Automata Cascades: Expressivity and Sample Complexity

Every automaton can be decomposed into a cascade of basic automata. This is the Prime Decomposition Theorem by Krohn and Rhodes. We show that cascades allow for describing the sample complexity of automata in terms of their components. In particular, we show that the sample complexity is linear in the number of components and the maximum complexity of a single component. This opens to the possibility for learning automata representing large dynamic 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 in turn implies that it is only possible to learn automata where the number of states and letters is linear in the amount of data available. Instead our results show that one can in principle learn automata with infinite input alphabets and a number of states that is exponential in the amount of data available.