The Arbitrarily Varying Broadcast Channel with Degraded Message Sets with Causal Side Information at the Encoder

In this work, we study the arbitrarily varying broadcast channel (AVBC), when state information is available at the transmitter in a causal manner. We establish inner and outer bounds on both the random code capacity region and the deterministic code capacity region with degraded message sets. The capacity region is then determined for a class of channels satisfying a condition on the mutual informations between the strategy variables and the channel outputs. As an example, we consider the arbitrarily varying binary symmetric broadcast channel with correlated noises. We show cases where the condition holds, hence the capacity region is determined, and other cases where there is a gap between the bounds. Index Terms Arbitrarily varying channel, broadcast channel, degraded message sets, causal state information, Shannon strategies, side information, minimax theorem, deterministic code, random code, symmetrizability. The arbitrarily varying channel (AVC) was first introduced by Blackwell et al. [5] to describe a communication channel with unknown statistics, that may change over time. It is often described as communication in the presence of an adversary, or a jammer, attempting to disrupt communication. The arbitrarily varying broadcast channel (AVBC) without side information (SI) was first considered by Jahn [13], who derived an inner bound on the random code capacity region, namely the capacity region achieved by encoder and decoders with a random experiment, shared between the three parties. As indicated by Jahn, the arbitrarily varying broadcast channel inherits some of the properties of its single user counterpart. In particular, the random code capacity region is not necessarily achievable using deterministic codes [5]. Furthermore, Jahn showed that the deterministic code capacity region either coincides with the random code capacity region or else, it has an empty interior [13]. This phenomenon is an analogue of Ahlswede’s dichotomy property [2]. Then, in order to apply Jahn’s inner bound, one has to verify whether the capacity region has nonempty interior or not. As observed in [12], this can be resolved using the results of Ericson [10] and Csiszár and Narayan [8]. Specifically, a necessary and sufficient condition for the capacity region to have a non-empty interior is that both user marginal channels are non-symmetrizable. Various models of interest involve SI available at the encoder. In [19], the arbitrarily varying degraded broadcast channel with non-causal SI is addressed, using Ahlswede’s Robustification and Elimination Techniques [1]. The single user AVC with causal SI is addressed in the book by Csiszár and Körner [7], while their approach is independent of Ahlswede’s work. A straightforward application of Ahlswede’s Robustification Technique (RT) would violate the causality requirement. In this work, we study the AVBC with causal SI available at the encoder. We extend Ahlswede’s Robustification and Elimination Techniques [2, 1], originally used in the setting of non-causal SI. In particular, we derive a modified version of Ahlswede’s RT, suited to the setting of causal SI. In a recent paper by the authors [15], a similar proof technique is applied to the arbitrarily varying degraded broadcast channel with causal SI. Here, we generalize those results, and consider a general broadcast channel with degraded message sets with causal SI. We establish inner and outer bounds on the random code and deterministic code capacity regions. Furthermore, we give conditions on the AVBC under which the bounds coincide, and the capacity region is determined. As an example, we consider the arbitrarily varying binary symmetric broadcast channel with correlated noises. We show that in some cases, the conditions hold and the capacity region is determined. Whereas, in other cases, there is a gap between the bounds. I. DEFINITIONS AND PREVIOUS RESULTS A. Notation We use the following notation conventions throughout. Calligraphic letters X ,S,Y, ... are used for finite sets. Lowercase letters x, s, y, . . . stand for constants and values of random variables, and uppercase letters X,S, Y, . . . stand for random variables. The distribution of a random variable X is specified by a probability mass function (pmf) PX(x) = p(x) over a finite set X . The set of all pmfs over X is denoted by P(X ). We use x = (x1, x2, . . . , xj) to denote a sequence of letters from X . A random sequence X and its distribution PXn(x) = p(x) are defined accordingly. For a pair of integers i and j, 1 ≤ i ≤ j, we define the discrete interval [i : j] = {i, i+ 1, . . . , j}. This research was supported by the Israel Science Foundation (grant No. 1285/16).