  In many control problems there is only limited information about the actions
that will be available at future stages. We introduce a framework where the
Controller chooses actions $a_{0}, a_{1}, \ldots$, one at a time. Her goal is
to maximize the probability that the infinite sequence $(a_{0}, a_{1}, \ldots)$
is an element of a given subset $G$ of $\mathbb{N}^{\mathbb{N}}$. The set $G$,
called the goal, is assumed to be a Borel tail set. The Controller's choices
are restricted: having taken a sequence $h_{t} = (a_{0}, \ldots, a_{t-1})$ of
actions prior to stage $t \in \mathbb{N}$, she must choose an action $a_{t}$ at
stage $t$ from a non-empty, finite subset $A(h_{t})$ of $\mathbb{N}$. The set
$A(h_{t})$ is chosen from a distribution $p_{t}$, independently over all $t \in
\mathbb{N}$ and all $h_{t} \in \mathbb{N}^{t}$. We consider several information
structures defined by how far ahead into the future the Controller knows what
actions will be available.
  In the special case where all the action sets are singletons (and thus the
Controller is a dummy), Kolmogorov's 0-1 law says that the probability for the
goal to be reached is 0 or 1. We construct a number of counterexamples to show
that in general the value of the control problem can be strictly between 0 and
1, and derive several sufficient conditions for the 0-1 ``law" to hold.
