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Structured d-DNNF Is Not Closed Under Negation

Harry Vinall-Smeeth

TL;DR

The paper addresses whether structured d-DNNF representations can match the succinctness and transformation capabilities of SDDs, and whether they are closed under key operations like negation. By linking knowledge compilation to best-partition communication complexity and employing a novel copying-and-permutation lifting construction, the authors prove that structured d-DNNF is not closed under negation, disjunction, or existential quantification, and that some functions admit poly-sized structured d-DNNFs but require superpolynomial size for any equivalent SDD. They obtain a quasi-polynomial gap of the form $n^{\tilde{\Omega}(\log n)}$ in size between these representations and extend the results to PSDD and the monotone arithmetic-circuit analogue $\textrm{AC}_m$, establishing a broader succinctness gap. The findings illuminate a fundamental trade-off between succinctness and support for transformations in knowledge- compilation formalisms and provide a framework that connects representation size to communication complexity, with open questions about potential exponential gaps and closure properties.

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.

Structured d-DNNF Is Not Closed Under Negation

TL;DR

The paper addresses whether structured d-DNNF representations can match the succinctness and transformation capabilities of SDDs, and whether they are closed under key operations like negation. By linking knowledge compilation to best-partition communication complexity and employing a novel copying-and-permutation lifting construction, the authors prove that structured d-DNNF is not closed under negation, disjunction, or existential quantification, and that some functions admit poly-sized structured d-DNNFs but require superpolynomial size for any equivalent SDD. They obtain a quasi-polynomial gap of the form in size between these representations and extend the results to PSDD and the monotone arithmetic-circuit analogue , establishing a broader succinctness gap. The findings illuminate a fundamental trade-off between succinctness and support for transformations in knowledge- compilation formalisms and provide a framework that connects representation size to communication complexity, with open questions about potential exponential gaps and closure properties.

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.
Paper Structure (20 sections, 14 theorems, 8 equations, 2 figures)

This paper contains 20 sections, 14 theorems, 8 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Formulas, NNF, structured d-DNNF and SDD
  3. Formulas and Boolean functions
  4. Negation Normal Form
  5. Decomposability, determinism and structuredness
  6. SDDs
  7. Succinctness
  8. Knowledge Compilation and Communication Complexity
  9. Proof of Theorems \ref{['thm: sep']} and \ref{['thm: main']}
  10. Proof Outline
  11. From Fixed Partition to Best Partition
  12. Idea of the Construction
  13. The Construction
  14. Step 1: Making Copies of Variables
  15. Step 2: Adding permutations
  16. ...and 5 more sections

Key Result

Theorem 1

For every $n \in \mathbb{N}$, there exists a function $f$ with an equivalent structured d-DNNF of size $n$ such that any SDD equivalent to $f$ has size $n^{\tilde{\Omega}(\log(n))}$.

Figures (2)

  • Figure 1: (left) A v-tree $T$. (right) A structured d-DNNF $\mathcal{C}$ that respects $T$. $\langle \mathcal{C} \rangle = (a \wedge b \wedge \neg c ) \vee (a \wedge b \wedge c \wedge e) \vee (a \wedge b \wedge c \wedge d \wedge \neg e))$
  • Figure 2: (left) A monotone arithmetic circuit, $\mathcal{C}$. Note that $\langle \mathcal{C} \rangle = (a+0) \times (b \times (\neg c +3))$. To evaluate $f_{\mathcal{C}}$ for $a=1, b=1, c=0$ we compute $(1+0) \times (1 \times ((1-0) +3)) = 4$. (right) $\phi(\mathcal{C})$, an NNF. Note that $\mathsf{supp}(\mathcal{C}) = \mathsf{sat}(\phi(\mathcal{C}))$.

Theorems & Definitions (32)

  • Theorem 1
  • Theorem 2
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Lemma 1
  • ...and 22 more