  Computational mechanics quantifies structure in a stochastic process via its
causal states, leading to the process's minimal, optimal predictor---the
$\epsilon$-machine. We extend computational mechanics to communication channels
between two processes, obtaining an analogous optimal model---the
$\epsilon$-transducer---of the stochastic mapping between them. Here, we lay
the foundation of a structural analysis of communication channels, treating
joint processes and processes with input. The result is a principled structural
analysis of mechanisms that support information flow between processes. It is
the first in a series on the structural information theory of memoryful
channels, channel composition, and allied conditional information measures.
