Structured d-DNNF Is Not Closed Under Negation

TL;DR

This paper demonstrates that structured d-DNNF representations are not closed under negation and are less succinct than SDD, highlighting fundamental differences in their computational properties and succinctness.

Contribution

It proves that structured d-DNNF cannot support polytime negation, unlike OBDD, and shows it is less succinct than SDD, establishing key differences in their expressive power.

Findings

Structured d-DNNF does not support polytime negation, disjunction, or existential quantification.

There exist functions with polynomial-sized structured d-DNNF but no polynomial-sized SDD.

A succinctness gap exists between PSDD and the monotone arithmetic circuit analogue.

Abstract

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.