Decomposable Negation Normal Forms \textsc{dnnf} [Darwiche, 'Decomposable Negation Normal Form', JACM, 2001] is a landmark Knowledge Compilation (\textsc{kc}) model, highly important both in \textsc{ai} and Theoretical Computer Science. Numerous restrictions of the model have been studied. In this paper we consider the restriction where all the gates are $\alpha$-imbalanced that is, at most one input of each gate depends on more than $n^{\alpha}$ variables (where $n$ is the number if variables of the function being represented). The concept of imbalanced gates has been first considered in [Lai, Liu, Yin 'New canonical representations by augmenting OBDDs with conjunctive decomposition', JAIR, 2017]. We consider the idea in the context of representation of \textsc{cnf}s of bounded primal treewidth. We pose an open question as to whether \textsc{cnf}s of bounded primal treewidth can be represented as \textsc{fpt}-sized \textsc{dnnf} with $\alpha$-imbalanced gates. We answer the question negatively for Decision \textsc{dnnf} with $\alpha$-imbalanced conjunction gates. In particular, we establish a lower bound of $n^{\Omega((1-\alpha) \cdot k)}$ for the representation size (where $k$ is the primal treewidth of the input \textsc{cnf}). The main engine for the above lower bound is a combinatorial result that may be of an independent interest in the area of parameterized complexity as it introduces a novel concept of bidimensionality.

翻译：可分解否定范式（Decomposable Negation Normal Forms, \textsc{dnnf}）[Darwiche, 'Decomposable Negation Normal Form', JACM, 2001]是知识编译（Knowledge Compilation, \textsc{kc}）领域的一个里程碑模型，在人工智能（\textsc{ai}）和理论计算机科学中均具有极高的重要性。该模型的多种限制形式已被广泛研究。本文考虑一种限制，即所有门电路均为$\alpha$-不平衡的，这意味着每个门电路至多有一个输入依赖于超过$n^{\alpha}$个变量（其中$n$为所表示函数的变量总数）。不平衡门电路的概念最初由[Lai, Liu, Yin 'New canonical representations by augmenting OBDDs with conjunctive decomposition', JAIR, 2017]提出。我们在有界原始树宽度的\textsc{cnf}表示背景下探讨这一概念。我们提出了一个开放性问题：有界原始树宽度的\textsc{cnf}是否能够表示为具有$\alpha$-不平衡门电路的\textsc{fpt}规模\textsc{dnnf}？针对具有$\alpha$-不平衡合取门的决策\textsc{dnnf}，我们给出了否定的答案。具体而言，我们建立了表示规模的下界为$n^{\Omega((1-\alpha) \cdot k)}$（其中$k$为输入\textsc{cnf}的原始树宽度）。上述下界证明的核心是一个组合学结果，该结果在参数化复杂度领域可能具有独立意义，因为它引入了一个新颖的双维度性概念。