In this paper we investigate the problem of quantifying the contribution of each variable to the satisfying assignments of a Boolean function based on the Shapley value. Our main result is a polynomial-time equivalence between computing Shapley values and model counting for any class of Boolean functions that are closed under substitutions of variables with disjunctions of fresh variables. This result settles an open problem raised in prior work, which sought to connect the Shapley value computation to probabilistic query evaluation. We show two applications of our result. First, the Shapley values can be computed in polynomial time over deterministic and decomposable circuits, since they are closed under OR-substitutions. Second, there is a polynomial-time equivalence between computing the Shapley value for the tuples contributing to the answer of a Boolean conjunctive query and counting the models in the lineage of the query. This equivalence allows us to immediately recover the dichotomy for Shapley value computation in case of self-join-free Boolean conjunctive queries; in particular, the hardness for non-hierarchical queries can now be shown using a simple reduction from the #P-hard problem of model counting for lineage in positive bipartite disjunctive normal form.

翻译：摘要：本文基于沙普利值研究了量化每个变量对布尔函数满足赋值贡献的问题。我们的主要成果是：对于任何在变量被新变量析取式替换下封闭的布尔函数类，计算沙普利值与模型计数之间存在多项式时间等价性。这一结果解决了先前工作中提出的一个开放性问题，该问题试图将沙普利值计算与概率查询评估联系起来。我们展示了该结果的两项应用。第一，在确定性和可分解电路上，由于其对OR替换封闭，沙普利值可在多项式时间内计算。第二，对于布尔合取查询答案中贡献元组的沙普利值计算，与查询谱系中模型计数之间存在多项式时间等价性。这一等价性使我们能直接恢复自连接自由布尔合取查询中沙普利值计算的二分法；特别地，对于非层级查询的难解性，现在可通过从正二分分离析取范式谱系模型计数的#P难问题简单归约来证明。