The field of knowledge compilation establishes the tractability of many tasks by studying how to compile them to Boolean circuit classes obeying some requirements such as structuredness, decomposability, and determinism. However, in other settings such as intensional query evaluation on databases, we obtain Boolean circuits that satisfy some width bounds, e.g., they have bounded treewidth or pathwidth. In this work, we give a systematic picture of many circuit classes considered in knowledge compilation and show how they can be systematically connected to width measures, through upper and lower bounds. Our upper bounds show that bounded-treewidth circuits can be constructively converted to d-SDNNFs, in time linear in the circuit size and singly exponential in the treewidth; and that bounded-pathwidth circuits can similarly be converted to uOBDDs. We show matching lower bounds on the compilation of monotone DNF or CNF formulas to structured targets, assuming a constant bound on the arity (size of clauses) and degree (number of occurrences of each variable): any d-SDNNF (resp., SDNNF) for such a DNF (resp., CNF) must be of exponential size in its treewidth, and the same holds for uOBDDs (resp., n-OBDDs) when considering pathwidth. Unlike most previous work, our bounds apply to any formula of this class, not just a well-chosen family. Hence, we show that pathwidth and treewidth respectively characterize the efficiency of compiling monotone DNFs to uOBDDs and d-SDNNFs with compilation being singly exponential in the corresponding width parameter. We also show that our lower bounds on CNFs extend to unstructured compilation targets, with an exponential lower bound in the treewidth (resp., pathwidth) when compiling monotone CNFs of constant arity and degree to DNNFs (resp., nFBDDs).

中文翻译：

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连接知识编辑类的宽度参数

领域知识汇编通过研究如何建立的许多任务的可追踪性编译它们遵循布尔化电路类，并满足一些要求，例如结构化，可分解性和确定性。但是，在其他设置（例如对数据库的查询查询评估）中，我们获得了满足某些宽度范围的布尔电路，例如，它们具有有限的树宽或路径宽度。在这项工作中，我们给出了知识汇编中考虑的许多电路类别的系统图片，并展示了它们如何可以通过上下限系统地与宽度度量联系起来。我们的上限表明，有界树宽电路可以被构造性地转换为d-SDNNF，其时间在电路大小上呈线性，而在树宽上则呈指数级；而且有界带宽电路可以类似地转换为uOBDD。我们展示了单调DNF或CNF公式与结构化目标的匹配下限，假设常数（子句的大小）和程度（每个变量的出现次数）的界限是恒定的：此类DNF（resp。，CNF）的任何d-SDNNF（resp。，SDNNF）在其DNF中必须是指数大小树宽，在考虑路径宽度时，对于uOBDD（resp。，n-OBDD）也是如此。与以前的大多数作品不同，我们的界限适用于此类的任何公式，而不仅仅是一个精心挑选的家庭。因此，我们表明，路径宽度和树宽度分别表征了将单调DNF编译为uOBDD和d-SDNNF的效率，并且在相应的width参数中编译成指数增长。我们还表明，当将恒定常数和程度恒定的单调CNF编译为DNNF（分别为nFBDD）时，CNF的下限可以扩展到非结构化编译目标，并且在树宽（指数，路径宽度）中具有指数下限。