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Computational Mechanics of Input-Output Processes: Structured transformations and the $ε$-transducer

Computational mechanics quantifies structure in a stochastic process via its causal states, leading to the process's minimal, optimal predictor---the $ε$-machine. We extend computational mechanics to communication channels between two processes, obtaining an analogous optimal model---the $ε$-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.