On Boolean Models for Quantified Boolean Horn Formulas

Abstract

For a Quantified Boolean Formula $({\it QBF })$ Φ=Qφ, an assignment is a function $\cal M$ that maps each existentially quantified variable of Φ to a Boolean function, where φ is a propositional formula and Q is a linear ordering of quantifiers on the variables of Φ. An assignment $\cal M$ is said to be proper, if for each existentially quantified variable y _i , the associated Boolean function f _i does not depend upon the universally quantified variables whose quantifiers in Q succeed the quantifier of y _i . An assignment $\cal M$ is said to be a model for Φ, if it is proper and the formula $\phi^{\cal M}$ is a tautology, where $\phi^{\cal M}$ is the formula obtained from φ by substituting f _i for each existentially quantified variable y _i . We show that any true quantified Horn formula has a Boolean model consisting of monotone monomials and constant functions only; conversely, if a QBF has such a model then it contains a clause–subformula in ${\it QHORN }\cap {\it SAT }$.