On balanced versus unbalanced computation trees

A great number of complexity classes between P and PSPACE can be defined via leaf languages for computation trees of nondeterministic polynomial-time machines. Jenner, McKenzie, and Therien (Proceedings of the 9th Conference on Structure in Complexity Theory, 1994) raised the issue of whether considering balanced or unbalanced trees makes any difference. For a number of leaf-language classes, coincidence of both models was shown, but for the very prominent example of leaf-language classes from the alternating logarithmic-time hierarchy the question was left open. It was only proved that in the balanced case these classes exactly characterize the classes from the polynomial-time hierarchy. Here, we show that balanced trees apparently make a difference: In the unbalanced case, a class from the logarithmic-time hierarchy characterizes the corresponding class from the polynomial-time hierarchy with a PP-oracle. Along the way, we get an interesting normal form for PP computations.