Rewriting (Dependency-)Quantified 2-CNF with Arbitrary Free Literals into Existential 2-HORN

We extend quantified 2-CNF formulas by also allowing literals over free variables which are exempt from the 2-CNF restriction. That means we consider quantified CNF formulas with clauses that contain at most two bound literals and an arbitrary number of free literals. We show that these Q2-CNF formulas can be transformed in polynomial time into purely existentially quantified CNF formulas in which the bound literals are in 2-HORN (∃2-HORN). Our result still holds if we allow Henkin-style quantifiers with explicit dependencies. In general, dependency quantified Boolean formulas (DQBF) are assumed to be more succinct at the price of a higher complexity. This paper shows that DQ2-CNF has a similar expressive power and complexity as ∃2-HORN. In the special case that the 2-CNF restriction is also applied to the free variables (DQ2-CNF∗), the satisfiability can be decided in linear time.