^n+Z_4^n|M_{12},M_{21},M_{43})\notag \\
& +H(Y_4^n|g_{32}X_3^n+Z_2^n,M_{43},M_{21})+H(g_{32}X_3^n+Z_2^n|M_{43},M_{21})-H(Y_4^n,g_{32}X_3^n+Z_2^n|M_{34},M_{21},M_{43})\notag \\
& \overset{(b)}{=}H(Y_2^n|g_{14}X_1^n+Z_4^n,M_{43},M_{21})+\sum_{i=1}^n[H(g_{14}X_{1,i}+Z_{4,i}|g_{14}X_1^{i-1}+Z_4^{i-1},M_{43},M_{21},M_{34})\notag \\
&-H(Y_{2,i},g_{14}X_{1,i}+Z_{4,i}|Y_2^{i-1},g_{14}X_1^{i-1}+Z_4^{i-1},M_{12},M_{21},M_{43},X_2^i,X_1^i)]\notag \\
&+H(Y_4^n|g_{32}X_3^n+Z_2^n,M_{43},M_{21})+\sum_{i=1}^n[H(g_{32}X_{3,i}+Z_{2,i}|g_{32}X_3^{i-1}+Z_2^{i-1},M_{43},M_{21},M_{12})\notag \\
&-H(Y_{4,i},g_{32}X_{3,i}+Z_{2,i}|Y_4^{i-1},g_{32}X_3^{i-1}+Z_2^{i-1},M_{34},M_{21},M_{43},X_4^i,X_3^i)]\notag 
\end{align}
\begin{align}
& \overset{(c)}{=}H(Y_2^n|g_{14}X_1^n+Z_4^n,M_{43},M_{21})+\sum_{i=1}^n[H(g_{14}X_{1,i}+Z_{4,i}|g_{14}X_1^{i-1}+Z_4^{i-1},M_{43},M_{21},M_{34},X_3^i,Y_4^{i-1},X_4^i,\notag \\
& \ \ g_{32}X_3^{i-1}+Z_2^{i-1})-H(g_{32}X_{3,i}+Z_{2,i},Z_{4,i}|Y_2^{i-1},g_{14}X_1^{i-1}+Z_4^{i-1},M_{12},M_{21},M_{43},X_2^i,X_1^i,g_{32}X_3^{i-1}+Z_2^{i-1})]\notag \\
&+H(Y_4^n|g_{32}X_3^n+Z_2^n,M_{43},M_{21})+\sum_{i=1}^n[H(g_{32}X_{3,i}+Z_{2,i}|g_{32}X_3^{i-1}+Z_2^{i-1},M_{43},M_{21},M_{12},X_1^i,Y_2^{i-1},X_2^i,\notag \\
& \ \ g_{14}X_1^{i-1}+Z_4^{i-1})-H(g_{14}X_{1,i}+Z_{4,i},Z_{2,i}|Y_4^{i-1},g_{32}X_3^{i-1}+Z_2^{i-1},M_{34},M_{21},M_{43},X_4^i,X_3^i,g_{14}X_1^{i-1}+Z_4^{i-1})]\notag \\
& \overset{(d)}{=}\sum_{i=1}^n[H(Y_{2,i}|Y_2^{i-1},g_{14}X_1^n+Z_4^n,M_{43},M_{21})-H(Z_{2,i})+H(Y_{4,i}|Y_4^{i-1},g_{32}X_3^n+Z_2^n,M_{43},M_{21})-H(Z_{4,i})]\notag \\
& \leq \sum_{i=1}^n [H(g_{12}X_{1,i}+g_{32}X_{3,i}+Z_{2,i}|g_{14}X_{1,i}+Z_{4,i},X_{2,i})-H(Z_{2,i})\notag \\
& \ \ +H(g_{34}X_{3,i}+g_{14}X_{1,i}+Z_{4,i}|g_{32}X_{3,i}+Z_{2,i},X_{4,i})-H(Z_{4,i})] \label{last}
\end{align}
In the first step, we have given $(g_{14}X_1^n+Z_4^n)$ and $(g_{32}X_3^n+Z_2^n)$ as side information. 
%\nrd{Can we have genie $g_{14}X_1^n+Z_4^n$ and $g_{32}X_3^n+Z_2^n$ at the beginning?} \textcolor{blue}{I don't think so now: in step (c), the 2nd and 6th, the 3rd and 5th terms cannot be cancelled in this case.}
Step (a) follows from the independence of the messages. In step (b), the 2nd and 5th terms follow since $g_{14}X_{1,i}$ and $g_{32}X_{3,i}$ are functions only of $M_{12}$ and $M_{34}$, and the 3rd and 6th terms follow from the definition of partial adaptation. For (c), in the conditioning of the 2nd term, we are able to add $(X_3^i,g_{32}X_3^{i-1}+Z_2^{i-1})$ due to partial adaptation constraints,  and $(Y_4^{i-1}, X_4^i)$ are constructed from $(g_{14}X_1^{i-1}+Z_4^{i-1},M_{43},X_3^i)$. The 5th term follows similarly. In step (d), $-H(Z_{2,i})$ and $-H(Z_{4,i})$ are obtained from a portion of the 6th and 3rd terms in (c) respectively using the chain rule (noises are independent from other terms), and the remainder (chain rule) of the 6th and 3rd terms are cancelled by the 2nd and 5th terms respectively.

%Note in the conditioning part of the third term in (a),$Y_2^{i-1}$ is removed since it is a function of $(g_{32}X_3^{i-1},Z_2^{i-1},M_{12},M_{21})$. Similar reason for the removal of $Y_4^{i-1}$ in the conditioning part of the sixth term.

To obtain \eqref{R_weak_E} we continue to outer bound \eqref{last} in terms of ${\tt SNR}$ and ${\tt INR}$, using the fact that Gaussians maximize entropy subject to variance constraints. Specifically, one may intuitively see that, if one defines $\lambda_{jk} = E[X_j X_k^*]$, that one may express \eqref{last} in terms of $\lambda_{12}, \lambda_{13}, \lambda_{14}, \lambda_{34}, \lambda_{23}$. One also notices from the conditional entropy expression in \eqref{last} that taking $ \lambda_{14} = \lambda_{23} = \lambda_{12}  = \lambda_{34} = 0$, and since $\lambda_{13} = 0$ (naturally, by partial adaptation) will maximize the outer bound. This may alternatively be worked out by calculating the conditional covariance matrices directly (as we will show for the next bound on $R_{\leftarrow}$). In this case then,  for each $i$, we may bound
\begin{align*}
H(g_{12}X_{1}&+g_{32}X_{3}+Z_{2}|g_{14}X_{1}+Z_{4},X_{2}) - H(Z_2)  \leq H(g_{12}X_{1}+g_{32}X_{3}+Z_{2}|g_{14}X_{1}+Z_{4}) - H(Z_2) \\
&\leq \log 2\pi e(\mbox{Var}(g_{12}X_{1}+g_{32}X_{3}+Z_{2}|g_{14}X_{1}+Z_{4}))-\log 2\pi e(\mbox{Var}(Z_2)) \\
& \leq \log\left(1+{\tt SNR}+{\tt INR} - \frac{{\tt SNR}\times {\tt INR}}{1+{\tt INR}}\right),
\end{align*}
which together with the symmetric expressions for the third and fourth terms in \eqref{last} yield \eqref{R_weak_E}.
% BELOW IS THE FORMAL PROOF WHICH WE'RE LEAVING OUT.
%Now we are going to evaluate our bound. Intuitively, to maximize the rate $R_{12}+R_{34}$ in the forward direction, we expect $X_2$ and $X_4$ in the backward direction are independent of $X_1$ and $X_3$ to obtain the maximal entropy. Indeed, choosing the trivial values of the covariance coefficients $\lambda_{21},\lambda_{23},\lambda_{41},\lambda_{43}$ to zero we could maximize the entropy, and by definition of partial adaptation, we have $\lambda_{13}=0$. Note it's sufficient to evaluate the first two entropy terms in the last step due to symmetry. First we compute the following conditional variance:
%\begin{align}
%& Q=Var(g_{12}X_1+g_{32}X_3+Z_2|g_{14}X_1+Z_4,X_2)
%\end{align}

%Consider in symmetirc case:
%\begin{align*}
%&A=\left[ 1+{\tt SNR}+{\tt INR}+2\lambda_{13}\sqrt{{\tt SNR}\times {\tt INR}}\right]\\
%&B=\left[ \begin{array}{cc} \sqrt{{\tt SNR}\times {\tt INR}}+\lambda_{13}{\tt INR} & \lambda_{12}\sqrt{{\tt SNR}}+\lambda_{23}\sqrt{{\tt INR}} \end{array} \right]\\
%&C=\begin{bmatrix} {\tt INR}+1 & \lambda_{12}\sqrt{{\tt INR}} \\ \lambda_{12}\sqrt{{\tt INR}} & 1 \end{bmatrix}\\
%&Q=A-BC^{-1}B^*
%\end{align*}

%Now we substitute $\lambda_{13}$ by 0 due to partial adaptation and further computation gives us the following expression:
%\begin{align*}
%&Q=1+{\tt SNR}+{\tt INR}-\frac{{\tt SNR}\times {\tt INR}(1-\lambda_{12}^2)}{1+{\tt INR}-\lambda_{12}^2{\tt INR}}\\
%&-\frac{\lambda_{23}^2{\tt INR}^2+\lambda_{23}^2{\tt INR}+\lambda_{12}^2{\tt SNR}+2\lambda_{12}\lambda_{23}\sqrt{{\tt SNR}\times {\tt INR}}}{1+{\tt INR}-\lambda_{12}^2{\tt INR}}
%\end{align*}
%To maximize $Q$, we just remove the second fraction (actually it can be shown that $\lambda_{12}=\lambda_{23}=0$ ) and take derivative of the rest with respect to $\lambda_{12}$, then we have $\lambda_{12}=0$ which maximizes the bound. Together with evaluation of $-H(Z_{2,i})$ and assuming a symmetric model $R_{sym\rightarrow}=\frac{R_{12}+R_{34}}{2}$ we obtain \eqref{R_weak_E}.

\bigskip
For the $\leftarrow$ direction, we are similarly able to obtain:
\begin{align}
&n(R_{21}+R_{43}-\epsilon)\leq I(M_{21};Y_1^n,g_{23}X_2^n+Z_3^n,M_{12},M_{34})+I(M_{43};Y_3^n,g_{41}X_4^n+Z_1^n,M_{12},M_{34})\notag \\
& = H(Y_1^n|g_{23}X_2^n+Z_3^n,M_{34},M_{12})+H(g_{23}X_2^n+Z_3^n|M_{34},M_{12})-H(Y_1^n,g_{23}X_2^n+Z_3^n|M_{21},M_{12},M_{34})\notag \\
& +H(Y_3^n|g_{41}X_4^n+Z_1^n,M_{34},M_{12})+H(g_{41}X_4^n+Z_1^n|M_{34},M_{12})-H(Y_3^n,g_{41}X_4^n+Z_1^n|M_{43},M_{12},M_{34})\notag 
\end{align}
\begin{align}
& \overset{(a)}{=}H(Y_1^n|g_{23}X_2^n+Z_3^n,M_{34},M_{12})+\sum_{i=1}^n[H(g_{23}X_{2,i}+Z_{3,i}|g_{23}X_2^{i-1}+Z_3^{i-1},M_{34},M_{12},M_{43})\notag \\
&-H(Y_{1,i},g_{23}X_{2,i}+Z_{3,i}|Y_1^{i-1},g_{23}X_2^{i-1}+Z_3^{i-1},M_{21},M_{12},M_{34},X_1^i,Z_2^{i-1},X_2^i)]\notag \\
&+H(Y_3^n|g_{41}X_4^n+Z_1^n,M_{34},M_{12})+\sum_{i=1}^n[H(g_{41}X_{4,i}+Z_{1,i}|g_{41}X_4^{i-1}+Z_1^{i-1},M_{34},M_{12},M_{21})\notag \\
&-H(Y_{3,i},g_{41}X_{4,i}+Z_{1,i}|Y_3^{i-1},g_{41}X_4^{i-1}+Z_1^{i-1},M_{43},M_{12},M_{34},X_3^i,Z_4^{i-1},X_4^i)]\notag \\
& =H(Y_1^n|g_{23}X_2^n+Z_3^n,M_{34},M_{12})+\sum_{i=1}^n[H(g_{23}X_{2,i}+Z_{3,i}|g_{23}X_2^{i-1}+Z_3^{i-1},M_{34},M_{12},M_{43},Z_4^{i-1},g_{41}X_4^{i-1}+Z_1^{i-1},\notag \\
& \ \ X_3^i,X_4^i,Y_3^{i-1})-H(g_{41}X_{4,i}+Z_{1,i},Z_{3,i}|Y_1^{i-1},g_{23}X_2^{i-1}+Z_3^{i-1},M_{21},M_{12},M_{34},Z_2^{i-1},X_1^i,X_2^i,g_{41}X_4^{i-1}+Z_1^{i-1})]\notag \\
&+H(Y_3^n|g_{41}X_4^n+Z_1^n,M_{34},M_{12})+\sum_{i=1}^n[H(g_{41}X_{4,i}+Z_{1,i}|g_{41}X_4^{i-1}+Z_1^{i-1},M_{34},M_{12},M_{21},Z_2^{i-1},g_{23}X_2^{i-1}+Z_3^{i-1},\notag \\
& \ \ X_1^i,X_2^i,Y_1^{i-1})-H(g_{23}X_{2,i}+Z_{3,i},Z_{1,i}|Y_3^{i-1},g_{41}X_4^{i-1}+Z_1^{i-1},M_{43},M_{12},M_{34},Z_4^{i-1},X_3^i,X_4^i,g_{23}X_2^{i-1}+Z_3^{i-1})]\notag \\
& = \sum_{i=1}^n [H(Y_{1,i}|Y_1^{i-1},g_{23}X_2^n+Z_3^n,M_{34},M_{12})-H(Z_{1,i})+H(Y_{3,i}|Y_3^{i-1},g_{41}X_4^n+Z_1^n,M_{34},M_{12})-H(Z_{3,i})]\notag \\
&\leq \sum_{i=1}^n [H(g_{21}X_{2,i}+g_{41}X_{4,i}+Z_{1,i}|g_{23}X_{2,i}+Z_{3,i},X_{1,i})-H(Z_{1,i})\notag \\
& \ \ +H(g_{43}X_{4,i}+g_{23}X_{2,i}+Z_{3,i}|g_{41}X_{4,i}+Z_{1,i},X_{3,i})-H(Z_{3,i})]\label{last2}
\end{align}
The slight differences compared to the previous outer bound are: in step (a), we add $M_{43}$ in the conditioning part of the second term since it is independent of $X_{2,i}$ according to Lemma \ref{lemma:partialG}; in the conditioning part of the third term we add $Z_2^{i-1}$ (independent), which, together with $(M_{12},M_{21},M_{34})$ allow us to construct $X_2^i$. Similar arguments can be made for the fifth and sixth term. 

We again proceed to outer bound \eqref{last2} to obtain \eqref{R_weak_BE}. 
It is sufficient to evaluate the first two terms in \eqref{last2} due to symmetry. Once again, we could outer bound \eqref{last2} in terms of the conditional covariance matrices and then proceed to select values of the correlation coefficients (complex) $\lambda_{ij} : = E[X_i X_j^*]$  which maximize this outer bound. A more intuitive method is to note that  the conditional entropies in \eqref{last2} will be maximized if $\lambda_{14} = \lambda_{32}=0$, and $\lambda_{12}=\lambda_{34}=0$ (similar to \eqref{r12r34}), which may also be obtained by dropping $X_{1,i}, X_{3,i}$ in the conditioning terms. At that point, we are only left with the coefficient $\lambda_{24} = E[X_2X_4^*]$, (which in contrast to the $\rightarrow$ bound is not automatically $0$ due to the possible adaptation in the $\leftarrow$ direction. Furthermore, setting it to zero cannot be argued intuitively as we see a tradeoff.) yielding the following bound for $R_{sym \leftarrow} = \frac{R_{21}+R_{43}}{2}$ by symmetry: 
%\nrd{ZHIYU: CHECK THE COMPLEX STUFF AND THAT I GOT THE ABSOLUTE VALUES IN THE RIGHT SPOTS}
\begin{align}
R_{sym\leftarrow}&\leq H(g_{21}X_{2}+g_{41}X_{4}+Z_{1}|g_{23}X_{2}+Z_{3},X_{1})-H(Z_{1})\notag \\
&\leq H(g_{21}X_{2}+g_{41}X_{4}+Z_{1}|g_{23}X_{2}+Z_{3})-H(Z_{1})\notag \\
&\leq \log 2\pi e\left(\mbox{Var}(g_{21}X_{2}+g_{41}X_{4}+Z_{1}|g_{23}X_{2}+Z_{3})\right)-\log 2\pi e (\mbox{Var}(Z_1))\notag \\
&\leq \log \left( 1+{\tt INR}+{\tt SNR}+2|\lambda_{24}|\cos \theta\sqrt{{\tt SNR}\times {\tt INR}}-\frac{{\tt SNR}\times {\tt INR} + {\tt INR}^2|\lambda_{24}|^2 + 2\sqrt{{\tt SNR}}{\tt INR}^{3/2}|\lambda_{24}|\cos\theta}{1+{\tt INR}}\right).\label{lambda}
%&\leq \log \left( 1+{\tt INR}+{\tt SNR}+2|\lambda_{24}|\sqrt{{\tt SNR}\times {\tt INR}}-\frac{(\lambda_{24}{\tt INR}+\sqrt{{\tt SNR}\times {\tt INR}})^2}{1+{\tt INR}}\right)
\end{align}
where $\theta$ is the angle of $g_{21}g_{41}^*\lambda_{24}$. To maximize \eqref{lambda}, we take the partials of the expression with respect to $|\lambda_{24}|$ and $\theta$ and set these to 0. For these to equal 0 for all ${\tt SNR}$ and ${\tt INR}$ we must have $\theta = 0$ and  $|\lambda_{24}|=\frac{\sqrt{{\tt SNR}\times {\tt INR}}}{{\tt INR}^2}$ (discussed next).  
Note that we must constrain $|\lambda_{24}|\in [0,1]$. In the interval $|\lambda_{24}| \in \left[0, \frac{\sqrt{\tt SNR\times INR}}{{\tt INR}^2}\right]$ one may verify that the function is increasing in $|\lambda_{24}|$. Thus, if $ \frac{\sqrt{\tt SNR\times INR}}{{\tt INR}^2} \leq 1$, $(|\lambda_{24}| =  \frac{\sqrt{\tt SNR\times INR}}{{\tt INR}^2}, \theta =0)$ maximizes \eqref{lambda}; this happens if ${\tt SNR}\leq {\tt INR}^3$, and yields the first bound in \eqref{R_weak_BE}. Otherwise, for ${\tt SNR}>{\tt INR}^3$, $(\lambda_{24} = 1, \theta=0)$ maximizes \eqref{lambda}, yielding the second equation in \eqref{R_weak_BE}. 
%
%\nbl{OLD VERSION:
%where $\theta$ is the angle between $|g_{21}|$ and $|g_{41}|$. To maximize \eqref{lambda}, one may take first derivative and second derivative with respect ro $|\lambda_{24}|$ and $\theta$ respectively to get $(|\lambda_{24}|=\frac{\sqrt{{\tt SNR}\times {\tt INR}}}{{\tt INR}^2}, \theta=0)$.  
%Note that we must constrain $|\lambda_{24}|\in [0,1]$. In the interval $|\lambda_{24}| \in \left[0, \frac{\sqrt{\tt SNR\times INR}}{{\tt INR}^2}\right]$ one may verify that the function is increasing in $|\lambda_{24}|\cos\theta$. Thus, if $ \frac{\sqrt{\tt SNR\times INR}}{{\tt INR}^2} \leq 1$, $(|\lambda_{24}| =  \frac{\sqrt{\tt SNR\times INR}}{{\tt INR}^2}, \theta =0)$ maximizes \eqref{lambda}; this happens if ${\tt SNR}\leq {\tt INR}^3$, and yields the first bound in \eqref{R_weak_BE}. Otherwise, for ${\tt SNR}>{\tt INR}^3$, $(\lambda_{24} = 1, \theta=0)$ maximizes \eqref{lambda}, yielding the second equation in \eqref{R_weak_BE}. 
%}

%Further observe that these two terms are symmetric (in the sense of directions) to the first two terms in the last step of the previous bound, therefore by repeating all the evaluation steps in the previous bound, and without loss of generality, we only consider the first two terms (the other two terms can be maximized in the same way but with different $\lambda$s), 
%then we obtain the following expression:
%\begin{align*}
% R_{sym\leftarrow} \leq &  \log \left( 1+{\tt INR}+{\tt SNR}+2\lambda_{24}\sqrt{{\tt SNR}\times {\tt INR}}
% \right. \notag\\
% &\left.
% -\frac{[{\tt INR}(\lambda_{24}-\lambda_{12}\lambda_{14})+\sqrt{{\tt SNR}\times {\tt INR}}(1-\lambda_{12}^2)](\lambda_{24}{\tt INR}+\sqrt{{\tt SNR}\times {\tt INR}})}{1+{\tt INR}-\lambda_{12}^2{\tt INR}}
% \right. \notag \\
% &\left.
% -\frac{[\lambda_{14}\sqrt{{\tt INR}}(1+{\tt INR})+\lambda_{12}(\sqrt{{\tt SNR}}-\lambda_{24}{\tt INR}^{3/2})](\lambda_{14}\sqrt{{\tt INR}}+\lambda_{12}\sqrt{{\tt SNR}})}{1+{\tt INR}-\lambda_{12}^2{\tt INR}}
% \right) \\
%\end{align*}
%To maximize this bound, it is easy to choose $\lambda_{14}=0$ since $X_{1,i}$ is in the conditioning and we hope it to be independent of $X_{4,i}$ to make the entropy larger. However, in this bound we can not assume the covariance coefficient $\lambda_{24}=0$ since $X_2$ and $X_4$ are not necessarily independent. 
%Only keep $\lambda_{24}$ (since it has tradeoff. Actually $\lambda_{12}$ also has tradeoff, but this is similar to equation \eqref{R_strong}, i.e. $\lambda_{12}=0$ maximizes the bound):
%\begin{align}
%R_{sym\leftarrow}\leq \log \left( 1+{\tt INR}+{\tt SNR}+2\lambda_{24}\sqrt{{\tt SNR}\times {\tt INR}}-\frac{(\lambda_{24}{\tt INR}+\sqrt{{\tt SNR}\times {\tt INR}})^2}{1+{\tt INR}}\right)\label{R_prime}
%\end{align}

%Take the derivative of the above equation and make it to zero:
%\begin{align*}
%\frac{2(\sqrt{{\tt SNR}\times {\tt INR}}-\lambda_{24}{\tt INR}^2)}{1+2{\tt INR}+{\tt SNR}+2\lambda_{24}\sqrt{{\tt SNR}\times {\tt INR}}-{\tt INR}^2(\lambda_{24}^2-1)}=0
%\end{align*}
% we have $\lambda_{24}=\frac{\sqrt{{\tt SNR}\times {\tt INR}}}{{\tt INR}^2}$. We also compute the second derivative of \eqref{R_prime}: 
%\begin{align*}
%-\frac{2{\tt INR}({\tt INR}^3(1+\lambda_{24}^2)+2{\tt INR}^2+{\tt INR}(1+{\tt SNR}-2\lambda_{24}\sqrt{{\tt SNR}\times {\tt INR}})+2{\tt SNR})}{({\tt INR}^2(1-\lambda_{24}^2)+2\lambda_{24}\sqrt{{\tt SNR}\times {\tt INR}}+2{\tt INR}+{\tt SNR}+1)^2}
%\end{align*} 
% which is negative at $\lambda_{24}=\frac{\sqrt{{\tt SNR}\times {\tt INR}}}{{\tt INR}^2}$. Therefore this value maximize \eqref{R_prime}, and by assuming a symmetric model we obtain the first term in \eqref{R_weak_BE}. %have a new outer bound for the backward direction of the Gaussian two-way interference channel with partial adaptation: %\nrd{change all this if bound changes!}

%%\begin{align*}
%%R_{sym\leftarrow}\leq \log 2\pi e(1+INR+SNR+\frac{2SNR}{INR}-\frac{(\frac{\sqrt{SNR\times INR}}{INR}+\sqrt{SNR\times INR})^2}{1+INR})\label{R_B}
%%\end{align*}
%Further notice that $|\lambda_{24}|\leq 1$, so the first term in \eqref{R_weak_BE} is based on the condition of ${\tt SNR}\leq {\tt INR}^3$. For the case of ${\tt SNR}> {\tt INR}^3$, it is easy to see that as ${\tt SNR}$ increasing, the maximal value $\lambda_{24}$ can take is 1. Thus, we have
%\begin{align*}
%\lambda_{24}=\min (1, \frac{\sqrt{{\tt SNR}\times {\tt INR}}}{{\tt INR}^2})
%\end{align*}
%And when $\lambda_{24}=1$, we obtain the second term in \eqref{R_weak_BE}. 
%%\begin{align*}
%%R_{sym\leftarrow}\leq  \log 2\pi e\left( 1+INR+SNR+2\sqrt{SNR\times INR}-\frac{(INR+\sqrt{SNR\times INR})^2}{1+INR}\right)
%%\end{align*}
\end{proof}


\begin{remark} %\nrd{Change remark if the bound actually changes!}
 The sum-rate bound for $R_{sym\rightarrow}$ of \eqref{R_weak_E} has the same form as Etkin, Tse and Wang's outer bound for one-way Gaussian interference channel \cite[(12)]{etkin_tse_wang} which is useful in weak interference.  The sum-rate bound for $R_{sym\leftarrow}$ is quite different, and we note that it may be verified that \eqref{R_weak_BE} is always at least as large as \eqref{R_weak_E}, as one might expect given the partial adaptation constraints on nodes in the $\rightarrow$ direction, but none on the nodes in the $\leftarrow$ direction.   
% \nrd{ZHIYU: I DID NOT UNDERSTAND YOUR REASONING HERE; I'D LIKE MORE INTUITION AS TO WHY WE DON'T GET AN ETW LIKE BOUND IN THE BACKWARDS DIRECTION AND WHY WE DO IN THE FORWARD... IT'S INTERESTING BUT I DON'T KNOW HOW TO EXPLAIN IT YET.  It is because $\lambda_{12}=\lambda_{23}=0$ renders the self-interference $X_2$ redundant and we further have $\lambda_{13}=0$ due to partial adaptation. These make our channel model similar to the regular interference channel and therefore our bound coincides with the ETW's bound.}
\end{remark}

\bigskip

We next show that these outer bounds, derived for the fully adaptive or partially adaptive models, may be achieved to within a constant gap or capacity by {\it non-adaptive} schemes -- i.e. the simultaneous decoding or the Han and Kobayashi scheme operating in the two directions independently. We break our analysis into three sub-sections: 1) very strong interference, 2) strong interference, and 3) weak interference. The overall finite gap results are summarized in Table \ref{table:gaps}.

\subsection{Very Strong Interference: ${\tt INR}\geq {\tt SNR}(1+{\tt SNR})$}
We first show that a non-adaptive scheme may achieve the capacity for the two-way Gaussian IC under a partially adaptive model in very strong interference. For the symmetric two-way Gaussian IC, define ``very strong interference'' as the class of channels for which ${\tt INR}\geq {\tt SNR}(1+{\tt SNR})$, as in \cite[below equation (21)]{etkin_tse_wang}.
%
%\begin{definition}
%\nrd{I changed according to ETW around equation (21): A two-way symmetric Gaussian interference channel is said to be in ``very strong interference'' }
%when ${\tt INR}\geq {\tt SNR}(1+{\tt SNR})$.
%\end{definition}
It is well known that the capacity region of the one-way Gaussian IC in very strong interference is that of two parallel Gaussian point-to-point channels \cite{Carleial1975}, which may be achieved by having each receiver first decode the interfering signal, treating its own as noise, subtracting off the decoded interference, and decoding its own message. Given that the interference is so strong,  this may be done without a rate penalty. We ask whether the same is true for the two-way Gaussian IC with partial adaptation. The answer is affirmative and the capacity region is given by the following theorem:

 
%First we state the capacity of Gaussian one-way interference channel when the interference is very strong:

%\begin{theorem}
%\cite{Carleial1975} The capacity region for the Gaussian one-way interference channel with very strong interference is the set of rate pairs ($R_{12},R_{34}$), such that 
%\begin{align}
%&R_{12}\leq \log (1+SNR_{12})\\
%&R_{34}\leq \log (1+SNR_{34})
%\end{align}
%\end{theorem}
%
%Now we derive our first bound with full adaptation, which intuitively is useless, because very strong interference is as good as no interference in one-way Gaussian interference channel, and if no interference in our channel, then the channel reduces to two point-to-point Gaussian two-way channels, which are proved by \cite{Han:1984} that adaptation is useless. 

%

%Achievability may be argued via mimicking the achievable scheme of one-way Gaussian interference channel without considering adaptation. We now prove the converse:


%However, this is cannot be done because we only could decode $g_{12}X_1^{i-1}$ from $Y_2^{i-1}$. If all the channel gains are the same, then it is okay, but in that case the interference is no longer very strong. 

%Fortunately, if we consider partial adaptation, then we are able to obtain the capacity and conclude that partial adaptation can not enlarge the capacity.

\begin{theorem}
\label{2wIC_very_strong}
The capacity region for the two-way Gaussian interference channel with partial adaptation in very strong interference is the set of rate pairs ($R_{12},R_{21},R_{34},R_{43}$), such that \eqref{partial_single_rate1}--\eqref{partial_singe-rate4} are satisfied. 
\end{theorem}
\begin{proof}
Each node may ignore its ability to adapt,  and rather transmit using a ${\cal CN}(0,1)$ Gaussian random code. Each receiver may cancel its own self-interference, and then proceed to first decode the single interfering term before decoding its own message. This standard non-adaptive scheme may achieve the outer bound in \eqref{partial_single_rate1}--\eqref{partial_singe-rate4} in Theorem \ref{thm:outer-partial}.
% , and the converse has been proved in Theorem \ref{thm:outer-partial}.
\end{proof}
% OLD THEOREM+PROOF
%\begin{theorem}
%The capacity region for the Gaussian two-way interference channel with very strong interference is the set of rate pairs ($R_{12},R_{21},R_{34},R_{43}$), such that
%\begin{align}
%&R_{12}\leq \log (1+SNR_{12})\\
%&R_{21}\leq \log (1+SNR_{21})\\
%&R_{34}\leq \log (1+SNR_{34})\\
%&R_{43}\leq \log (1+SNR_{43})
%\end{align}
%\end{theorem}
%\begin{proof}
%We could achieve the capacity by mimicking the achievable scheme for one-way Gaussian interference channel with very strong interference without considering adaptation. Now we prove the converse:
%\begin{align*}
%& n(R_{12}-\epsilon)\\
%&\leq I(M_{12};Y_2^n,Y_3^n|M_{21},M_{34},M_{43},Z_2^{n-1},Z_3^n,Z_4^{n-1})\\
%&\leq H(Y_2^n,Y_3^n|M_{21},M_{34},M_{43},Z_2^{n-1},Z_3^n,Z_4^{n-1})-H(Z_2^n)\\
%& \leq \sum_{i=1}^n [H(g_{12}X_{1,i}+g_{22}X_{2,i}+g_{32}X_{3,i}+Z_{2,i}, g_{23}X_{2,i}+g_{33}X_{3,i}+g_{43}X_{4,i}+Z_{3,i}|M_{21},M_{34},M_{43},Z_3^n,Y_2^{i-1},Y_3^{i-1},\\
%& \ \ \ \ \ \ \ X_2^i,X_3^i,Z_2^{i-1},Z_4^{i-1})-H(Z_{2,i})]\\
%& \overset{(a)}{\leq} \sum_{i=1}^n [H(g_{12}X_{1,i}+Z_{2,i},g_{43}X_{4,i}+Z_{3,i}|M_{21},M_{34},M_{43},Z_3^n,Y_2^{i-1},Y_3^{i-1},X_2^i,X_3^i,Z_2^{i-1},Z_4^{i-1},X_1^{i-1},X_{4,i})-H(Z_{2,i})]\\
%%& \leq \sum_{i=1}^n [H(g_{12}X_{1,i}+Z_{2,i}|X_{2,i},X_{3,i}+Z_{4,i})-H(Z_{2,i})]\\
%&\leq \sum_{i=1}^n [H(g_{12}X_{1,i}+Z_{2,i})-H(Z_{2,i})]
%\end{align*}
%In (a), $X_1^{i-1}$ can be decoded from $Y_2^{i-1}$ since we know $X_2^i,X_3^i,Z_2^{i-1}$, and $X_{4,i}$ can be constructed since we know $X_1^{i-1},X_3^{i-1},Z_4^{i-1},M_{43}$. 

%%\begin{align*}
%%& n(R_{12}-\epsilon)\\
%%&\leq I(M_{12};Y_2^n|M_{21},M_{34})\\
%%&=H(Y_2^n|M_{21},M_{34})-H(Z_2^n)\\
%%& \overset{(a)}{\leq} \sum_{i=1}^n [H(g_{12}X_{1,i}+g_{22}X_{2,i}+g_{32}X_{3,i}+Z_{2,i}|M_{21},M_{34},Y_2^{i-1},X_{2,i},X_{3,i})-H(Z_{2,i})]\\
%%& \leq \sum_{i=1}^n [H(g_{12}X_{1,i}+Z_{2,i}|M_{21},M_{34},Y_2^{i-1},X_{2,i},X_{3,i})-H(Z_{2,i})]\\
%%&\leq \sum_{i=1}^n [H(g_{12}X_{1,i}+Z_{2,i})-H(Z_{2,i})]
%%\end{align*}
%From here we could easily evaluate our bound as $R_{12}\leq \log (1+SNR_{12})$ by using Gaussian maximizes entropy. The other three bounds follow by symmetry.
%%The key observation is that in step (a), we could add $X_{3,i}$ by definition of partial adaptation in the conditioning of the first entropy term because we have $M_{34}$. Now we prove the single rate bound in the opposite direction:
%%\begin{align*}
%%& n(R_{21}-\epsilon)\\
%%&\leq I(M_{21};Y_1^n|M_{12},M_{34},M_{43},Z_4^n)\\
%%&=H(Y_1^n|M_{12},M_{34},M_{43},Z_4^n)-H(Z_1^n)\\
%%& \overset{(b)}{\leq} \sum_{i=1}^n [H(g_{21}X_{2,i}+g_{11}X_{1,i}+g_{41}X_{4,i}+Z_{1,i}|M_{12},M_{34},M_{43},Y_1^{i-1},X_{1,i},Z_4^n,X_{4,i})-H(Z_{1,i})]\\
%%& \leq \sum_{i=1}^n [H(g_{21}X_{2,i}+Z_{1,i}|M_{21},M_{34},M_{43},Y_1^{i-1},X_{1,i},Z_4^n,X_{4,i})-H(Z_{1,i})]\\
%%&\leq \sum_{i=1}^n [H(g_{21}X_{2,i}+Z_{1,i})-H(Z_{1,i})]
%%\end{align*}
%%In step (b), we add $X_{4,i}$ in the conditioning of the first entropy term is due to Lemma \ref{lemma:partial}. By Gaussian maximizes entropy we have the desired bound $R_{21}\leq \log (1+SNR_{21})$. The other two single rate bounds follow by symmetry.
%\end{proof}

Interestingly, the capacity region of the two-way Gaussian interference channel with partial adaptation in very strong interference, is equivalent to the capacity regions of two one-way Gaussian interference channels with very strong interference in parallel and is achieved using a non-adaptive scheme. This allows us to conclude that {\it partial} adaptation is useless in this symmetric and very strong interference regime.

\subsection{Strong Interference: ${\tt SNR}\leq {\tt INR}\leq {\tt SNR}(1+{\tt SNR})$}
In this regime, we are able to show that a non-adaptive scheme may achieve capacity to within a constant gap of any {\it fully} adaptive scheme (in contrast to any {\it partially} adaptive scheme in the last subsection).  A symmetric two-way Gaussian IC,  as in \cite{etkin_tse_wang}, is said to be in ``strong interference'' when ${\tt INR}\geq {\tt SNR}$.

The capacity region of one-way Gaussian interference channel in strong interference is given by \cite{sato_strong}, and for symmetric channels, the capacity region when the interference is strong but not very strong, i.e. ${\tt SNR}\leq {\tt INR}\leq {\tt SNR}(1+{\tt SNR})$,  may be written as 
\begin{align}
R_{sym} = \frac{R_{12}+R_{34}}{2} \leq \frac{1}{2}\log (1+{\tt SNR}+{\tt INR}).\label{R_sato}
\end{align} 
We note that this rate is achievable for the two-way Gaussian IC  by using the simultaneous non-unique decoding scheme for the interference channel in strong interference \cite{ElGamalKim:book, sato_strong, ahlswede1974}) in the $\rightarrow$ and $\leftarrow$ directions, and noting that any self-interference may be canceled. This is a non-adaptive scheme.

We will show that this non-adaptive scheme which achieves \eqref{R_sato} in each direction also achieves to within 1 bit (per user, per direction) of our fully adaptive outer bound \eqref{R_strong_E} in strong but not very strong interference. 

%We will show that the capacity region for two-way Gaussian interference channel with full adaptation is within 1 bit to \eqref{R_sato} in both 
%directions. 
\begin{theorem}
The capacity region for two-way symmetric Gaussian interference channel with full adaptation in strong (but not very strong) interference is within 1 bit to \eqref{R_sato} (per user, per direction)
\end{theorem}
\begin{proof}
%We use \eqref{R_strong_E} to bound \eqref{R_sato} to within 1 bit:
\begin{align*}
 \eqref{R_strong_E} - \eqref{R_sato}
&= \frac{1}{2} \log (1+{\tt SNR}+{\tt INR}+2\sqrt{{\tt SNR}\times {\tt INR}})+\frac{1}{2}\log \left(1+\frac{{\tt SNR}}{1+{\tt INR}}\right)-\frac{1}{2}\log \left(1+{\tt SNR}+{\tt INR}\right)\\
&\overset{(a)}{\leq} \frac{1}{2} \log 2(1+{\tt SNR}+{\tt INR})+\frac{1}{2}\log \left(1+\frac{{\tt SNR}}{1+{\tt INR}}\right)-\frac{1}{2}\log (1+{\tt SNR}+{\tt INR})\\
&\overset{(b)}{\leq} \frac{1}{2}+\frac{1}{2}\log \left(1+\frac{{\tt INR}}{{\tt INR}}\right)\\
&=1
\end{align*}
In step (a), we use the fact that $1+{\tt SNR}+{\tt INR}+2\sqrt{{\tt SNR}\times {\tt INR}}\leq 2(1+{\tt SNR}+{\tt INR})$. Step (b) follows from the condition of strong interference ${\tt INR}\geq {\tt SNR}$. Since our bound \eqref{R_strong_E} is valid for the symmetric assumptions of full adaptation, we conclude that the non-adaptive schemes' gap to the fully adaptive outer bound for each user, for each direction is at most 1 bit.
\end{proof}
%First we state the capacity of Gaussian one-way interference channel with strong interference:

%\begin{theorem}
%\cite{sato_strong} The capacity region for the Gaussian one-way interference channel with strong interference is the set of rate pairs ($R_{12}, R_{34}$), such that 
%\begin{align}
%&R_{12}\leq \log (1+SNR_{12})\\
%&R_{34}\leq \log (1+SNR_{34})\\
%&R_{12}+R_{34}\leq \min \{\log (1+INR_{32}+SNR_{12}),\log (1+INR_{14}+SNR_{34})\}
%\end{align}
%\end{theorem}

%For symmetric case, the capacity (when the interference is strong but not very strong) is 
%\begin{align}
%R_{Sato}\leq 1/2\log (1+SNR+INR)
%\end{align} 


%Now we show that our bound is within 1 bit to the symmetric capacity of one-way Gaussian interference channel with strong interference (but not very strong)\cite{sato_strong} $R_{Sato}\leq 1/2\log (1+SNR+INR)$:


\begin{remark} Note that if we were to evaluate the fully adaptive outer bound of \eqref{R_strong} under partial adaptation constraints instead, i.e., $X_1$ and $X_3$ are only functions of $M_{12}$ and $M_{34}$ respectively,  then we would be able to set $\lambda_{13}$ in \eqref{R_strong} equal to 0, yielding a new outer bound $R_{sym\rightarrow} \leq \frac{1}{2}\log(1+{\tt SNR}+{\tt INR}) + \frac{1}{2}\log\left(1+\frac{{\tt SNR}}{1+{\tt INR}}\right)$. In this case a gap of $\frac{1}{2}$ bit instead of 1 bit may be shown for  $R_{sym \rightarrow}$. However, due to the asymmetry of partial adaptation ($\lambda_{24}$ in general not equal to 0), in the opposite direction,  we would still have a 1 bit gap for $R_{sym\leftarrow}$. 
%But if we consider the opposite direction, the gap would still be 1 bit since $\lambda_{24}\neq 0$.
\end{remark}

\subsection{Weak Interfererence: ${\tt INR}\leq {\tt SNR}$}
We now show that the well known Han and Kobayashi scheme employed in the $\rightarrow$ and $\leftarrow$ directions may achieve to within a constant number of bits of the fully or partially adaptive (depends on the channel regimes, or relative ${\tt SNR}$ and ${\tt INR}$ values) capacity region for the two-way Gaussian IC. 

%a fully or partially adaptive model for the two-way symmetric Gaussian IC in weak interference to within a constant gap. 
%We have showed that our bound \ref{R_strong} in full adaptation for Gaussian two-way interference channel with strong interference is within 1 bit to the capacity of one-way Gaussian interference channel with strong interference in both directions, now we proceed to consider Gaussian two-way interference channel with weak interference. 
\begin{theorem}
 A non-adaptive scheme may achieve to within a $2$ bit per user per direction of partially adaptive capacity region for the two-way Gaussian IC in weak interference. In some channel regimes, this non-adaptive scheme also achieves to within a constant gap of any {\it fully} adaptive scheme.
%The capacity region for two-way symmetric Gaussian interference channel in weak interference with full or partial adaptation is within at most 2 bits to Han and Kobayashi region in both directions.
\end{theorem}
\begin{proof}
As for the one-way IC \cite{etkin_tse_wang}, we break our proof into two regimes: ${\tt INR}\geq 1$ or ${\tt INR}<1$. % as \cite{etkin_tse_wang} did for one-way Gaussian IC, we discuss these two cases seperately.

\subsubsection{${\tt INR}\geq 1$}
Outer bounds have already been derived. Consider now using the specific choice of the Han and Kobayashi  (HK) strategy utilized for the symmetric one-way IC as in \cite[(4)]{etkin_tse_wang}  in each direction. That is, view nodes 1,2 as transmitters and 3,4 as receivers in the $\rightarrow$ direction and employ the particular choice of the HK scheme where private messages are encoded at the level of the noise, and similarly for the $\leftarrow$ direction consider nodes 3,4 as transmitters and 1,2 as receivers. Due to the additive nature of the channel and each node's ability to first cancel out their self-interference, one may achieve the following rates per user, per node for each direction when ${\tt INR}\geq 1$ for the symmetric two-way Gaussian IC:
%The best known inner bound for one-way Gaussian interference channel is based on Han and Kobayashi scheme \cite{H+K} and paraphrased by \cite{etkin_tse_wang} when ${\tt INR}\geq 1$ in symmetric case:
\begin{align}
& R_{HK}=\min \left\{\frac{1}{2}\log (1+{\tt INR}+{\tt SNR})+\frac{1}{2}\log \left(2+\frac{{\tt SNR}}{{\tt INR}}\right)-1, \log \left(1+{\tt INR}+\frac{{\tt SNR}}{{\tt INR}}\right)-1\right\} \label{HK} \\
& =: \min\{R_{HK1}, R_{HK2}\}. \label{HKdef}
\end{align}

\smallskip
\noindent {\bf If the first term in \eqref{HK} is active} we show a constant gap to the outer bound \eqref{R_strong_E},
\begin{align*}
& \eqref{R_strong_E}-R_{HK1}\\
&=\frac{1}{2} \log \left(1+{\tt SNR}+{\tt INR}+2\sqrt{{\tt SNR}\times {\tt INR}}\right)+\frac{1}{2}\log \left(1+\frac{{\tt SNR}}{1+{\tt INR}}\right)\\
&-\frac{1}{2}\log (1+{\tt INR}+{\tt SNR})-\frac{1}{2}\log \left(2+\frac{{\tt SNR}}{{\tt INR}}\right)+1\\
&\leq \frac{1}{2} \log 2(1+{\tt SNR}+{\tt INR})-\frac{1}{2}\log (1+{\tt INR}+{\tt SNR})+\frac{1}{2}\log \left(1+\frac{{\tt SNR}}{{\tt INR}}\right)-\frac{1}{2}\log \left(2+\frac{{\tt SNR}}{{\tt INR}}\right)+1\\
&\leq \frac{1}{2}\log (2)+\frac{1}{2}\log (1)+1\\
&=1.5
\end{align*}

\begin{remark} Since our bound \eqref{R_strong_E} is derived assuming full adaptation,  we may conclude that this gap holds for both $R_{sym\rightarrow}$ and $R_{sym\leftarrow}$ (i.e. holds for $R_{sym}$). 
%for the opposite direction of our channel model, the constant gap is also 1.5 bits. In addition, 
If  we were to consider partial adaptation ($\lambda_{13}=0$), this gap could be reduced to 1 bit instead of 1.5 bits for $R_{sym\rightarrow}$, but would remain 1.5 bits for $R_{sym\leftarrow}$ as $\lambda_{24}\neq 0$ in general for partial adaptation.\end{remark}

 \smallskip
% But can we therefore claim that partial adaptation is useless since this bound coincide with the ETW's bound?
\noindent {\bf If the second term in \eqref{HK} is active,} {we use outer bound \eqref{R_weak_E} for the forward direction, to bound the gap for $R_{sym\rightarrow}$ as }
\begin{align*}
 \eqref{R_weak_E}-R_{HK2}& =\log \left(1+{\tt INR}+{\tt SNR}-\frac{{\tt INR}\times {\tt SNR}}{1+{\tt INR}}\right)-\log \left(1+{\tt INR}+\frac{{\tt SNR}}{{\tt INR}}\right)+1\\\
& =\log \left(\frac{(1+{\tt INR})^2+{\tt SNR}}{1+{\tt INR}}\frac{{\tt INR}}{{\tt INR}(1+{\tt INR})+{\tt SNR}}\right)+1\\
&= \log \left(\frac{{\tt INR}(1+{\tt INR})^2+{\tt SNR}\times {\tt INR}}{{\tt INR}(1+{\tt INR})^2+{\tt SNR}(1+{\tt INR})}\right)+1\\
&\leq \log (1)+1\\
&=1
\end{align*}
Since our bound \eqref{R_weak_E}  has the same form as the ETW bound \cite{etkin_tse_wang}, the capacity of the Gaussian two-way interference channel with partial adaptation in the forward direction is also to within 1 bit of the specific HK rate \eqref{HK}, \eqref{HKdef} when ${\tt INR}\geq 1$.

%Although we do not assume weak interference $INR\leq SNR$ when we bounding the gap, it is only good for $0<\alpha <2/3$, since $R_{HK2}$ is corresponding to this regime.

{We use outer bound \eqref{R_weak_BE} for the backward direction, to bound the gap for $R_{sym\leftarrow}$,} noting that we need to consider both cases separately. 
{If the first term in \eqref{R_weak_BE} is relevant (${\tt SNR}\leq {\tt INR}^3$):}
%\nrd{ZHIYU: PLEASE PUT LEFT ANF RIGHT BRACKETS ( ) TO MAKE THE EQUATIONS LOOK GOOD -- I.E. TO MAKE BIG PARENTHESIS}
%Now we proceed to consider the backward direction of Gaussian two-way interference channel with partial adaptation, and we first use the first term in \eqref{R_weak_BE} to bound the second term in \eqref{HK}:
%However, since we apply partial adaptation in deriving bound $R_{12}+R_{34}$, it is necessary to consider bound $R_{21}+R_{43}$ since it is not symmetric to bound $R_{12}+R_{34}$.
%Now we show that this new bound \eqref{R_B}  is within $\log (2\pi e)+1$ bits to the second term of \ref{HK}:
\begin{align*}
  \eqref{R_weak_BE} -R_{HK2}
%& =\log  \left(1+{\tt INR}+{\tt SNR}+\frac{2{\tt SNR}}{{\tt INR}}-\frac{\left(\frac{\sqrt{{\tt SNR}\times {\tt INR}}}{{\tt INR}}+\sqrt{{\tt SNR}\times {\tt INR}}\right)^2}{1+{\tt INR}}\right)-\log \left(1+{\tt INR}+\frac{{\tt SNR}}{{\tt INR}}\right)+1\\
%& =\log  \left(\left(1+{\tt INR}+{\tt SNR}-\frac{{\tt SNR}\times {\tt INR}}{1+{\tt INR}}\right)+\left(\frac{2{\tt SNR}}{{\tt INR}}-\frac{\frac{{\tt SNR}}{{\tt INR}}+2{\tt SNR}}{1+{\tt INR}}\right)\right)-\log \left(1+{\tt INR}+\frac{{\tt SNR}}{{\tt INR}}\right)+1\\
%&= \log  \left(\left(1+{\tt INR}+{\tt SNR}-\frac{{\tt SNR}\times {\tt INR}}{1+{\tt INR}}\right)+\left(\frac{2{\tt SNR}(1+{\tt INR})-{\tt SNR}-2{\tt SNR}\times {\tt INR}}{{\tt INR}(1+{\tt INR})}\right)\right)\\
%& \ \ \ \ -\log \left(1+{\tt INR}+\frac{{\tt SNR}}{{\tt INR}}\right)+1\\
%&=\log \left(\frac{(1+{\tt INR})^2{\tt INR}+{\tt SNR}\times {\tt INR}+{\tt SNR}}{{\tt INR}(1+{\tt INR})}\right)-\log \left(1+{\tt INR}+\frac{{\tt SNR}}{{\tt INR}}\right)+1\\
&=\log \left(1+{\tt INR}+\frac{{\tt SNR}}{{\tt INR}}\right)-\log \left(1+{\tt INR}+\frac{{\tt SNR}}{{\tt INR}}\right)+1 =1
%&=\log \left(\frac{((1+{\tt INR})^2{\tt INR}+{\tt SNR}\times {\tt INR}+{\tt SNR}){\tt INR}}{{\tt INR}(1+{\tt INR})((1+{\tt INR}){\tt INR}+{\tt SNR})}\right)+1\\
%&=\log \left(\frac{{\tt INR}^2(1+{\tt INR})^2+{\tt SNR}\times {\tt INR}^2+{\tt SNR}\times {\tt INR}}{{\tt INR}^2(1+{\tt INR})^2+{\tt SNR}\times {\tt INR}(1+{\tt INR})}\right)+1\\
%&=\log (1)+1\\
\end{align*}

%\nrd{I GET -- please double check!} \begin{align*}
%&  \eqref{R_weak_BE} -R_{HK2}\\
%& =\log  (1+{\tt INR}+{\tt SNR}+\frac{2{\tt SNR}}{{\tt INR}}-\frac{(\frac{\sqrt{{\tt %SNR}\times {\tt INR}}}{{\tt INR}}+\sqrt{{\tt SNR}\times {\tt INR}})^2}{1+{\tt INR}})-\log (1+{\tt INR}+\frac{{\tt SNR}}{{\tt INR}})+1\\
%& =\log  ((1+{\tt INR}+{\tt SNR}-\frac{{\tt SNR}\times {\tt INR}}{1+{\tt INR}})+(\frac{2{\tt SNR}}{{\tt INR}}-\frac{\frac{{\tt SNR}}{{\tt INR}}+2{\tt SNR}}{1+{\tt INR}}))-\log (1+{\tt INR}+\frac{{\tt SNR}}{{\tt INR}})+1\\
%&= \log  ((1+{\tt INR}+{\tt SNR}-\frac{{\tt SNR}\times {\tt INR}}{1+{\tt INR}})+(\frac{2{\tt SNR}(1+{\tt INR})-{\tt SNR}-2{\tt SNR}\times {\tt INR}}{{\tt INR}(1+{\tt INR})}))\\
%& \ \ \ \ -\log (1+{\tt INR}+\frac{{\tt SNR}}{{\tt INR}})+1\\
%&=\log (\frac{(1+{\tt INR})^2{\tt INR}+{\tt SNR}\times \nrd{{\tt INR}^2}+{\tt SNR}}{{\tt INR}(1+{\tt INR})})-\log (1+{\tt INR}+\frac{{\tt SNR}}{{\tt INR}})+1\\
%&=\log (\frac{((1+{\tt INR})^2{\tt INR}+{\tt SNR}\times \nrd{{\tt INR}^2}+{\tt SNR}){\tt INR}}{{\tt INR}(1+{\tt INR})((1+{\tt INR}){\tt INR}+{\tt SNR})})+1\\
%&=\log (\frac{{\tt INR}^2(1+{\tt INR})^2+{\tt SNR}\times \nrd{{\tt INR}^3}+{\tt SNR}\times {\tt INR}}{{\tt INR}^2(1+{\tt INR})^2+{\tt SNR}\times {\tt INR}(1+{\tt INR})})+1\\
%&=\nrd{\mbox{now what? should be do-able..... hopefully?}}
%\end{align*}


%Further notice that $|\lambda_{24}|\leq 1$, so the above constant gap is based on the condition of $SNR\leq INR^3$. For the case of $SNR> INR^3$, it is easy to see that as $SNR$ increasing, the maximal value $\lambda_{24}$ can take is 1. Thus, we have
%\begin{align*}
%\lambda_{24}=\min (1, \frac{\sqrt{SNR\times INR}}{INR^2})
%\end{align*}
{If the second term in \eqref{R_weak_BE} is relevant (${\tt SNR}\geq {\tt INR}^3$):} %\nrd{fix brackets! I again get something different.....}
%Now we use the second term in \eqref{R_weak_BE} to bound the second term in \eqref{HK}: 
\begin{align*}
&  \eqref{R_weak_BE} -R_{HK2}\\
%&=\log \left(1+{\tt INR}+{\tt SNR}+2\sqrt{{\tt SNR}\times {\tt INR}}-\frac{({\tt INR}+\sqrt{{\tt SNR}\times {\tt INR}})^2}{1+{\tt INR}}\right)-\log \left(1+{\tt INR}+\frac{{\tt SNR}}{{\tt INR}}\right)+1\\
%& = \log \left(\frac{1+2{\tt INR}+{\tt SNR}+2\sqrt{{\tt SNR}\times {\tt INR}}}{1+{\tt INR}}\right)-\log \left(1+{\tt INR}+\frac{{\tt SNR}}{{\tt INR}}\right)+1\\
&=\log \left(1+\frac{(\sqrt{{\tt SNR}}+\sqrt{{\tt INR}})^2}{1+{\tt INR}}\right)-\log \left(1+{\tt INR}+\frac{{\tt SNR}}{{\tt INR}}\right)+1\\
&=\log \left(\frac{(1+2{\tt INR}+{\tt SNR}+2\sqrt{{\tt SNR}\times {\tt INR}}){\tt INR}}{(1+{\tt INR})((1+{\tt INR}){\tt INR}+{\tt SNR})}\right)+1\\
&\overset{(a)}{\leq }\log \left(\frac{(2(1+{\tt INR}+{\tt SNR})+{\tt INR}){\tt INR}}{(1+{\tt INR})((1+{\tt INR}){\tt INR}+{\tt SNR})}\right)+1\\
&= \log \left(\frac{2{\tt INR}+2{\tt SNR}\times {\tt INR}+3{\tt INR}^2}{{\tt INR}+2{\tt INR}^2+{\tt SNR}+{\tt INR}^3+{\tt SNR}\times {\tt INR}}\right)+1\\
&\leq \log\left(\frac{2({\tt INR}+{\tt SNR}\times {\tt INR}+2{\tt INR}^2+{\tt SNR}+{\tt INR}^3)}{{\tt INR}+{\tt SNR}\times {\tt INR}+2{\tt INR}^2+{\tt SNR}+{\tt INR}^3}\right)+1\\
& = \log (2)+1\\
&=2
\end{align*} 
where (a) follows the fact that $1+{\tt SNR}+{\tt INR}+2\sqrt{{\tt SNR}\times {\tt INR}}\leq 2(1+{\tt SNR}+{\tt INR})$.
%\nrd{I GET: please check!!}
%\begin{align*}
%&  \eqref{R_weak_BE} -R_{HK2}\\
%&=\log (1+{\tt INR}+{\tt SNR}+2\sqrt{{\tt SNR}\times {\tt INR}}-\frac{({\tt INR}+\sqrt{{\tt SNR}\times {\tt INR}})^2}{1+{\tt INR}})-\log (1+{\tt INR}+\frac{{\tt SNR}}{{\tt INR}})+1\\
%& =\nrd{ \log (\frac{1+2{\tt INR}+{\tt INR}\times{\tt SNR}+2\sqrt{{\tt SNR}\times {\tt INR}}}{1+{\tt INR}})}-\log (1+{\tt INR}+\frac{{\tt SNR}}{{\tt INR}})+1\\
%&=\nrd{\mbox{now what? from here on very different!}}........\log (\frac{(1+{\tt INR}+{\tt SNR}+2\sqrt{{\tt SNR}\times {\tt INR}}){\tt INR}}{(1+{\tt INR})((1+{\tt INR}){\tt INR}+{\tt SNR})})+1\\
%&\overset{(a)}{\leq }\log (\frac{2(1+{\tt INR}+{\tt SNR}){\tt INR}}{(1+{\tt INR})(1+{\tt INR}+{\tt SNR})})+1\\
%& \leq \log (2)+1\\
%&=2
%\end{align*} 
%where (a) follows the condition of ${\tt INR}\geq 1$ and the fact that $1+{\tt SNR}+{\tt INR}+2\sqrt{{\tt SNR}\times {\tt INR}}\leq 2(1+{\tt SNR}+{\tt INR})$.


\begin{remark} We have shown that the capacity region of the Gaussian two-way interference channel with partial adaptation (fix $X_1$ and $X_3$) is within at most 2 bits per user per direction to the region achieved by two simultaneous HK schemes in opposite directions when ${\tt INR}\geq 1$ for both directions. Again,  we may conclude that partial adaptation cannot significantly increase the capacity for Gaussian two-way IC with weak interference. %Further observe that if we define partial adaptation in the opposite way, i.e., fix $X_2$ and $X_4$, we will have the same results.
\end{remark}  


\subsubsection{${\tt INR}<1$} In this case, a symmetric version of the HK scheme may be obtained 
%Now we proceed to consider the case of ${\tt INR}<1$ for the Han and Kobayashi region, and this inner bound is paraphrased by
from  \cite[(69)]{etkin_tse_wang}, for which each of the four users may achieve the following rate:
\begin{align}
& R_{{\tt INR}<1}\leq \log \left(1+\frac{{\tt SNR}}{1+{\tt INR}}\right)\label{R_INR<1}
\end{align}
%\begin{theorem}
%The capacity region for two-way symmetric Gaussian interference channel in weak interference when ${\tt INR}< 1$ is within 1 bit to \eqref{R_INR<1} in both directions.
%\end{theorem}
%\begin{proof}
{\bf We show that this achieves to within 1 bit of the outer bound \eqref{R_strong_E}} %(which is valid for full adaptation) %\nrd{fix brackets!}% to bound \eqref{R_INR<1} to within 1 bit:
%\begin{align*}
%&R_{sym2}-R_{INR<1}\\
%&=\log (1+INR+SNR-\frac{INR\times SNR}{1+INR})-\log (1+\frac{SNR}{1+INR})\\
%& =\log (1+\frac{INR(1+INR)}{1+INR+SNR})\\
%&\overset{(a)}{\leq} \log (1+1)\\
%&=1 
%\end{align*}
%where (a) we use the condition of $INR<1$.
\begin{align*}
\eqref{R_strong_E}-R_{{\tt INR}<1}
&\leq \frac{1}{2} \log \left(1+{\tt SNR}+{\tt INR}+2\sqrt{{\tt SNR}\times {\tt INR}}\right)+\frac{1}{2}\log \left(1+\frac{{\tt SNR}}{1+{\tt INR}}\right)-\log \left(1+\frac{{\tt SNR}}{1+{\tt INR}}\right)\\
& =\frac{1}{2} \log \left(1+{\tt SNR}+{\tt INR}+2\sqrt{{\tt SNR}\times {\tt INR}}\right)-\frac{1}{2}\log \left(1+\frac{{\tt SNR}}{1+{\tt INR}}\right)\\
& \leq \frac{1}{2}\log \left(\frac{2(1+{\tt SNR}+{\tt INR})(1+{\tt INR})}{1+{\tt SNR}+{\tt INR}}\right)\\
& \overset{(a)}{\leq} \frac{1}{2}\log (4)\\
&=1
\end{align*}
where (a) uses the condition ${\tt INR}<1$. Since \eqref{R_strong_E} was obtained for full adaptation, we conclude that the capacity of the Gaussian two-way IC is to within 1 bit to the HK region when ${\tt INR}<1$ for both directions.
\end{proof}

We summarize our results for constant gaps in Table \ref{table:gaps}.
\begin{table}
\resizebox{18cm}{!} {
\begin{tabular}{|c|c|c|c|c|c|}
\hline
{\bf Two-way Interference} & \multicolumn{5}{r|}{{\bf Constant Gaps per user per direction, in bits (to outer bound)}}\\ \hline
Very Strong & \multicolumn{5}{r|}{0 (partial)}\\ \hline
Strong & \multicolumn{5}{r|}{1 (full)}\\ \hline
 & ${\tt INR}<1$ & \multicolumn{4}{r|}{1 (full)}\\ \cline{2-6}
 &  & HK1 is active & \multicolumn{3}{r|}{1.5 (full)}\\ \cline{3-6}
 Weak & ${\tt INR}\geq 1$ &  & $\rightarrow$ direction & \multicolumn{2}{r|}{1 (partial)}\\ \cline{4-6}
  & & HK2 is active & $\leftarrow$ direction & ${\tt SNR}\leq {\tt INR}^3$ & 1 (partial)\\ \cline{5-6}
  & & & &  ${\tt SNR}> {\tt INR}^3$ & 2 (partial)\\ \hline
\end{tabular}
}
\caption{Constant gaps between non-adaptive symmetric Han and Kobayashi schemes in each direction and partially or fully adaptive outer bounds for the two-way Gaussian IC.}
\label{table:gaps}
\end{table}

%We summarize our results of constant gaps in Fig. \ref{fig:gaps}
%\begin{figure*}
%\begin{figure}
%\includegraphics[width=19cm]{Gaps}
%\vspace{-0.9cm}
%\caption{A summary of constant gaps. (Gaps assuming partial adaptation in red)}
%\label{fig:gaps}
%\end{figure}
%\end{figure*}

\subsection{Final comments on adaptation versus no-adaptation, and versus perfect output feedback}

In the above, we have highlighted classes of two-way interference channels for which adaptation is useless. For the Gaussian channel, only highly symmetric scenarios were considered. The conclusions made for such 
symmetric scenarios, while insightful, do not tell the whole story. That is, it must be noted that adaptation can provide unbounded gains over non-adaptation for certain channels. A simple example of a channel in which we 
see this is shown in Fig. \ref{fig:ex1}. Here the forward channel has no direct links, while the reverse channel has no cross-over links. In this scenario, a non-adaptive scheme would not be able to achieve any positive rate 
for $R_{12}$ and $R_{34}$. However, the adaptive scheme would be able to achieve positive rates for $R_{12}$ and $R_{34}$ by ``routing'' the messages. For example, for message $M_{12}$ to travel from Tx 1 to Rx 2, 
using adaptation it may do so by taking the path: Tx 1 $\rightarrow$  cross-over link to Rx 4 $\rightarrow$  direct reverse link from Tx 4 to Rx 3 $\rightarrow$  cross-over link Tx 3 to Rx 2. An adaptive network may thus 
provide unbounded gain over a non-adaptive network for at least some of the rates. Note however that if the reverse direction is ``routing'' messages for the forward direction, its own message rates will decrease. %Characterizing this tradeoff is an interesting. 

\begin{figure}
\centerline{\includegraphics[width=7cm]{response-fig2}}
\caption{Example of a channel in which adaptation yields unbounded gain over non-adaptation. The non-adaptive scheme would not be able to achieve any non-trivial rates for $R_{12}$ and $R_{34}$, while the adaptive scheme would be able to achieve strictly positive rates (depending on the noise in the different links).}
\label{fig:ex1}
\end{figure}

%\textcolor{blue}{I think this unbounded gain is in terms of the forward rates only, and whether the overall performance of a two-way networks is good (achieve capacity gain) or not depends on rates in both directions (the backward links look like two clean channels, but actually they also routing $M_{12}$ and $M_{34}$, thus, if we want unbounded gain for the forward channel, then the rates in the reverse channel maybe zero, which is not a good overall performance for the two-way networks). Nevertheless, this is a good example to show adaptation can be very useful (for increasing rates in one direction).}

One can also find examples of networks where perfect output feedback provides unbounded gain over adaptation.
Note that in most scenarios where adaptation is useless, perfect output feedback is known to be useless as well. For example, for the symmetric linear deterministic two-way IC shown in Fig. \ref{fig:curves}, the symmetric sum-rate of the linear deterministic one-way IC and one-way IC {with feedback} are identical for $\frac{2}{3}\leq \alpha \leq 2$, indicating that feedback is useless. In this regime, adaptation was also shown to be useless. In the two-way MAC/BC, for all deterministic models, perfect output feedback may be shown to be useless, and adaptation was also useless.  
%\nrd{For example... need a lot of citations for the regimes in which adaptation is useless but also feedback is useless.}
One might ask whether feedback and adaptation being useless always go hand in hand. The following example shows the intuitive fact that adaptation being useless does not imply that feedback is useless. % for general networks. % holds for certain deterministic networks. 
%... need to figure out if giving feedback is always an other bound or if can even actually do more than two one-way channels with FB.
To show that feedback may provide an unbounded gain over pure adaptation, consider the two-way IC in Fig. \ref{fig:ex2}. For message $M_{12}$ to travel from Tx 1 to Rx 2, using feedback it may do so by taking the path: Tx 1 $\rightarrow$  cross-over link to Rx 4 $\rightarrow$  feedbacks to Tx 3 $\rightarrow$  cross-over link to Rx 2. However, clearly if we employ only adaptation over these forward and reverse links, $M_{12}$ is only able to be decoded by Rx 4 and even with adaptation has no possible way to reach Rx 2. So, feedback may improve the capacity in an unbounded way over adaptation, at least when feedback is ``free'', or perfect (not over other interfering links in the reverse direction). An alternative example of when feedback may outperform adaptation is in the symmetric linear deterministic IC: for $\alpha>2$, as seen in Fig. \ref{fig:curves}, feedback outperforms adaptation. This is intuitive, as feedback is provided over perfect, infinite capacity links, whereas adaptation must take place over the same links over which the data travels. 
Whether  feedback being useless necessarily implies that adaptation is also useless is an interesting open question; all known examples for additive channels seem to suggest this but it has not been rigorously shown.% \textcolor{blue}{Intuitively yes, since adaptation cannot outperform perfect feedback. }

\begin{figure}
\centerline{\includegraphics[width=14cm]{response-fig1}}
\caption{An example of a channel in which perfect output feedback (denoted by the dashed arrows in the left figure) would be able to achieve strictly positive rates for $R_{12}$ and $R_{34}$, but the adaptive scheme for the same channel on the right would not be able to achieve non-zero rates for any of the links.}
\label{fig:ex2}
\end{figure}

\section{Conclusion}
\label{conclusion}
In this work, we have demonstrated a few examples of two-way multi-user channels for which adaptation, or the ability of nodes to adapt their current channel inputs based on previously received channel outputs, is useless from a capacity region perspective, i.e. non-adaptive schemes achieve outer bounds derived for the fully adaptive models. 
Specifically, we obtained the capacity regions of the two-way MAC/BC channel, the two-way Z channel, and the two-way IC of binary modulo-2 addition model, the ``deterministic, invertible and cardinality constrained''  model, and the linear deterministic model. Interestingly, adaptation (full or partial) is not needed to attain the capacity regions even though it is permitted. For noisy channels, we first showed that adaptation can only increase the sum-rate of the two-way Gaussian MAC/BC by up to $\frac{1}{2}$ bit per direction. We then considered the Gaussian two-way IC with 4 terminals and 4 messages. There, it was shown that partial adaptation is useless in very strong interference, and for all other regimes non-adaptive schemes achieved to within constant gaps of fully, or partially, adaptive schemes.
We have demonstrated several examples where adaptation is useless -- the question of when adaptation is useless in general networks remains a challenging open question. However, based on some of the examples seen here, we believe that the following properties may be needed to make the claim that ``adaptation is useless'' for a particular network: 1) the self-interference can be cancelled (excludes the binary multiplier channel), 2)  no loop in the networks (excludes the relaying of data along stronger paths), and 3)  no ``coherent'' gains (excluding possible gains by having users use adaptation to create joint input distributions in for example Gaussian networks). %Finding the general conditions for two-way networks when adaptation is useless is the subject of on-going work.
%The bounds presented here may be generalized to general discrete memoryless channels and result in cut-set-like (but not cut-set) outer bounds. Determining the capacity region of two-way linear deterministic interference channel with full adaptation and the capacity region of multi-user Gaussian two-way channels is the subject of on-going work. 

\section{Appendix}
%\subsection{Proof of Theorem \ref{thm:3user}}
%\label{MACBC:mod2}
%\begin{proof}
%We first prove the converse; achievability follows via (non-adaptive) time-sharing between nodes in the same direction. 
%
%\smallskip
%\noindent {\it Proof of bound \eqref{3user1}:} 
%The first bound follows from the sum-rate outer bound of the multiple-access channel. %By Fano's inequality, 
%\begin{align*}
%&n(R_{12}+R_{32}-\epsilon)\\
% &\leq I(M_{12},M_{32};Y_2^n)= H(Y_2^n)-H(Y_2^n|M_{12},M_{32})\\
%&\leq \sum_{i=1}^n [H(Y_{2,i}|Y_2^{i-1})]\leq \sum_{i=1}^n [H(Y_{2,i})]\leq n
%\end{align*}
%where the last inequality follows as $Y_{2,i}$ is a binary random variable whose entropy is thus bounded by 1. 
%
%\begin{figure}[h]
%\begin{center}
%\includegraphics[width=3.3in]{3usermarkovchain}
%\caption{The Markov chain used in the outer bound proof of Theorem \ref{thm:3user}.} 
%\label{fig:3userchain}
%\end{center}
%\end{figure} 
%\smallskip
%\noindent {\it Proof of bound \eqref{3user2}:}
%\begin{align*}
%& n(R_{21}+R_{23}-\epsilon)\\
%& \overset{(a)}{\leq} I(M_{21};Y_1^n|M_{12},M_{32})+I(M_{23};Y_3^n|M_{12},M_{32},M_{21})\\
%& \overset{(b)}{\leq} H(Y_1^n|M_{12},M_{32})-H(Y_1^n|M_{21},M_{12},M_{32})+H(Y_3^n|M_{12},M_{32},M_{21})\\
%&\overset{(c)}{\leq} \sum_{i=1}^n [H(Y_{1,i})-H(Y_{1,i}|M_{21},M_{12},M_{32},Y_1^{i-1})+H(Y_{3,i}|M_{12},M_{32},M_{21},Y_3^{i-1})]\\
%& \overset{(d)}{=} \sum_{i=1}^n [H(Y_{1,i})-H(X_{1,i}\oplus X_{2,i}|M_{21},M_{12},M_{32},Y_1^{i-1},X_1^i)+H(X_{2,i}\oplus X_{3,i}|M_{12},M_{32},M_{21},Y_3^{i-1},X_3^i)]\\
%& \overset{(e)}{\leq} \sum_{i=1}^n [H(Y_{1,i}) -H(X_{2,i}|M_{21},M_{12},M_{32},Y_1^{i-1},X_1^i,X_2^{i-1},X_3^i,Y_3^{i-1})\\
%& \ \  +H(X_{2,i}|M_{21},M_{12},M_{32},Y_1^{i-1},X_1^i,X_2^{i-1},X_3^i,Y_3^{i-1})]\\
%& =\sum_{i=1}^n [H(Y_{1,i})]\leq n
%\end{align*}
%
%where (a) follows from Fano's inequality, (b) drops a negative entropy term, (c) uses the chain rule and conditioning reduces entropy. In (d), $X_1^i=f_1(M_{12},Y_1^{i-1})$ and $X_3^i=f_3(M_{32},Y_3^{i-1})$. We cancel $X_{1,i}$ and $X_{3,i}$ in the entropy term in (e) since we know $X_1^i$ and $X_3^i$  respectively. In addition, we introduce genies $X_2^{i-1}, Y_3^{i-1}, X_3^i$ in the negative entropy term. For the third entropy term, $X_2^{i-1}$ may be obtained from $Y_3^{i-1}=X_2^{i-1}\oplus X_3^{i-1}$ (bit-wise modulo 2) since we know $X_3^{i-1}$. We may obtain $Y_1^{i-1}$ using the Markov chain illustrated in Fig. \ref{fig:3userchain}. Finally, $X_1^i$ is given by the encoding function $X_1^i=f_1(M_{12},Y_1^{i-1})$. % (with slight abuse of notation, as vectors are denoted here).
% We again bound $H(Y_{1,i})\leq 1$.
%%Similarly, since the random variable $Y_1$ is binary, we proved the desired bound $R_{21}+R_{23}\leq 1$.
%
%
%We are able to achieve this outer bound using two time-sharing random variables {\it without adaptation}: $\alpha$ time-shares between channel inputs $X_1$ and $X_3$ for the MAC channel in the $\rightarrow$ direction, while $\beta$ time-shares between the messages $M_{21}$ and $M_{23}$ (encoded as $X_2(M_{21})$ and $X_2(M_{23})$ respectively) in the BC in the $\leftarrow$ direction. Both directions ignore the received signals and use i.i.d. Bernoulli(1/2) codebooks.  
%
%%For clarity, we illustrate who transmits when in Figure  \ref{fig:timesharing}, where we note the inputs and outputs in order to emphasize that at all times, both receivers may decode 1 bit per channel use after canceling their own transmitted codeword. 
%
%
%
%
%\end{proof}

%\subsection{Proof of Theorem \ref{thm:generalMABC}}
%\label{generalMABC}
%
%\begin{proof}
%Outer bound \eqref{general1} follows by the cut-set bound, exactly as shown for the binary modulo 2 adder MAC/BC. Outer bound \eqref{general2} may be shown using the cut-set bound, but may alternatively be shown as:
% \begin{align*}
%& n(R_{21}+R_{23}-\epsilon)\\
%& \leq I(M_{21};Y_1^n|M_{12},M_{32})+I(M_{23};Y_3^n|M_{12},M_{32},M_{21})\\
%& \leq H(Y_1^n|M_{12},M_{32})-H(Y_1^n|M_{21},M_{12},M_{32}) +H(Y_3^n|M_{12},M_{32},M_{21})\\
%&\leq \sum_{i=1}^n [H(Y_{1,i})-H(Y_{1,i}|M_{21},M_{12},M_{32},Y_1^{i-1})+H(Y_{3,i}|M_{12},M_{32},M_{21},Y_3^{i-1})]\\
%& \overset{(a)}{=} \sum_{i=1}^n [H(Y_{1,i})-H(X_{2,i}|M_{21},M_{12},M_{32},Y_1^{i-1},X_1^i)+H(X_{2,i}|M_{12},M_{32},M_{21},Y_3^{i-1},X_3^i)]\\
%& \overset{(b)}{\leq} \sum_{i=1}^n [H(Y_{1,i})  -H(X_{2,i}|M_{21},M_{12},M_{32},Y_1^{i-1},X_1^i,X_2^{i-1},X_3^i,Y_3^{i-1}) \\
%& \ \ +H(X_{2,i}|M_{21},M_{12},M_{32},Y_1^{i-1},X_1^i,X_2^{i-1},X_3^i,Y_3^{i-1})]\\
%& =\sum_{i=1}^n [H(Y_{1,i})]\leq n\log\kappa
%\end{align*}
%where (a) follows from the invertibility condition {\bf P2}. %, i.e. remove $X_{1,i}$ from $Y_{1,i}$. Similar for $Y_{3,i}$. \nrd{(not sure about this step)}.
% In (b), $X_2^{i-1}$ in the conditioning of the third term is decoded from $Y_3^{i-1}$ using the invertibility condition {\bf P2}.
%
%Our achievability scheme consists of time-sharing between user 1 and user 3 in the $\rightarrow$ (MAC) direction, while simultaneously time-sharing between sending data to user 1 and 3 in the $\leftarrow$ (BC) direction, as in 
%%\nrd{NOT SURE ABOUT THIS PART: The first sum-rate bound can be achieved using time-sharing as follows.
%%Let $X_2$ always be selected to be uniform over its $\kappa$ possible symbols. 
%% For $\alpha\in [0,1]$ for $n\alpha$ channel uses set $X_3 = x_3^*$ 
%%(hence $R_{32}=0$), in which case selecting $X_1$ to be uniform over its $\kappa$ symbols will achieve $H(Y_2) = \log\kappa$, by property \eqref{P3} (noting that $X_2$ is also uniform). For the remaining 
%%$n(1-\alpha)$ channel uses set $X_1 = x_1^*$ (hence $R_{12} = 0$) and select $X_3$ to be uniform over its $\kappa$ symbols, again 
%%achieving the outer bound. The sum-rate bound for the BC direction \eqref{general2} is achieved via time-sharing using conditions {\bf P2} and {\bf P3}. 
%%
%%
% Fig. \ref{fig:TS}. There, we see two time-sharing coefficients $0\leq \alpha, \beta\leq 1$, where $\alpha$ time-shares in the $\rightarrow$ direction and $\beta$ in the $\leftarrow$ direction. Let us consider the rates achieved in time slot (1), of duration $\alpha$ (WLOG we have taken $\alpha<\beta$). Node 1 encodes $M_{12}$ into $X_1$ uniformly distributed over the $\log(\kappa)$ input symbols; node 2 encodes $M_{21}$ into $X_2$ uniformly distributed over $\log(\kappa)$ input symbols and node 3 fixes $X_3=x_3^*$ (rate 0). We claim this scheme achieves the rates $R_{12} = R_{21} =  \alpha \log(\kappa)$, $R_{23} = R_{32} = 0$. Consider $R_{12}$: node 2 receives $Y_2=F_2(X_1,X_2,x_3^*)$. Since node 2 knows $X_2$ and knows that $X_3=x_3^*$,  by {\bf P2},  it may construct $X_1 = G_{21}(X_2,x_3^*,Y_2)$ to decode $M_{12}$. By {\bf P3} this may be done at full rate $\alpha \log(\kappa)$. Similar arguments for time slots (2) and (3) demonstrate that the rates in \eqref{general1}-\eqref{general2} are achievable. 
%
%
%\begin{figure}[h]
%\begin{center}
%\includegraphics[width=3in]{TS}
%\caption{Time-sharing based achievability for the proof of Theorem \ref{thm:generalMABC}.} 
%\label{fig:TS}
%\end{center}
%\end{figure}
%
%
%%The converse:
%%\begin{align*}
%%n(R_{12}+R_{32}-\epsilon) &\leq I(M_{12},M_{32};Y_2^n)\\
%%&= H(Y_2^n)-H(Y_2^n|M_{12},M_{32})\\
%%& \leq \sum_{i=1}^n [H(Y_{2,i}|Y_2^{i-1})]\\
%%& \leq \sum_{i=1}^n [H(Y_{2,i})]\\
%%&\leq n\log\kappa
%%\end{align*}
%
%\end{proof}


\subsection{Proof of Theorem \ref{mod2thm:z}}
\label{mod2Z}

\begin{proof}
Time-sharing may again be used to achieve this region.   For the converse,

\smallskip
\noindent{\it Proof of bound \eqref{z1}:}
\begin{align*}
& n(R_{12}+R_{32}+R_{34}-\epsilon)\\
 & \leq I(M_{12};Y_2^n|M_{21},M_{23},M_{43})+I(M_{32},M_{34};Y_4^n,Y_2^n|M_{43},M_{12},M_{21},M_{23})\\
&\overset{(a)}{\leq} I(M_{12};Y_2^n|M_{21},M_{23},M_{43})+I(M_{32},M_{34};Y_2^n|M_{43},M_{12},M_{21},M_{23})\\
& \ \ +I(M_{32},M_{34};Y_4^n|M_{43},M_{12},M_{21},M_{23},Y_2^n)\\
&\overset{(b)}{\leq} H(Y_2^n|M_{21},M_{23},M_{43})-H(Y_2^n|M_{12},M_{21},M_{23},M_{43})+H(Y_2^n|M_{12},M_{21},M_{23},M_{43})\\
& \ \ +H(Y_4^n|M_{43},M_{12},M_{21},M_{23},Y_2^n)\\
&\overset{(c)}{\leq} \sum_{i=1}^n [H(Y_{2,i})+H(Y_{4,i}|M_{12},M_{21},M_{23},M_{43},Y_4^{i-1},Y_2^n)]\\
& \overset{(d)}{=} \sum_{i=1}^n [H(Y_{2,i})+H(X_{3,i}\oplus X_{4,i}|M_{12},M_{21},M_{23},M_{43},Y_4^{i-1},X_4^i,X_3^{i-1},Y_2^n,X_2^n)]\\
& \overset{(e)}{=} \sum_{i=1}^n [H(Y_{2,i})+H(X_{3,i}|M_{12},M_{21},M_{23},M_{43},Y_4^{i-1}, X_4^i,X_3^{i-1},X_1^n \oplus X_2^n\oplus X_3^n,X_2^n,X_1^n)]\\
&=\sum_{i=1}^n [H(Y_{2,i})]\leq n
\end{align*}

where (a) follows from the chain rule. We drop two negative entropy terms in inequality (b) and notice that the second and the third entropy terms cancel each other. In  (c), we apply the chain rule first, then we drop the conditioning part of the first entropy term. In (d), we construct $X_4^i=f_4(M_{43},Y_4^{i-1})$ and  note that $X_3^{i-1}$ may be obtained from $Y_4^{i-1}=X_3^{i-1}\oplus X_4^{i-1}$, given  $X_4^{i-1}$. Adding $X_2^n$ follows from the fact $X_2^n=f_2(M_{21},M_{23},Y_2^{n-1})$. In (e), we cancel $X_{4,i}$ in the second entropy term since we know $X_4^i$. In addition, given $M_{12}$ and $X_2^n$, we may construct $X_1^n$ as illustrated in Fig. \ref{fig:zchain1}. Now, we may obtain $X_3^n$ from $Y_2^n=X_1^n\oplus X_2^n\oplus X_3^n$, so that the second entropy term in zero. Bound  \eqref{z2} follows by symmetry.


\begin{figure}
\begin{center}
\includegraphics[width=3.6in]{zmarkovchain1}
\caption{The Markov chain used in the outer bound proof of Theorem \ref{mod2thm:z}.} 
\label{fig:zchain1}
\end{center}
\end{figure}


\end{proof}

%\subsection{Proof of Theorem \ref{mod2thm:z}}
%\begin{proof}
%Clearly we may achieve this region using two time-sharing random variables; one between nodes 1 and 3, and a second between nodes 2 and 4. For the converse,

%\smallskip
%\noindent{\it Proof of bound \eqref{intf1}:}
%\begin{align*}
%&n(R_{12}+R_{34}-\epsilon)\\
%&\overset{(a)}{\leq} I(M_{12};Y_2^n|M_{21},M_{43})+I(M_{34};Y_4^n,Y_2^n|%M_{12},M_{21},M_{43})\\
%&=I(M_{12};Y_2^n|M_{21},M_{43})+I(M_{34};Y_2^n|M_{21},M_{12},M_{43})\\
%&+I(M_{34};Y_4^n|M_{21},M_{12},M_{43},Y_2^n)\\
%&\overset{(b)}{\leq} I(M_{12};Y_2^n|M_{21},M_{43})+I(M_{34};Y_2^n|M_{21},M_{12},M_{43})\\
%&+H(Y_4^n|M_{21},M_{12},M_{43},Y_2^n)\\
%&\overset{(c)}{=}I(M_{12};Y_2^n|M_{21},M_{43})+I(M_{34};Y_2^n|M_{21},M_{12},M_{43})\\
%&+\sum_{i=1}^n[H(X_{1,i}\oplus X_{3,i}\oplus X_{4,i}|M_{21},M_{12},M_{43},Y_4^{i-1},X_4^i,Y_2^n,X_2^n)]\\
%&\overset{(d)}{=}I(M_{12};Y_2^n|M_{21},M_{43})+I(M_{34};Y_2^n|M_{21},M_{12},M_{43})\\
%&+\sum_{i=1}^n[H(X_{1,i}\oplus X_{3,i}|M_{21},M_{12},M_{43},Y_4^{i-1},X_4^i,X_1^n\oplus X_3^n,X_2^n)]\\
%&=I(M_{12};Y_2^n|M_{21},M_{43})+I(M_{34};Y_2^n|M_{21},M_{12},M_{43})\\
%&\overset{(e)}{\leq} \sum_{i-1}^n [H(Y_{2,i})-H(Y_{2,i}|Y_2^{i-1},M_{12},M_{21},M_{43})\\
%&+H(Y_{2,i}|Y_2^{i-1},M_{12},M_{21},M_{43})]\\
%&=\sum_{i=1}^n [H(Y_{2,i})]\\
%&\leq n
%\end{align*}

%where (a) follows from Fano's inequality and the introduction of a genie $Y_2^n$ in the second mutual information term. We drop a negative entropy term in inequality (b). In (c), we apply the chain rule on the entropy term and we add $X_4^i$ and $X_2^n$ in the conditioning part since $X_4^i=f_4(M_{43},Y_4^{i-1})$ and $X_2^n=f_2(M_{21},Y_2^{n-1})$. In (d), we cancel $X_{4,i}$ in the entropy term since we know $X_4^i$. In addition, $X_1^n\oplus X_3^n$ is decoded from $Y_2^n=X_1^n\oplus X_2^n\oplus X_3^n$ since $X_2^n$ is known. Thus, the last entropy term is zero. We apply the chain rule again in step (e) and drop the conditioning part of the first entropy term and a negative entropy term.  The bound \eqref{intf2} follows by symmetry.
%\end{proof}



%\subsection{Proof of Theorem \ref{mod2thm:intf}}
%\label{mod2IC}
%\begin{proof}
%We may achieve this region using two time-sharing random variables; one between nodes 1 and 3, and a second between nodes 2 and 4. 
%For the converse,
%
%\noindent{\it Proof of bound \eqref{intf1}:}
%\begin{align*}
%&n(R_{12}+R_{34}-\epsilon)\\
%&\overset{(a)}{\leq} I(M_{12};Y_2^n|M_{21},M_{43})+I(M_{34};Y_4^n,Y_2^n|M_{12},M_{21},M_{43})\\
%&=I(M_{12};Y_2^n|M_{21},M_{43})+I(M_{34};Y_2^n|M_{21},M_{12},M_{43})+I(M_{34};Y_4^n|M_{21},M_{12},M_{43},Y_2^n)\\
%&\overset{(b)}{\leq} I(M_{12};Y_2^n|M_{21},M_{43})+I(M_{34};Y_2^n|M_{21},M_{12},M_{43})+H(Y_4^n|M_{21},M_{12},M_{43},Y_2^n)\\
%&\overset{(c)}{=}I(M_{12};Y_2^n|M_{21},M_{43})+I(M_{34};Y_2^n|M_{21},M_{12},M_{43})\\
%&+\sum_{i=1}^n[H(X_{1,i}\oplus X_{3,i}\oplus X_{4,i}|M_{21},M_{12},M_{43},Y_4^{i-1},X_4^i,Y_2^n,X_2^n)]\\
%&\overset{(d)}{=}I(M_{12};Y_2^n|M_{21},M_{43})+I(M_{34};Y_2^n|M_{21},M_{12},M_{43})\\
%&+\sum_{i=1}^n[H(X_{1,i}\oplus X_{3,i}|M_{21},M_{12},M_{43},Y_4^{i-1},X_4^i,X_1^n\oplus X_3^n,X_2^n)]\\
%&=I(M_{12};Y_2^n|M_{21},M_{43})+I(M_{34};Y_2^n|M_{21},M_{12},M_{43})\\
%&{\leq} \sum_{i-1}^n [H(Y_{2,i})-H(Y_{2,i}|Y_2^{i-1},M_{12},M_{21},M_{43})+H(Y_{2,i}|Y_2^{i-1},M_{12},M_{21},M_{43})]\\
%&=\sum_{i=1}^n [H(Y_{2,i})]\leq n
%\end{align*}
%
%where (a) follows from Fano's inequality and the introduction of a genie $Y_2^n$ in the second mutual information term. We drop a negative entropy term in inequality (b). In (c), we construct $X_4^i=f_4(M_{43},Y_4^{i-1})$ and $X_2^n=f_2(M_{21},Y_2^{n-1})$. In (d), we cancel $X_{4,i}$ in the entropy term since we know $X_4^i$. In addition, $X_1^n\oplus X_3^n$ is decoded from $Y_2^n=X_1^n\oplus X_2^n\oplus X_3^n$ since $X_2^n$ is known. The bound \eqref{intf2} follows by symmetry.
%\end{proof}

%\subsection{A more general model for deterministic MAC/BC}
%Consider a class of deterministic two-way MAC/BC without self-interference, which are described by:
%\begin{align*}
%&Y_1=Y_3=g(X_2)\\
%&Y_2=f(X_1,X_3)
%\end{align*}
%where $g()$ is an identity function, i.e. $g(X_2)=X_2$, and $f()$ is a deterministic function which satisfies the "invertibility" constraints, i.e. $H(Y_2|X_1)=H(X_3)$ and $H(Y_2|X_3)=H(X_1)$. Furthermore, we assume the following restrictions:
%\begin{itemize}
%\item Restrict $|{\cal X}_1| = |{\cal X}_2|= |{\cal X}_3| = |{\cal X}| = |{\cal Y}_1|= |{\cal Y}_2| = |{\cal Y}_3| = |{\cal Y}| = k$. 
%\item Let $f(x_1,x_3)$ be a function such that $\exists x_3^*$ such that $x_1$ and $y_2  =  f(x_1,x_3^*)$ are in 1-to-1 correspondence, and such that $\exists x_1^*$ such that $x_3$ and $y_2 = f(x_1^*,x_3)$ are in 1-to-1 correspondence.
%\end{itemize}
%If these conditions are met, then clearly
%\begin{align*}
%R_{12}+R_{32}\leq \max H(Y_2)\leq \log (k)
%\end{align*}
%Now we claim that the capacity region of this class of deterministic MAC/BC is the following:
%\begin{align*}
%&R_{12}\leq H(X_1)\\
%&R_{32}\leq H(X_3)\\
%&R_{12}+R_{32}\leq H(Y_2)\\
%&R_{21}+R_{23}\leq H(Y_1)=H(Y_3)=H(X_2)
%\end{align*}
%The single-rate bounds can be achieved in a straightforward way, and the first sum-rate bound can be achieved using time-sharing as follows. For $\alpha\in [0,1]$ for $n\alpha$ channel uses set $X_3 = x_3^*$ 
%(hence $R_{32}=0$), in which case selecting $x_1$ to be uniform over its $k$ symbols will achieve $H(Y_2) = k$. For the remaining 
%$n(1-\alpha)$ channel uses set $X_1 = x_1^*$ (hence $R_{12} = 0$) and select $x_3$ to be uniform over its $k$ symbols, again 
%achieving the outer bound, even if the input distributions are not jointly chosen. The last sum-rate bound for the BC direction also can be achieved via time-sharing; the single-rate bounds in BC direction are redundant. The converse also can be proved straightforwardly.

\subsection{Evaluation of the sum-rate outer bound with full adaptation in Gaussian two-way IC of Theorem \ref{thm:outer-full}}
\label{Gsum}
% Letting $E[X_jX_k^*] = \lambda_{jk}$, 
%suppressing the subscript $i$, and assuming a symmetric channel, the first two terms  in \eqref{(c)} may be bounded as
%\begin{align*}
%& H(g_{12}X_1+g_{32}X_3+Z_2|X_2)-H(Z_2)\\
%& \leq \log 2\pi e (\mbox{Var}(g_{12}X_1|X_2)+\mbox{Var}(g_{32}X_3|X_2)+2\mbox{Cov}(g_{12}X_1,g_{32}X_3|X_2)+1)-\log 2\pi e(1)\\
%%& = \log 2\pi e (g_{12}^2P_1(1-\frac{\lambda^2_{12}}{P_1P_2})+g_{32}^2P_3(1-\frac{\lambda_{23}^2}{P_3P_2})+2g_{12}g_{32}(\lambda_{13}-\frac{\lambda_{12}\lambda_{32}}{P_2})+\sigma_2^2)-\log 2\pi e(\sigma_2^2)\\
%& \nrd{\leq}\log  ({\tt SNR}(1-|\lambda_{12}|^2)+{\tt INR}(1-|\lambda_{23}|^2)+2\sqrt{{\tt SNR}\times {\tt INR}}(|\lambda_{13}|-|\lambda_{12}||\lambda_{32}|)+1).
%\end{align*}
%\nrd{I think the above should actually be (since we have complex and in the minus term the cosine should be bounded by -1 not by 1!! See my scan
%\begin{align*}
%& H(g_{12}X_1+g_{32}X_3+Z_2|X_2)-H(Z_2)\\
%& \leq \log 2\pi e (\mbox{Var}(g_{12}X_1|X_2)+\mbox{Var}(g_{32}X_3|X_2)+2\mbox{Cov}(g_{12}X_1,g_{32}X_3|X_2)+1)-\log 2\pi e(1)\\
%& = \log ({\tt SNR}(1-|\lambda_{12}|^2)+{\tt INR}(1-|\lambda_{23}|^2)\\ 
%& \;\;\;\;\;\;\;\;\;\; +2\sqrt{{\tt SNR}\times {\tt INR}}(|\lambda_{13}|\cos(\theta_1) - |\lambda_{12}||\lambda_{32}|\cos(\theta_2)+1)\\
%%& = \log 2\pi e (g_{12}^2P_1(1-\frac{\lambda^2_{12}}{P_1P_2})+g_{32}^2P_3(1-\frac{\lambda_{23}^2}{P_3P_2})+2g_{12}g_{32}(\lambda_{13}-\frac{\lambda_{12}\lambda_{32}}{P_2})+\sigma_2^2)-\log 2\pi e(\sigma_2^2)\\
%& \overset{(a)}{\leq} \log  ({\tt SNR}(1-|\lambda_{12}|^2)+{\tt INR}(1-|\lambda_{23}|^2)+2\sqrt{{\tt SNR}\times {\tt INR}}(|\lambda_{13}|+|\lambda_{12}||\lambda_{32}|)+1),
%\end{align*}
%where $\theta_1$ is the angle of $g_{12}g_{32}^*\lambda_{13}$ and $\theta_2$ is the angle of $g_{12}g_{32}^*\lambda_{12}\lambda_{32}$. 
%In this case (a) follows by selecting the angles of the correlation coefficients such that $\theta_1=0$ and $\theta_2 = -\pi$.}

Letting $E[X_jX_k^*] = \lambda_{jk}$, 
suppressing the subscript $i$, and assuming a symmetric channel, the first two terms  in \eqref{(c)} may be bounded as
\begin{align*}
& H(g_{12}X_1+g_{32}X_3+Z_2|X_2)-H(Z_2)\\
&\leq H(g_{12}X_1+g_{32}X_3+Z_2)-H(Z_2)\\
& \leq \log 2\pi e (\mbox{Var}(g_{12}X_1)+\mbox{Var}(g_{32}X_3)+2\mbox{Cov}(g_{12}X_1,g_{32}X_3)+1)-\log 2\pi e(1)\\
& =\log  ({\tt SNR}+{\tt INR}+2|\lambda_{13}|\cos \theta\sqrt{{\tt SNR}\times {\tt INR}}+1)
\end{align*} 
where $\theta$ is the angle of $g_{12}g_{32}^*\lambda_{13}$.

Similarly, the last two terms may be bounded as
\begin{align}
H(g_{34}X_3+Z_4|g_{32}X_3+Z_2,X_4)-H(Z_4)&\leq \log \left(\frac{\mbox{Var}(g_{34}X_3+Z_4|g_{32}X_3+Z_2,X_4)}{\sigma_4^2}\right)\notag \\
&\leq\log \left(1+\frac{{\tt SNR}(1-|\lambda_{34}|^2)}{{\tt INR}(1-|\lambda_{34}|^2)+1}\right).\label{r12r34}
\end{align}

Combining all terms, we obtain
\begin{align}
R_{sym}=\frac{R_{12}+R_{34}}{2}&\leq \frac{1}{2}\log  ({\tt SNR}+{\tt INR}+2|\lambda_{13}|\cos \theta\sqrt{{\tt SNR}\times {\tt INR}}+1) \notag \\
&+\frac{1}{2}\log \left(1+\frac{{\tt SNR}(1-|\lambda_{34}|^2)}{{\tt INR}(1-|\lambda_{34}|^2)+1}\right) \label{R_strong} 
\end{align}
To obtain \eqref{R_strong_E} one may verify  that \eqref{R_strong} is maximized at $\lambda_{34} = 0$ and $\lambda_{13}=1, \theta=0$.

%Combining all terms, we obtain
%\begin{align}
%R_{sym}=\frac{R_{12}+R_{34}}{2}&\leq \frac{1}{2}\log  ({\tt SNR}(1-|\lambda_{12}|^2)+{\tt INR}(1-|\lambda_{23}|^2)+2\sqrt{{\tt SNR}\times {\tt INR}}(|\lambda_{13}|\nrd{+}|\lambda_{12}||\lambda_{32}|)+1) \notag \\
%&+\frac{1}{2}\log \left(1+\frac{{\tt SNR}(1-|\lambda_{34}|^2)}{{\tt INR}(1-|\lambda_{34}|^2)+1}\right) \label{R_strong} 
%\end{align}
%To obtain \eqref{R_strong_E} one may verify  that \eqref{R_strong} is maximized at $\lambda_{12}=\lambda_{23}= \lambda_{34} = 0$ and $\lambda_{13}=1$.
%\nrd{This may be seen as the expression is increasing in $|\lambda_{13}|$ and by setting the partials with respect to the remaining $|\lambda_{ij}|$ to zero and solving for solutions in $[0,1]$ (which hold for all values of ${\tt SNR}$ and ${\tt INR}$). ZHIYU: Can you again verify the plus sign and using Matlab, plot this to verify that it is indeed maximizes at $\lambda_{12}=\lambda_{23}= \lambda_{34} = 0$ and $\lambda_{13}=1$? I think so, but want to double check, as if this changes many bounds would change!}

\bibliographystyle{IEEEtran}
\bibliography{refs}

\end{document}
Cheng_Devroye_revision_final.bbl
% Generated by IEEEtran.bst, version: 1.13 (2008/09/30)
\begin{thebibliography}{10}
\providecommand{\url}[1]{#1}
\csname url@samestyle\endcsname
\providecommand{\newblock}{\relax}
\providecommand{\bibinfo}[2]{#2}
\providecommand{\BIBentrySTDinterwordspacing}{\spaceskip=0pt\relax}
\providecommand{\BIBentryALTinterwordstretchfactor}{4}
\providecommand{\BIBentryALTinterwordspacing}{\spaceskip=\fontdimen2\font plus
\BIBentryALTinterwordstretchfactor\fontdimen3\font minus
  \fontdimen4\font\relax}
\providecommand{\BIBforeignlanguage}[2]{{%
\expandafter\ifx\csname l@#1\endcsname\relax
\typeout{** WARNING: IEEEtran.bst: No hyphenation pattern has been}%
\typeout{** loaded for the language `#1'. Using the pattern for}%
\typeout{** the default language instead.}%
\else
\language=\csname l@#1\endcsname
\fi
#2}}
\providecommand{\BIBdecl}{\relax}
\BIBdecl

\bibitem{zcheng_Allerton}
Z.~Cheng and N.~Devroye, ``Multi-user two-way deterministic modulo 2 adder
  channels -- when adaptation is useless,'' in \emph{Proc. Allerton Conf.
  Commun., Control and Comp.}, Sep. 2011.

\bibitem{zcheng_ISIT}
------, ``On the capacity of multi-user two-way linear deterministic
  channels,'' in \emph{Proc. IEEE Int. Symp. Inf. Theory}, Cambridge, Jul.
  2012.

\bibitem{zcheng_Allerton2012}
------, ``On constant gaps for the two-way gaussian interference channel,'' in
  \emph{Proc. Allerton Conf. Commun., Control and Comp.}, Sep. 2012.

\bibitem{Shannon:1961}
C.~E. Shannon, ``Two-way communications channels,'' in \emph{4th Berkeley Symp.
  Math. Stat. Prob.}, Chicago, IL, Jun. 1961, pp. 611--644.

\bibitem{Kramer:thesis}
G.~Kramer, ``Directed information for channels with feedback,'' Ph.D.
  dissertation, Swiss Federal Institute of Technology Zurich, 1998.

\bibitem{Han:1984}
T.~Han, ``A general coding scheme for the two-way channel,'' \emph{IEEE Trans.
  Inf. Theory}, vol. IT-30, pp. 35--44, Jan. 1984.

\bibitem{Avestimehr2009}
A.~Avestimehr, S.~Diggavi, and D.~Tse, ``Wireless network information flow: a
  deterministic approach,'' \emph{IEEE Trans. Inf. Theory}, vol.~57, no.~4, pp.
  1872--1905, 2011.

\bibitem{etkin_tse_wang}
R.~Etkin, D.~Tse, and H.~Wang, ``Gaussian interference channel capacity to
  within one bit,'' \emph{IEEE Trans. Inf. Theory}, vol.~54, no.~12, pp.
  5534--5562, Dec. 2008.

\bibitem{Dueck:1979}
G.~Dueck, ``The capacity region of the two-way channel can exceed the inner
  bound,'' \emph{Inform. Contr.}, vol.~40, pp. 258--266, Mar. 1979.

\bibitem{Schalkwijk:1982}
L.~Schalkwijk, ``The binary multiplying channel - a coding scheme that operates
  beyond {S}hannon's inner bound region,'' \emph{IEEE Trans. Inf. Theory}, vol.
  IT-28, pp. 107--110, Jan. 1982.

\bibitem{Schalkwijk:1983}
------, ``On an extension of an achievable rate region for the binary
  multiplying channel,'' \emph{IEEE Trans. Inf. Theory}, vol. IT-29, pp.
  445--448, May 1983.

\bibitem{Zhang:1986}
Z.~Zhang, T.~Berger, and J.~Schalkwijk, ``New outer bounds to capacity regions
  of two-way channels,'' \emph{IEEE Trans. Inf. Theory}, vol. IT-32, pp.
  383--386, May 1986.

\bibitem{Hekstra:1989}
A.~Hekstra and F.~Willems, ``Dependence balance bounds for single output
  two-way channels,'' \emph{IEEE Trans. Inf. Theory}, vol.~35, no.~1, pp.
  44--53, Jan. 1989.

\bibitem{bresler_tse}
G.~Bresler and D.~Tse, ``The two-user gaussian interference channel: A
  deterministic view,'' \emph{European Transactions in Telecommunications},
  vol.~19, pp. 333--354, Apr. 2008.

\bibitem{Dash:allerton}
D.~Dash and A.~Sabharwal, ``An outer bound for a multiuser two-way channel,''
  in \emph{Proc. Allerton Conf. Commun., Control and Comp.}, Monticello, Sep.
  2010.

\bibitem{Dash:asilomar}
D.~Dash, A.~Khoshnevis, and A.~Sabharwal, ``An achievable rate region for a
  multiuser half-duplex two-way channel,'' in \emph{Proc. Asilomar Conf.
  Signals, Systems and Computers}, Pacific Grove, Oct. 2006, pp. 707--711.

\bibitem{ElGamalKim:book}
A.~{El Gamal} and Y.-H. Kim, \emph{Network Information Theory}.\hskip 1em plus
  0.5em minus 0.4em\relax Cambridge University Press, 2011.

\bibitem{willems_FB}
F.~Willems, ``The feedback capacity region of a class of discrete memoryless
  multiple-access channels with feedback,'' \emph{IEEE Trans. Inf. Theory},
  vol. IT-28, no.~1, pp. 93--95, Jan. 1982.

\bibitem{elgamal_deg_BC}
{A. El Gamal}, ``The feedback capacity of degraded broadcast channels,''
  \emph{IEEE Trans. Inf. Theory}, vol. IT-24, no.~3, pp. 379--381, May 1978.

\bibitem{Vishwanath2003Z}
S.~Vishwanath, N.~Jindal, and A.~Goldsmith, ``The {Z} channel,'' in \emph{IEEE
  Proc. of Globecom}, San Francisco, 2003, pp. 1726--1730.

\bibitem{elgamal_det_IC}
{A. El Gamal and M.H.M. Costa}, ``The capacity region of a class of
  deterministic interference channels,'' \emph{IEEE Trans. Inf. Theory},
  vol.~28, no.~2, pp. 343--346, Mar. 1982.

\bibitem{VCadambe2009Z}
V.~Cadambe, S.~A. Jafar, and S.~Vishwanath, ``The capacity region of a class of
  deterministic {Z} channels,'' in \emph{Proc. of ISIT}, Seoul, 2009, pp. 2634
  -- 2638.

\bibitem{sahai2009channel}
A.~Sahai, V.~Aggarwal, M.~Yuksel, and A.~Sabharwal, ``On channel output
  feedback in deterministic interference channels,'' in \emph{Proc. IEEE Inf.
  Theory Workshop}, 2009, pp. 298--302.

\bibitem{Changho2010}
C.~Suh and D.~Tse, ``Feedback capacity of the gaussian interference channel to
  within 2 bits,'' \emph{IEEE Trans. Inf. Theory}, vol.~57, no.~5, pp.
  2667--2685, May 2011.

\bibitem{Vahid-IC-rate-FB}
A.~Vahid, C.~Suh, and A.~S. Avestimehr, ``Interference channels with
  rate-limited feedback,'' \emph{IEEE Trans. Inf. Theory}, vol.~58, no.~5, pp.
  2788--2812, 2012.

\bibitem{tuninetti2010outer}
D.~Tuninetti, ``{An outer bound region for Interference Channels with
  Generalized Feedback},'' in \emph{Information Theory and Applications
  Workshop (ITA), 2010}, 2010.

\bibitem{Suh:ISIT2012}
C.~Suh, I.-H. Wang, and D.~Tse, ``Two-way interference channels,'' in
  \emph{Proc. IEEE Int. Symp. Inf. Theory}, Cambridge, Jul. 2012, pp.
  2811--2815.

\bibitem{zcheng_CISS}
Z.~Cheng and N.~Devroye, ``An outer bound region for the parallel two-way
  channel with interference,'' in \emph{45th Annual Conf. on Information
  Sciences and Systems (CISS)}, Mar. 2011.

\bibitem{Suh2012}
C.~Suh, I.-H. Wang, and D.~Tse, ``Two-way interference channels,''
  arXiv:1202.5014v1, 2012.

\bibitem{Sahai:2009}
\BIBentryALTinterwordspacing
A.~Sahai, V.~Aggarwal, M.~Yuksel, and A.~Sabharwal, ``On channel output
  feedback in deterministic interference channels,'' 2009. [Online]. Available:
  \url{http://arxiv.org/abs/0905.2392}
\BIBentrySTDinterwordspacing

\bibitem{H+K}
T.~Han and K.~Kobayashi, ``A new achievable rate region for the interference
  channel,'' \emph{IEEE Trans. Inf. Theory}, vol. IT-27, no.~1, pp. 49--60,
  1981.

\bibitem{Ozarow:MAC-FB}
L.~Ozarow, ``The capacity of the white gaussian multiple access channel with
  feedback,'' \emph{IEEE Trans. Inf. Theory}, vol.~30, no.~4, pp. 623--629,
  Jul. 1984.

\bibitem{Ozarow:1984}
L.~Ozarow and S.~Leung-Yan-Cheong, ``An achievable region and outer bound for
  the gaussian broadcast channel with feedback,'' \emph{IEEE Trans. Inf.
  Theory}, vol.~30, no.~7, pp. 667--671, Jul. 1984.

\bibitem{Carleial1975}
A.~Carleial, ``A case where interference does not reduce capacity (corresp.),''
  \emph{IEEE Trans. Inf. Theory}, vol.~21, no.~5, pp. 569 -- 570, sep 1975.

\bibitem{sato_strong}
H.~Sato, ``The capacity of {G}aussian interference channel under strong
  interference,'' \emph{IEEE Trans. Inf. Theory}, vol. IT-27, no.~6, Nov. 1981.

\bibitem{ahlswede1974}
R.~Ahlswede, ``\BIBforeignlanguage{English}{The capacity region of a channel
  with two senders and two receivers},'' \emph{\BIBforeignlanguage{English}{The
  Annals of Probability}}, vol.~2, no.~5, pp. 805--814, 1974.

\end{thebibliography}
