Query evaluation on probabilistic databases is generally intractable (#P-hard). Existing dichotomy results have identified which queries are tractable (or safe), and connected them to tractable lineages. In our previous work, using different tools, we showed that query evaluation is linear-time on probabilistic databases for arbitrary monadic second-order queries, if we bound the treewidth of the instance. In this paper, we study limitations and extensions of this result. First, for probabilistic query evaluation, we show that MSO tractability cannot extend beyond bounded treewidth: there are even FO queries that are hard on any efficiently constructible unbounded-treewidth class of graphs. This dichotomy relies on recent polynomial bounds on the extraction of planar graphs as minors, and implies lower bounds in non-probabilistic settings, for query evaluation and match counting in subinstance-closed families. Second, we show how to explain our tractability result in terms of lineage: the lineage of MSO queries on bounded-treewidth instances can be represented as bounded-treewidth circuits, polynomial-size OBDDs, and linear-size d-DNNFs. By contrast, we can strengthen the previous dichotomy to lineages, and show that there are even UCQs with disequalities that have superpolynomial OBDDs on all unbounded-treewidth graph classes; we give a characterization of such queries. Last, we show how bounded-treewidth tractability explains the tractability of the inversion-free safe queries: we can rewrite their input instances to have bounded-treewidth.

翻译：概率数据库上的查询评估通常是难解的（#P-难）。现有的二分性结果识别了哪些查询是可处理的（或安全的），并将其与可处理的谱系联系起来。在我们先前的工作中，使用不同的工具，我们证明了如果限定实例的树宽，任意一元二阶查询在概率数据库上的查询评估是线性时间的。本文研究了该结果的极限与扩展。首先，对于概率查询评估，我们证明了MSO可处理性无法扩展到有界树宽之外：存在甚至一阶查询在任意有效可构造的无界树宽图类上都是难解的。这一二分性依赖于近期关于平面图作为子式提取的多项式界限，并蕴含了非概率设置中查询评估与子实例闭族中匹配计数的下界。其次，我们展示了如何从谱系角度解释可处理性结果：有界树宽实例上MSO查询的谱系可表示为有界树宽电路、多项式大小的有序二元决策图以及线性大小的确定性分解否定范式。相反地，我们可以将先前的二分性强化到谱系层面，并且证明存在甚至包含不等式的并合合取查询在所有无界树宽图类上具有超多项式大小的有序二元决策图；我们给出了此类查询的特征刻画。最后，我们展示了有界树宽可处理性如何解释无逆安全查询的可处理性：我们可以将其输入实例重写为具有有界树宽的形式。