Models and quantifier elimination for quantified Horn formulas

Abstract

In this paper, quantified Horn formulas ( QHORN) are investigated. We prove that the behavior of the existential quantifiers depends only on the cases where at most one of the universally quantified variables is zero. Accordingly, we give a detailed characterization of QHORN satisfiability models which describe the set of satisfying truth assignments to the existential variables. We also consider quantified Horn formulas with free variables ( QHORN * ) and show that they have monotone equivalence models. The main application of these findings 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 quantified Horn formula of length O ( | ∀ | · | Φ | ) which contains only existential quantifiers. We also obtain a new algorithm for solving the satisfiability problem for quantified Horn formulas with or without free variables in time O ( | ∀ | · | Φ | ) by transforming the input formula into a satisfiability-equivalent propositional formula. Moreover, we show that QHORN satisfiability models can be found with the same complexity.