  Computability, in the presence of asynchrony and failures, is one of the
central questions in distributed computing. The celebrated asynchronous
computability theorem (ACT) charaterizes the computing power of the read-write
shared-memory model through the geometric properties of its protocol complex: a
combinatorial structure describing the states the model can reach via its
finite executions. This characterization assumes that the memory is of
unbounded capacity, in particular, it is able to store the exponentially
growing states of the full-information protocol.
  In this paper, we tackle an orthogonal question: what is the minimal memory
capacity that allows us to simulate a given number of rounds of the
full-information protocol? In the iterated immediate snapshot model (IIS), we
determine necessary and sufficient conditions on the number of bits an IIS
element should be able to store so that the resulting protocol is equivalent,
up to isomorphism, to the full-information protocol. Our characterization
implies that $n\geq 3$ processes can simulate $r$ rounds of the
full-information IIS protocol as long as the bit complexity per process is
within $\Omega(r n)$ and $O(r n \log n)$. Two processes, however, can simulate
any number of rounds of the full-information protocol using only $2$ bits per
process, which implies, in particular, that just $2$ bits per process are
sufficient to solve $\varepsilon$-agreement for arbitrarily small
$\varepsilon$.
