Quantifier Rewriting and Equivalence Models for Quantified Horn Formulas

In this paper, quantified Horn formulas with free variables (QHORN∗) are investigated. The main result is that any quantified Horn formula Φ of length |Φ| with free variables, |∀| universal quantifiers and an arbitrary number of existential quantifiers can be transformed into an equivalent formula of length O(|∀| · |Φ|) which contains only existential quantifiers. Moreover, it is shown that quantified Horn formulas with free variables have equivalence models where every existential quantifier is associated with a monotone Boolean function. The results allow a simple representation of quantified Horn formulas as purely existentially quantified Horn formulas (∃HORN∗). An application described in the paper is to solve QHORN∗-SAT in O(|∀| · |Φ|) by using this transformation in combination with a linear-time satisfiability checker for propositional Horn formulas.