AAP 2026 CONFERENCE

The Krohn-Rhodes theorem, the cornerstone of algebraic automata theory, has fallen into relative obscurity. That's not a huge surprise:  most presentations of the theorem are mathematically forbidding, the payoff is not obvious,  and relatively few philosophers (or mathematicians, for that matter)  care about semigroup theory. However, several recent articles have suggested that the Krohn-Rhodes is useful for understanding hierarchically organised complex systems, gene regulatory networks, and Large Language Models, all things that philosophers do care about. I will present an accessible introduction to the Krohn-Rhodes theorem, focusing on its use in the cascaded decomposition of finite-state automata. I also gesture at a simple proof. Time permitting, I will then outline Eilenberg’s holonomy decomposition---the most common procedure for generating cascaded decompositions---and connect this back to the theorem's use in understanding complex systems.