  Every automaton can be decomposed into a cascade of basic prime automata.
This is the Prime Decomposition Theorem by Krohn and Rhodes. Guided by this
theory, we propose automata cascades as a structured, modular, way to describe
automata as complex systems made of many components, each implementing a
specific functionality. Any automaton can serve as a component; using specific
components allows for a fine-grained control of the expressivity of the
resulting class of automata; using prime automata as components implies
specific expressivity guarantees. Moreover, specifying automata as cascades
allows for describing the sample complexity of automata in terms of their
components. We show that the sample complexity is linear in the number of
components and the maximum complexity of a single component, modulo logarithmic
factors. This opens to the possibility of learning automata representing large
dynamical systems consisting of many parts interacting with each other. It is
in sharp contrast with the established understanding of the sample complexity
of automata, described in terms of the overall number of states and input
letters, which implies that it is only possible to learn automata where the
number of states is linear in the amount of data available. Instead our results
show that one can learn automata with a number of states that is exponential in
the amount of data available.
