On OBDDs for CNFs of Bounded Treewidth

Knowledge compilation is a rewriting approach to propositional knowledge representation. The ‘knowledge base’ is initially represented as a cnf for which many important types of queries are np-hard to answer. Therefore, the cnf is compiled into another representation for which the minimal requirement is that the clausal entailment query (can the given partial assignment be extended to a complete satisfying assignment?) can be answered in a polynomial time [5]. Such transformation can result in exponential blow up of the representation size. A possible way to circumvent this issue is to identify a structural parameter of the input cnf such that the resulting transformation is exponential in this parameter and polynomial in the number of variables. A notable result in this direction is an O(2n) upper bound on the size of Decomposable Negation Normal Form (dnnf) [3], where n is the number of variables of the given CNF and k is the treewidth of its primal graph. Quite recently this upper bound has been shown to hold for Sentential Decision Diagrams (sdd) [4], a subclass of dnnf that can be considered as a generalization of the famous Ordered Binary Decision Diagrams (obdd) and shares with the obdd the key nice features (e.g. poly-time equivalence testing). Under the treewidth parameterization, the best known upper bound for an obdd is O(n) [6]. A natural question is whether, similarly to sdd, a fixed parameter upper bound holds for obdd. We provide a negative answer to the above question. In particular, for every fixed k, we demonstrate an infinite class of cnfs of the primal graph treewidth at most k for which the obdd size is Ω(n), essentially matching the upper bound of [6]. This result establishes a parameterized separation of obdd from sdd. We further show that the considered class of instances can be transformed into one for which the obdd size is at least n n) and the sdd size is O(n) thus separating obdd from sdd in the classical sense. We also provide a more optimistic version of the O(n) upper bound for the obdd showing that it in fact holds when k is the treewidth of the incidence graph of the given cnf. 1 ar X iv :1 30 8. 38 29 v3 [ cs .L O ] 3 0 Ju l 2 01 4