A problem dating back to Boole [Laws of Thought, Walton & Maberly,1854] is what can be computed about the probability of a finite union of events when given as input the probabilities of intersections of some of the events. The modern geometric study of the problem can be traced back to Hailperin [Amer. Math. Monthly 2 (1965) 343--359] who phrased the problem in the language of linear programming and generalized it to logical formulas of the events other than disjunction, heralding a substantial body of work in probabilistic logic [Nilsson, Artif.\ Intell.\ 28 (1986) 71--87], including the probabilistic satisfiability problem of Georgakopoulos, Kavvadis, and Papadimitriou [J.Complexity 4 (1988) 1--11], as well as fundamental connections to the geometry of metrics via cut and correlation polytopes [Deza and Laurent, Geometry of Cuts and Metrics, Springer, 1997] and to the study of marginal polytopes in graphical models of machine learning [Wainwright and Jordan, Found.\ Trends Mach.\ Learn. 1 (2008) 1--305]. This paper (i) describes the pertinent geometry of Boole's problem via coordinate projections of an elementary polytope arising essentially from Hailperin's linear program on the atoms of a Venn diagram, and (ii) shows that computing the optimal interval for the union probability is NP-hard, resolving an apparent gap in the literature highlighted by Pitowsky [Math.\ Programming 50 (1991) 395--414] and Boros et al. [Math.\ Oper.\ Res. 39 (2014) 1311--1329 and 51 (2026) 134--148].

翻译：追溯到布尔（《思想定律》，Walton & Maberly，1854年）的一个问题是：当输入某些事件交的概率时，能够计算出有限事件并的概率的哪些信息。该问题的现代几何研究可追溯至Hailperin（Amer. Math. Monthly 2 (1965) 343--359），他利用线性规划语言表述该问题，并将其推广至除析取之外的事件逻辑公式，预示了概率逻辑领域的大量后续工作[Nilsson，Artif. Intell. 28 (1986) 71--87]，包括Georgakopoulos、Kavvadis和Papadimitriou的概率可满足性问题[J. Complexity 4 (1988) 1--11]，以及通过割与相关多面体[Deza and Laurent，《割与度量的几何》，Springer，1997]与度量几何的基本联系，和机器学习图模型中边缘多面体的研究[Wainwright and Jordan，Found. Trends Mach. Learn. 1 (2008) 1--305]。本文（i）通过一个基本源自Hailperin关于文氏图原子线性规划的初等多面体的坐标投影，描述了布尔问题的相关几何；并且（ii）证明了计算并概率最优区间是NP-难的，解决了由Pitowsky[Math. Programming 50 (1991) 395--414]和Boros等人[Math. Oper. Res. 39 (2014) 1311--1329 及 51 (2026) 134--148]所强调的文献中的明显空白。