On complexity of restricted fragments of Decision DNNF

Decision \textsc{dnnf} (a.k.a. $\wedge_d$-\textsc{fbdd}) is an important special case of Decomposable Negation Normal Form (\textsc{dnnf}), a landmark knowledge compilation model. Like other known \textsc{dnnf} restrictions, Decision \textsc{dnnf} admits \textsc{fpt} sized representation of \textsc{cnf}s of bounded \emph{primal} treewidth. However, unlike other restrictions, the complexity of representation for \textsc{cnf}s of bounded \emph{incidence} treewidth is wide open. In[arxiv:1708.07767], we resolved this question for two restricted classes of Decision \textsc{dnnf} that we name $\wedge_d$-\textsc{obdd} and Structured Decision \textsc{dnnf}. In particular, we demonstrated that, while both these classes have \textsc{fpt}-sized representations for \textsc{cnf}s of bounded primal treewidth, they need \textsc{xp}-size for representation of \textsc{cnf}s of bounded incidence treewidth. In the main part of this paper we carry out an in-depth study of the $\wedge_d$-\textsc{obdd} model. We formulate a generic methodology for proving lower bounds for the model. Using this methodology, we reestablish the \textsc{xp} lower bound provided in [arxiv:1708.07767]. We also provide exponential separations between \textsc{fbdd} and $\wedge_d$-\textsc{obdd} and between $\wedge_d$-\textsc{obdd} and an ordinary \textsc{obdd}. We study the complexity of Apply operation for $\wedge_d$-\textsc{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 \textsc{dnnf} that we name Structured $\wedge_d$-\textsc{fbdd} and demonstrate that this model is quite powerful for \textsc{cnf}s of bounded incidence treewidth.