On the Structure of the Monadic Logic of the Binary Tree

Since the work of Rabin [9], it has been known that any monadic second order property of the (labeled) binary tree with successor functions (and not the prefix ordering) is a monadic ∆3 property. In this paper, we show this upper bound is optimal in the sense that there is a monadic Σ2 formula, stating the existence of a path where a given predicate holds infinitely often, which is not equivalent to any monadic Π2 formula. We even show that some monadic second order definable properties of the binary tree are not definable by any boolean combination of monadic Σ2 and Π2 formulas. These results rely in particular on applications of Ehrenfeucht-Fräıssé like game techniques to the case of monadic Σ2 formulas.