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

Abstract

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 ∃P→Φ, where P→is a tuple of relation symbols. and Φ is a first order formula. Fagin was also the first to study monadic NP: the class of properties expressible by existential second order sentences where all quantified relations are unary. In their very difficult 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 quantifiers can interleave with first order quantifiers. Among other results they show that directed reachability is expressible by a formula of the form ∃P¬x∃P1 Φ, where P and P1 are unary relation symbols and Φ is first order. They state the question if this property is in the positive first order closure of monadic NP, that is if it is expressible by a sentence of the form Q→x∃P→Φ, where Q→x is a tuple of first order quantifiers and PΦis a tuple of unary relation symbols. In this paper we give a negative solution to the problem.