Dependency Quantified Horn Formulas: Models and Complexity

Dependency quantified Boolean formulas (DQBF ) extend quantified Boolean formulas with Henkin-style partially ordered quantifiers. It has been shown that this is likely to yield more succinct representations at the price of a computational blow-up from PSPACE to NEXPTIME. In this paper, we consider dependency quantified Horn formulas (DQHORN ), a subclass of DQBF, and show that the computational simplicity of quantified Horn formulas is preserved when adding partially ordered quantifiers. We investigate the structure of satisfiability models for DQHORN formulas and prove that for both DQHORN and ordinary QHORN formulas, the behavior of the existential quantifiers depends only on the cases where at most one of the universally quantified variables is zero. This allows us to transform DQHORN formulas with free variables into equivalent QHORN formulas with only a quadratic increase in length. An application of these findings is to determine the satisfiability of a dependency quantified Horn formula Φ with |∀| universal quantifiers in time O(|∀| · |Φ|), which is just as hard as QHORN -SAT.