Structured d-DNNF Is Not Closed Under Negation

Both structured d-DNNF and SDD can be exponentially more succinct than OBDD. Moreover, SDD is essentially as tractable as OBDD. But this has left two important open questions. Firstly, does OBDD support more tractable transformations than structured d-DNNF? And secondly, is structured d-DNNF more succinct than SDD? In this paper, we answer both questions in the affirmative. For the first question we show that, unlike OBDD, structured d-DNNF does not support polytime negation, disjunction, or existential quantification operations. As a corollary, we deduce that there are functions with an equivalent polynomial-sized structured d-DNNF but with no such representation as an SDD, thus answering the second question. We also lift this second result to arithmetic circuits (AC) to show a succinctness gap between PSDD and the monotone AC analogue to structured d-DNNF.