The Monadic Quantifier Alternation Hierarchy over Graphs is Infinite

We show that in monadic second-order logic over finite directed graphs, a strict hierarchy of expressiveness is obtained by increasing the (second-order) quantifier alternation depth of formulas. Thus, the “monadic analogue” of the polynomial hierarchy is found to be strict, which solves a problem of Fagin. The proof is based on automata theoretic concepts (rather than Ehrenfeucht-Fraı̈ssé games) and starts from a restricted class of graph-like structures, namely finite two-dimensional grids. We investigate monadic second-order definable sets of grids where the width of grids is a function of the height. In this context, the infiniteness of the quantifier alternation hierarchy is witnessed by n-fold exponential functions for increasing n. It is notable that these witness sets of the monadic hierarchy all belong to the complexity class NP, the first level of the polynomial hierarchy.