On Compiling Cnf Into Decision-Dnnf

Decision-DNNF is a strict subset of decomposable negation normal form (DNNF) that plays a key role in analyzing the complexity of model counters (the searches performed by these counters have their traces in Decision-DNNF). This paper presents a number of results on Decision-DNNF. First, we introduce a new notion of CNF width and provide an algorithm that compiles CNFs into Decision-DNNFs in time and space that are exponential only in this width. The new width strictly dominates the treewidth of the CNF primal graph: it is no greater and can be bounded when the treewidth of the primal graph is unbounded. This new result leads to a tighter bound on the complexity of model counting. Second, we show that the output of the algorithm can be converted in linear time to a sentential decision diagram (SDD), which leads to a tighter bound on the complexity of compiling CNFs into SDDs.