In this paper we give exponential upper and lower bounds on the size of the cascaded (Krohn-Rhodes) decomposition of automata. These results are used for giving elementary algorithms for various translations between automata and temporal logic, where the previously-known translations were non-elementary.
Proceedings of the 17th International Colloquium on Automata, Languages, and Programming (ICALP 1990), Lecture Notes in Computer Science 443, pages 553-571. Springer-Verlag, 1990.