Lower Bounds for Approximate Knowledge Compilation
This work provides theoretical lower bounds for approximate knowledge compilation, which is important for researchers and practitioners working with d-DNNF circuits in areas like probabilistic reasoning.
This paper formalizes two notions of approximation for knowledge compilation into deterministic decomposable negation normal form (d-DNNF): weak and strong approximation. It then establishes lower bounds for the size of d-DNNF circuits under these approximations, complementing existing positive results in the literature.
Knowledge compilation studies the trade-off between succinctness and efficiency of different representation languages. For many languages, there are known strong lower bounds on the representation size, but recent work shows that, for some languages, one can bypass these bounds using approximate compilation. The idea is to compile an approximation of the knowledge for which the number of errors can be controlled. We focus on circuits in deterministic decomposable negation normal form (d-DNNF), a compilation language suitable in contexts such as probabilistic reasoning, as it supports efficient model counting and probabilistic inference. Moreover, there are known size lower bounds for d-DNNF which by relaxing to approximation one might be able to avoid. In this paper we formalize two notions of approximation: weak approximation which has been studied before in the decision diagram literature and strong approximation which has been used in recent algorithmic results. We then show lower bounds for approximation by d-DNNF, complementing the positive results from the literature.