The past decade has witnessed a surge of interest in practical techniques for projected model counting. Despite significant advancements, however, performance scaling remains the Achilles' heel of this field. A key idea used in modern counters is to count models projected on an \emph{independent support} that is often a small subset of the projection set, i.e. original set of variables on which we wanted to project. While this idea has been effective in scaling performance, the question of whether it can benefit to count models projected on variables beyond the projection set, has not been explored. In this paper, we study this question and show that contrary to intuition, it can be beneficial to project on variables beyond the projection set. In applications such as verification of binarized neural networks, quantification of information flow, reliability of power grids etc., a good upper bound of the projected model count often suffices. We show that in several such cases, we can identify a set of variables, called upper bound support (UBS), that is not necessarily a subset of the projection set, and yet counting models projected on UBS guarantees an upper bound of the true projected model count. Theoretically, a UBS can be exponentially smaller than the smallest independent support. Our experiments show that even otherwise, UBS-based projected counting can be more efficient than independent support-based projected counting, while yielding bounds of very high quality. Based on extensive experiments, we find that UBS-based projected counting can solve many problem instances that are beyond the reach of a state-of-the-art independent support-based projected model counter.
翻译:过去十年来,人们对预测模型计算的实际技术表现出了浓厚的兴趣。 然而,尽管取得了显著的进步,绩效缩放仍然是该领域的缩脚。 现代计数器使用的一个关键想法是将预测的模型进行计算,这种模型通常是预测数据集的一个小子集,即我们想要预测的最初一组变量。 尽管这一想法在提高绩效方面是有效的,但是没有探讨用预测数据集以外的变量来计算预测模型是否有益的问题。 在本文中,我们研究这一问题,并表明与直觉相反的是,对预测数据集以外的变量进行预测是有好处的。 现代计数器所使用的一个关键想法是,将预测模型中预测的神经网络的核查、信息流量化、电网的可靠性等应用中,预测模型的正确上限往往就足够了。 在这样的例子中,我们可以确定一套变量,称为基于上限支持的模型(UBS),这不一定是基于预测数据集的一个独立子集,而在UBS上所预测的模型中,而预测的模型可以保证在真实预测模型的高度支持之上,我们所预测的预测的预测的预测模型的预测模型中可以进行最大幅度地计算。