  We consider the communication problem over binary causal adversarial erasure
channels. Such a channel maps $n$ input bits to $n$ output symbols in
$\{0,1,\wedge\}$, where $\wedge$ denotes erasure. The channel is causal if, for
every $i$, the channel adversarially decides whether to erase the $i$th bit of
its input based on inputs $1,...,i$, before it observes bits $i+1$ to $n$. Such
a channel is $p$-bounded if it can erase at most a $p$ fraction of the input
bits over the whole transmission duration. Causal channels provide a natural
model for channels that obey basic physical restrictions but are otherwise
unpredictable or highly variable. For a given erasure rate $p$, our goal is to
understand the optimal rate (the capacity) at which a randomized (stochastic)
encoder/decoder can transmit reliably across all causal $p$-bounded erasure
channels. In this paper, we introduce the causal erasure model and provide new
upper bounds and lower bounds on the achievable rate. Our bounds separate the
achievable rate in the causal erasures setting from the rates achievable in two
related models: random erasure channels (strictly weaker) and fully adversarial
erasure channels (strictly stronger). Specifically, we show:
  - A strict separation between random and causal erasures for all constant
erasure rates $p\in(0,1)$.
  - A strict separation between causal and fully adversarial erasures for
$p\in(0,\phi)$ where $\phi \approx 0.348$.
  - For $p\in[\phi,1/2)$, we show codes for causal erasures that have higher
rate than the best known constructions for fully adversarial channels.
  Our results contrast with existing results on correcting causal bit-flip
errors (as opposed to erasures) [Dey et. al 2008, 2009], [Haviv-Langberg 2011].
For the separations we provide, the analogous separations for bit-flip models
are either not known at all or much weaker.
