Directed Reachability: From Ajtai-Fagin to Ehrenfeucht-Fraïssé Games

In 1974 Ronald Fagin proved that properties of structures which are in NP are exactly the same as those expressible by existential second order sentences, that is sentences of the form 9 ~ P, where ~ P is a tuple of relation symbols. and is a rst order formula. Fagin was also the rst to study monadic NP: the class of properties expressible by existential second order sentences where all quantiied relations are unary. In their very diicult paper AF90] Ajtai and Fagin show that directed reachability is not in monadic NP. In AFS97] Ajtai, Fagin and Stockmeyer introduce closed monadic NP: the class of properties which can be expressed by a kind of monadic second order existential formula, where the second order quantiiers can interleave with rst order quantiiers. Among other results they show that directed reachability is expressible by a formula of the form 9P8x9P 1 ; P 2 , where P; P 1 ; P 2 are unary relation symbols and is rst order. They state the question if this property is in the rst order closure of monadic NP, that is if it is expressible by a sentence of the form ~ Qx9 ~ P, where ~ Qx is a tuple of rst order quantiiers and ~ P is a tuple of unary relation symbols. In this paper we give a negative solution to the problem.