On the capacity of a channel with action-dependent state and reversible input

Abstract

We consider a problem of coding for channels with action-dependent states available noncausally to the encoder where the decoder is additionally required to be able to decode the channel input reliably. Lower and upper bounds on the channel capacity are derived. It is shown that the capacity is determined if there exists a maximizing joint probability distribution in the upper bound which satisfies the two-stage coding condition, and it, in turn, reveals the formula duality between this problem and that of source coding with common reconstruction and action-dependent side information. We also state two simple coding schemes and the corresponding achievable rates for the cases where the two-stage coding condition is not fulfilled.

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On the Capacity of a Channel With Action-dependent
State and Reversible Input
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KITTIPONG KITTICHOKECHAI, TOBIAS J. OECHTERING, AND MIKAEL
SKOGLUND
Stockholm 2011
Communication Theory
School of Electrical Engineering
KTH Royal Institute of Technology
IR-EE-KT 2011:035

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On the Capacity of a Channel With
Action-dependent State and Reversible Input
Kittipong Kittichokechai, Tobias J. Oechtering, and Mikael Skoglund
School of Electrical Engineering and the ACCESS Linnaeus Center
Royal Institute of Technology (KTH), Stockholm, Sweden
Abstract—We consider a problem of coding for channels with
action-dependent states available noncausally to the encoder
where the decoder is additionally required to be able to decode the
channel input reliably. Lower and upper bounds on the channel
capacity are derived. It is shown that the capacity is determined
if there exists a maximizing joint probability distribution in the
upper bound which satisfies the two-stage coding condition, and
it, in turn, reveals the formula duality between this problem and
that of source coding with common reconstruction and action-
dependent side information. We also state two simple coding
schemes and the corresponding achievable rates for the cases
where the two-stage coding condition is not fulfilled.
I. INTRODUCTION
The problem of coding for channels with random states and
source coding with side information have received consider-
able attention due to their broad sets of potential applications.
Gel’fand and Pinsker [1] solved the problem of coding for
channels with noncausal states available to the encoder, while
the problem of rate distortion coding for source with side
information at the decoder was solved by Wyner and Ziv [2].
Recently, there were some interesting extensions based on
the Gel’fand-Pinsker problem. In [3] Weissman considered
the case where the channel state is allowed to depend on a
message-dependent action sequence (action-dependent state).
Furthermore, Sumszyk and Steinberg in [4] considered a
variant of problem in the context of information embedding
where there is an additional requirement of decoding the
stegotext reliably at the decoder (reversible stegotext). On the
other hand, there was also a similar set of dual extensions
based on the Wyner-Ziv source coding problem, e.g., the work
by Permuter and Weissman [5] where the side information
can be influenced by the node in the system, and the work by
Steinberg [6] where the encoder estimates the decoder’s recon-
struction as an additional constraint (common reconstruction).
The concept of reversible stegotext in information embed-
ding [4] is closely related to that of common reconstruction
in Wyner-Ziv source coding [6]. If we neglect the distortion
constraint in the information embedding problem, these two
problems can be considered as dual problems in the sense that
there exists one-to-one correspondence between variables in
the two problems [7]. In [8] we studied the problem of source
coding with action-dependent side information and common
reconstruction where the encoder is allowed to both control
This work was supported in part by the European Commission through the
FP7 project FeedNetBack and the Swedish Research Council.
M
Action
encoder
encoder
An(M)
PSe|A
Sn
e
Channel
PY |X,Se
Xn
Yn
Decoder
ˆ
M,
ˆ Xn
Fig. 1.
channel input.
Channel with action-dependent state information and reversible
the side information and monitor the decoder’s reconstruction.
To complete the study on the potentially dual relationship to
[8], in this work we extend the problem in [3] by adding
the requirement that the decoder is able to estimate the
channel input sequence with arbitrarily small error probability
(reversible input). One potential application of this setup is in
multi-terminal communication scenario where the decoder has
incentive to estimate Xn, e.g., Xncould be an interference
created to other terminals in the network.
The paper is organized as follows. In Section II we formu-
late the problem of coding for a channel with action-dependent
states available noncausally to the encoder and reversible input
at the decoder. We provide lower and upper bounds on the
capacity of the channel and introduce an additional constraint
that we need to prove successful two-stage coding in the
achievable scheme termed as the two-stage coding condition.
The proofs of the results are provided in Section III. In
Section IV we further discuss the implication of the two-
stage coding condition and point out that if there exists a
maximizing joint probability distribution in the upper bound
such that the condition is satisfied, then the channel capacity
is determined and the duality to the source coding problem in
[8] is recognized.
II. CHANNEL CODING WITH ACTION-DEPENDENT STATE
AND REVERSIBLE CHANNEL INPUT
We consider channel coding with action-dependent states
where the states are known noncausally to the encoder as
depicted in Figure 1. The channel input Xnis estimated
with arbitrarily small error probability at the decoder and
Notation: We denote the discrete random variables, their corresponding
realizations or deterministic values, and their alphabets by the upper case,
lower case, and calligraphic letters, respectively. The term Xn
sequence {Xm,...,Xn} when m ≤ n, and the empty set otherwise. Also,
we use the shorthand notation Xnfor Xn
{X1,...,Xi−1,Xi+1,...,Xn}.
mdenotes the
1. The term Xn\idenotes the set

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the corresponding estimate is termed as reversible input. Let
n denote the block length and A,Se,X and Y be finite
sets. A state information channel is described by a triple
(A,PSe|A,Se), where A is the action alphabet, Seis the state
alphabet, and PSe|Ais the transition probability from A to
Se. A state-dependent channel is described by a quadruple
(X,Se,PY |X,Se,Y), where X is the input alphabet, Y is the
output alphabet and PY |X,Seis the transition probability from
X × Seto Y. Both channels are assumed to be memoryless
with transition probabilities,
PSn
e|An(sn
e|an) =
n
?
i=1
PSe|A(se,i|ai),
PYn|Xn,Sn
e(yn|xn,sn
e) =
n
?
i=1
PY |X,Se(yi|xi,se,i).
Let M(n)= {1,2,...,2nR} be the set of messages. An
(|M(n)|,n) code for the channels PSe|Aand PY |X,Seconsists
of an action encoder
f(n)
a
: M(n)→ An,
an encoder
f(n): M(n)× Sn
e→ Xn,
a message decoder
g(n)
m : Yn→ M(n),
and a channel input decoder
g(n)
x
: Yn→ Xn.
Definition 1: A rate R is said to be achievable if there exists
a sequence of (|M(n)|,n)-codes?f(n)
that for any δ > 0 there is an n0 such that for all block
length n > n0,
δ, where P(n)
x,e are the average error probabilities
that M ?= g(n)
channel capacity is the supremum of all achievable rates.
Theorem 2: For all joint distribution PA,PX|A,Sesuch that
I(X;Y |A)−I(X;Se|A) > 0, the rate R is achievable for the
channel with action-dependent states available noncausally to
the encoder and reversible input at the decoder if
a ,f(n),g(n)
m ,g(n)
x
?such
x,e ≤
1
nlog|M(n)| ≥ R−δ, P(n)
m,e and P(n)
m (Yn) and Xn?= g(n)
m,e≤ δ, and P(n)
x (Yn), respectively. The
R ≤ I(A,X;Y ) − I(X;Se|A)
= I(A;Y ) + I(X;Y |A) − I(X;Se|A),
(1)
where the joint distribution of (A,Se,X,Y ) is of the form
PA(a)PSe|A(se|a)PX|A,Se(x|a,se)PY |X,Se(y|x,se).
Proof: The proof is provided in Section III.
Theorem 3: If the rate R is achievable, it follows that
R ≤max
PA,PX|A,Se[I(A,X;Y ) − I(X;Se|A)].
(2)
Proof: The proof is provided in Section III.
Remark 4: For given channels PSe|Aand PY |X,Se, if there
exists a maximizing joint distribution PA,PX|A,Seof (2)
which satisfies I(X;Y |A) − I(X;Se|A) > 0, then the lower
and upper bounds on the capacity in Theorem 2 and 3
coincide and the capacity of the channel is determined by
C = maxPA,PX|A,Se[I(A,X;Y ) − I(X;Se|A)].
One trivial example is when the channel output is inde-
pendent of the states given the input. In this case it can be
shown that for any joint distribution P(1)
such that I(X;Y |A) − I(X;Se|A) ≤ 0, there always exists
another joint distribution P(2)
isfies I(X;Y |A) − I(X;Se|A) ≥ 0 and achieves a higher
rate. One possible choice is to let P(2)
P(2)
X|A,Se(x|a,se) =
?
sequently, the maximizing joint distribution of (2) in this case
will result in I(X;Y |A)−I(X;Se|A) ≥ 0 and the capacity of
such a channel is given by C = maxPA,PX|A,Se[I(A,X;Y )−
I(X;Se|A)].
In the following statements we give the lower bounds
on the channel capacity if the maximizing joint distribution
PA,PX|A,Sein (2) does not satisfy the condition I(X;Y |A)−
I(X;Se|A) > 0.
Corollary 5: The lower bound on the capacity of the chan-
nel with action-dependent states available noncausally to the
encoder and reversible input at the decoder is given by the
maximum of all R satisfying
A(a),P(1)
X|A,Se(x|a,se)
A(a),P(2)
X|A,Se(x|a,se) which sat-
A(a) = P(1)
X|A,Se(x|a,se). Con-
A(a) and
sePSe|A(se|a)P(1)
R ≤ I(X;Y ) − I(X;Se|A),
(3)
where the joint distribution of (A,Se,X,Y ) is of the form
PA(a)PSe|A(se|a)PX|A,Se(x|a,se)PY |X,Se(y|x,se).
Proof: The proof follows as a special case of the proof of
Theorem 2 where the decoding process only checks the joint
typicality between the received signal Ynand the codeword
Xn, neglecting the action codeword An. Therefore, the coding
scheme works without the extra condition on I(X;Y |A) −
I(X;Se|A).
Proposition 6: Another lower bound on the capacity of
the channel is given by considering the channel input X
independent of the states Se. The achievable rate is given by
R ≤ I(A,X;Y ),
(4)
where the joint distribution of (A,Se,X,Y ) is of the form
PA(a)PSe|A(se|a)PX|A(x|a)PY |X,Se(y|x,se).
Proof: When the state Sn
the encoder, the channel reduces to the equivalent channel
with two inputs (An,Xn), an output Yn, and the transition
probability PY |A,X=?
then follows directly from the point-to-point channel case.
Corollary 7: When the action alphabet size is one (the
degenerate case), the capacity of the channel is given by
eis not taken into account at
sePY |X,SePSe|A. The achievable rate
C = max[I(X;Y ) − I(X;Se)],
(5)
where the joint distribution of (Se,X,Y ) is of the form
PSe(se)PX|Se(x|se)PY |X,Se(y|x,se)

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and the maximization is over all PX|Se. Note that the result
recovers a special case of the results in [4] where there is no
distortion constraint.
We call the condition I(X;Y |A) − I(X;Se|A) > 0 which
appears in Theorem 2 the two-stage coding condition since it
represents the sufficient condition for a successful two-stage
coding. This shall be seen and discussed in detail in Section
III and IV.
III. PROOFS OF THEOREM 2 AND 3
A. Proof of Theorem 2: Achievability
The proof follows from a standard random coding argument
where we use the definition and properties of ǫ-typicality as
in [9], i.e., the set of ǫ-typical sequence for ǫ > 0 with respect
to PX(·) is denoted by
T(n)
ǫ
(X) =
?
xn∈ Xn:
????
1
nN(x|xn) − PX(x)
????≤ ǫPX(x)
?
,
where N(x|xn) is the number of occurrences of x in xn. We
use the technique of rate splitting, i.e., the message M of rate
R is split into two messages M1 and M2 of rates R1 and
R2. Two-stage coding is then considered, i.e., a first stage
for communicating the identity of the action sequence, and a
second stage for communicating the identity of Xnbased on
the known action sequence.
For given channels with transition probabilities PSe|A(se|a)
and PY |X,Se(y|x,se) we can assign the joint probability to any
random vector (A,X,Se) by
PA,Se,X,Y(a,se,x,y) = PA(a)PSe|A(se|a)PX|A,Se(x|a,se)·
PY |X,Se(y|x,se)
Fix PAand PX|A,Se.
Random
{1,2,...,2nR1} and M(n)
m1∈ M(n)
to PA. For each m1∈ M(n)
{xn(m1,m2,j)}1≤m2≤2nR2,1≤j≤2nR′
PX|A. Then the codebooks are revealed to the action
encoder, the channel encoder and the decoder. Consider
0 < ǫ0< ǫ1< ǫ ≤ 1.
Encoding: Given the message m = (m1,m2) ∈ M(n),
the action codeword an(m1) is chosen and the channel state
information sn
eis generated as an output of the memoryless
channel, PSn
looks for the smallest value of j, 1 ≤ j ≤ 2nR′such that
?sn
exists, set j = 1. The channel input sequence is then chosen
to be xn(m1,m2,j).
Decoding: Upon receiving yn, the decoder in the first
step looks for the smallest ˜ m1 such that
T(n)
ǫ
(Y,A). If successful, then set ˆ m1= ˜ m1. Otherwise, set
ˆ m1 = 1. Then, based on the known an(ˆ m1), the decoder
looks for a pair (˜ m2,˜j) with the smallest ˜ m2and˜j such that
?yn,an(ˆ m1),xn(ˆ m1, ˜ m2,˜j)?∈ T(n)
such a pair, the decoded message is set to be ˆ m = (ˆ m1, ˜ m2),
CodebookGeneration:
=
Let
M(n)
1
=
2
{1,2,...,2nR2}. For all
1, generate an(m1) ∈ An, n-tuples i.i.d. according
1, generate 2n(R2+R′)codewords,
i.i.d.accordingto
e|An(sn
e|an) =?n
i=1PSe|A(se,i|ai). The encoder
e,an(m1),xn(m1,m2,j)?∈ T(n)
ǫ1(Se,A,X). If no such j
?yn,an(˜ m1)?
∈
ǫ
(Y,A,X). If there exists
and ˆ xn= xn(ˆ m1, ˜ m2,˜j). Otherwise, ˆ m = (1,1) and ˆ xn=
xn(1,1,1).
Analysis of probability of error: Due to the symmetry of
the random code construction, the error probability does not
depend on which message m was sent. Assuming that m =
(¯ m1, ¯ m2) and j =¯j were sent and chosen at the encoder. We
define the error events as follows.
E1= {An(¯ m1) / ∈ T(n)
E2=??Sn
E3=??Sn
∀j ∈ {1,2,...,2nR′}?
E41=??Yn,An(¯ m1)?/ ∈ T(n)
E42=??Yn,An(˜ m1)?∈ T(n)
E51=??Yn,An(¯ m1),Xn(¯ m1, ¯ m2,¯j)?/ ∈ T(n)
E52=??Yn,An(¯ m1),Xn(¯ m1, ˜ m2,˜j)?∈ T(n)
for some (˜ m2,˜j) ?= (¯ m2,¯j)?.
The probability of error events can be bounded by
ǫ0(A)}
e,An(¯ m1)?/ ∈ T(n)
e,An(¯ m1),Xn(¯ m1, ¯ m2,j)?/ ∈ T(n)
ǫ1(Se,A)?
ǫ1(Se,A,X),
ǫ
(Y,A)?
(Y,A),for some ˜ m1?= ¯ m1
ǫ
?
ǫ
(Y,A,X)?
(Y,A,X),
ǫ
Pr(E) ≤ Pr(E1) + Pr(E2∩ Ec
+ Pr(E41∩ Ec
+ Pr(E51∩ Ec
1) + Pr(E3∩ Ec
3) + Pr(E42∩ Ec
3∩ Ec
2)
3)
4) + Pr(E52∩ Ec
3∩ Ec
4),
where E4? E41∪ E42 and Ec
event Ei.
E1) Since An(¯ m1) is i.i.d. according to PA, by the law of
large numbers we have Pr(E1) → 0 as n → ∞.
E2) Consider the event Ec
T(n)
e
is drawn independently according to
?n
[9], we have that Pr?E2∩ Ec
E3) By the covering lemma [9] where Xnis i.i.d. according
to PX|A, we have Pr?E3∩ Ec
I(X;Se|A) + δǫ1, where δǫ1→ 0 as ǫ1→ 0.
E41)Considertheevent
?sn
have the Markov chain A − (X,Se) − Y and Ynis i.i.d. ac-
cording to PY |X,Se, by using the conditional typicality lemma
[9], it follows that Pr??Yn,sn
T(n)
ǫ
(Y,A,Se,X)?
Pr(E41∩ Ec
E42) By the packing lemma [9], we have Pr?E42∩Ec
as n → ∞ if R1< I(A;Y ) − δǫ, where δǫ→ 0 as ǫ → 0.
E51) As in E41) we have Pr(E51∩Ec
E52) By the packing lemma where Xnis i.i.d. according
to PX|A, we have Pr?E52∩ Ec
R2+ R′< I(X;Y |A) − δǫ.
Finally, by combining the bounds on the rates,
idenotes the complement of
1where we have an(¯ m1) ∈
ǫ0(A). Since Sn
i=1PSe|A
?se,i|ai(¯ m1)?, by the conditional typicality lemma
1
?→ 0 as n → ∞.
?
Ec
e,an(¯ m1),xn(¯ m1, ¯ m2,¯j)?
2
→ 0 as n → ∞ if R′>
3
wherewe have
∈ T(n)
ǫ1(Se,A,X). Since we
e,an(¯ m1),xn(¯ m1, ¯ m2,¯j)?
→ 1 as n → ∞. This also implies that
3) → 0 as n → ∞.
∈
3
?→ 0
3∩Ec
4) → 0 as n → ∞.
3∩ Ec
4
?
→ 0 as n → ∞ if
R′> I(X;Se|A) + δǫ1
R1< I(A;Y ) − δǫ
R2+ R′< I(X;Y |A) − δǫ,

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and taking ǫ → 0 so that δǫ1,δǫ → 0, we have shown that
Pr(E) → 0 as n → ∞ when R = R1+ R2 < I(A;Y ) +
I(X;Y |A) − I(X;Se|A) and I(X;Y |A) − I(X;Se|A) > 0.
Note that the latter condition is for the two-stage coding to be
successful, i.e., we can split the message into two parts with
positive rates. This concludes the achievability proof.
?
B. Proof of Theorem 3: Upper bound on the capacity
We show that for any achievable rate R, it follows that
R ≤ maxPA,PX|A,Se[I(A,X;Y )−I(X;Se|A)]. According to
the joint probability mass function,
PM,An,Sn
1{f(n)
e,Xn,Yn,ˆ
M,ˆ
Xn(m,an,sn
e,xn,yn, ˆ m, ˆ xn)
=
a
(m)=an,f(n)(m,sn
e)=xn,g(n)
|M|
x
(yn)=ˆ xn,g(n)
m (yn)= ˆ m}
·
n
?
i=1
PSe|A(se,i|ai)PY |X,Se(yi|xi,se,i),
where M is chosen uniformly at random from the set M =
{1,2,...,2nR}, let us assume that a sequence of (2nR,n)
codes exists such that the average error probabilities P(n)
and P(n)
x,e → 0 as n → ∞. Then standard properties of the
entropy function give
m,e→ 0
nR = H(M)
= H(M) − H(Xn,M|Yn) + H(M|Yn) + H(Xn|M,Yn)
≤ H(M) − H(Xn,M|Yn) + H(M|Yn) + H(Xn|Yn)
Consider the last two terms in the above inequality. Similarly
to [4], by Fano’s inequality, we get
H(M|Yn) ≤ h(P(n)
H(Xn|Yn) ≤ h(P(n)
m,e) + P(n)
x,e) + P(n)
m,e· log(|M|n− 1) = nδ(m)
x,e· log(|X|n− 1) = nδ(x)
n
,
n,
where h(·) is the binary entropy function, and δ(m)
0,δ(x)
n
→ 0 as n → ∞.
Let nδ(m)
n
+nδ(x)
n
= nδn, where δnsatisfies limn→∞δn=
0. Now we continue the chain of inequalities and get
n
→
△
nR ≤ H(M) − H(Xn,M|Yn) + nδn
= H(M,Sn
e) − H(Sn
e|M) − H(Xn,M|Yn) + nδn
(a)
= H(M,Sn
− H(Xn,M|Yn) + nδn
e,Xn) − H(Sn
e|M,An)
(b)
= H(M,Sn
= I(Xn,M;Yn) − H(Sn
e,Xn) − H(Sn
e|An) − H(Xn,M|Yn) + nδn
e|An) + H(Sn
e|Xn,M) + nδn
(c)
= I(Xn,M;Yn) − H(Sn
= I(Xn,M;Yn) − I(Xn,M;Sn
e|An) + H(Sn
e|An) + nδn,
e|Xn,M,An) + nδn
where (a) follows from the fact that Xn= f(n)(M,Sn
An= f(n)
given An, and (c) from An= f(n)
e) and
a (M), (b) follows since Sn
eis independent of M
a (M).
Continuing the chain of inequalities, we get
n(R − δn)
n
?
i=1
n
?
i=1
≤I(Xn,M;Yi|Yi−1) − I(Xn,M;Se,i|Sn
e,i+1,An)
=[I(Xn,M,Sn
e,i+1,An;Yi|Yi−1)
− I(Sn
− [I(Xn,M,Yi−1;Se,i|Sn
− I(Yi−1;Se,i|Xn,M,Sn
n
?
i=1
− I(Xn,M,Yi−1;Se,i|Sn
n
?
i=1
− [H(Se,i|Sn
− H(Se,i|Sn
n
?
i=1
+ H(Se,i|Zi,Ai)
n
?
i=1
n
?
i=1
n
?
i=1
where (a) follows from the identity which is similar to the
Csisz´ ar’s sum in [10],?n
I(Yi−1;Se,i|Xn,M,Sn
fact that (Sn
△
= (M,An\i,Sn
follows from the definition of Zi.
Consider the sum of the last two terms,
e,i+1,An;Yi|Xn,M,Yi−1)]
e,i+1,An)
e,i+1,An)]
(a)
=I(Xn,M,Sn
e,i+1,An;Yi|Yi−1)
e,i+1,An)
=[H(Yi|Yi−1) − H(Yi|Yi−1,Xn,M,Sn
e,i+1,An)]
e,i+1,An)
e,i+1,An,Xn,M,Yi−1)]
(b)
≤
H(Yi) − H(Yi|Zi,Ai) − H(Se,i|Ai)
=I(Ai,Zi;Yi) − I(Zi;Se,i|Ai)
(c)
=
I(Ai,Zi,Xi;Yi) − I(Zi,Xi;Se,i|Ai)
=
I(Ai;Yi) + I(Zi,Xi;Yi|Ai) − I(Zi,Xi;Se,i|Ai),
i=1I(Sn
e,i+1,An) = 0, (b) follows from the
e,i+1,An\i) − Ai− Se,i forms a Markov chain
and by defining Zi
e,i+1,An;Yi|Xn,M,Yi−1)−
e,i+1,Xn,Yi−1), and (c)
n
?
i=1
I(Zi,Xi;Yi|Ai) − I(Zi,Xi;Se,i|Ai)
(a)
=
n
?
i=1
n
?
i=1
− I(Xi;Se,i|Ai,Yi) − I(Zi;Se,i|Ai,Yi,Xi)
n
?
i=1
n
?
i=1
I(Zi,Xi;Yi|Ai,Se,i) − I(Zi,Xi;Se,i|Ai,Yi)
(b)
=H(Yi|Ai,Se,i) − H(Yi|Ai,Se,i,Xi)
≤I(Yi;Xi|Ai,Se,i) − I(Xi;Se,i|Ai,Yi)
(c)
=I(Xi;Yi|Ai) − I(Xi;Se,i|Ai),