Classifying regular languages by a split game

In this paper, we introduce the split game, a variant of the Ehrenfeucht–Fraïssé game from logic, which is useful for analyzing the expressive power of classes of generalized regular expressions. An extension of the split game to generalized ω-regular expressions is also established. To gain insight into how the split game can be applied to attack the long-standing generalized star height 2 problem, we propose and solve the omega power problem, a similar but tractable problem in the context of ω-languages. Namely we show that omega powers, together with boolean combinations and concatenations, are not sufficient to express the class of ω-regular languages.