Classifying regular languages by a split game

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