Automata Cascades: Expressivity and Sample Complexity

TL;DR

This paper introduces automata cascades, a modular framework based on the Prime Decomposition Theorem, to analyze automata expressivity and sample complexity, enabling learning of complex systems with many interacting components.

Contribution

It proposes automata cascades as a structured approach to describe automata, linking their expressivity and learnability to their component structure, and provides new bounds on sample complexity.

Findings

Sample complexity is linear in the number of components

Automata with exponential states can be learned from data

Component-based automata enable scalable learning of complex systems

Abstract

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.