On complexity of restricted fragments of Decision DNNF
Abstract
Decision dnnf (a.k.a. d-fbdd) is an important special case of Decomposable Negation Normal Form (dnnf), a landmark knowledge compilation model. Like other known dnnf restrictions, Decision dnnf admits fpt sized representation of cnfs of bounded primal treewidth. However, unlike other restrictions, the complexity of representation for cnfs of bounded incidence treewidth is wide open. In[arxiv:1708.07767], we resolved this question for two restricted classes of Decision dnnf that we name d-obdd and Structured Decision dnnf. In particular, we demonstrated that, while both these classes have fpt-sized representations for cnfs of bounded primal treewidth, they need xp-size for representation of cnfs of bounded incidence treewidth. In the main part of this paper we carry out an in-depth study of the d-obdd model. We formulate a generic methodology for proving lower bounds for the model. Using this methodology, we reestablish the xp lower bound provided in [arxiv:1708.07767]. We also provide exponential separations between fbdd and d-obdd and between d-obdd and an ordinary obdd. We study the complexity of Apply operation for d-obdd. While, in general, the Apply operation leads to exponential blow up of the resulting model, we identify a special restricted case where the Apply operation can be carried out efficiently. We introduce a relaxed version of Structured Decision dnnf that we name Structured d-fbdd and demonstrate that this model is quite powerful for cnfs of bounded incidence treewidth.
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