A Lower Bound on the Size of Decomposable Negation Normal Form

Abstract

We consider in this paper the size of a Decomposable Negation Normal Form (DNNF) that respects a given vtree (known as structured DNNF). This representation of propositional knowledge bases was introduced recently and shown to include OBDD as a special case (an OBDD variable ordering is a special type of vtree). We provide a lower bound on the size of any structured DNNF and discuss three particular instances of this bound, which correspond to three distinct subsets of structured DNNF (including OBDD). We show that our lower bound subsumes the influential Sieling and Wegener's lower bound for OBDDs, which is typically used for identifying variable orderings that will cause a blowup in the OBDD size. We show that our lower bound allows for similar usage but with respect to vtrees, which provide structure for DNNFs in the same way that variable orderings provide structure for OBDDs. We finally discuss some of the theoretical and practical implications of our lower bound.