Computer programs, so-called solvers, for solving the well-known Boolean satisfiability problem (Sat) have been improving for decades. Among the reasons, why these solvers are so fast, is the implicit usage of the formula's structural properties during solving. One of such structural indicators is the so-called treewidth, which tries to measure how close a formula instance is to being easy (tree-like). This work focuses on logic-based problems and treewidth-based methods and tools for solving them. Many of these problems are also relevant for knowledge representation and reasoning (KR) as well as artificial intelligence (AI) in general. We present a new type of problem reduction, which is referred to by decomposition-guided (DG). This reduction type forms the basis to solve a problem for quantified Boolean formulas (QBFs) of bounded treewidth that has been open since 2004. The solution of this problem then gives rise to a new methodology for proving precise lower bounds for a range of further formalisms in logic, KR, and AI. Despite the established lower bounds, we implement an algorithm for solving extensions of Sat efficiently, by directly using treewidth. Our implementation is based on finding abstractions of instances, which are then incrementally refined in the process. Thereby, our observations confirm that treewidth is an important measure that should be considered in the design of modern solvers.
翻译:解决众所周知的布利安相向性问题(Sat)的所谓计算机程序(Sat)几十年来一直在改善。原因之一是,这些解决问题者之所以如此迅速,其原因之一是在解决过程中暗含使用公式的结构属性。这种结构指标之一是所谓的树枝,试图衡量公式实例的接近程度,以利于容易(像树一样)。这项工作侧重于基于逻辑的问题和基于树枝的方法和工具。其中许多问题也与知识的表述和推理(KR)以及一般的人工智能(AI)有关。我们提出了一种新的问题减少类型,用分解定位制导(DG)来参考。这种缩减类型构成了解决自2004年以来开放的量化布利安约束树枝公式(QBFFs)问题的基础。这个问题的解决,从而产生了一种新的方法,用以证明在逻辑、逻辑和人工智能方面一系列进一步的形式主义(KR)和人工智能(AI)的精确的下限值。尽管已经确立了较低的观察,但我们提出了一种新型的问题减少类型,通过分解制制(Dwirestration)设计,我们用一种基于不断升级的模型的方法,从而有效地测量了我们随后在树上找到一种不断改进的模型。