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.