Integrity constraints such as functional dependencies (FD) and multi-valued dependencies (MVD) are fundamental in database schema design. Likewise, probabilistic conditional independences (CI) are crucial for reasoning about multivariate probability distributions. The implication problem studies whether a set of constraints (antecedents) implies another constraint (consequent), and has been investigated in both the database and the AI literature, under the assumption that all constraints hold {\em exactly}. However, many applications today consider constraints that hold only {\em approximately}. In this paper we define an approximate implication as a linear inequality between the degree of satisfaction of the antecedents and consequent, and we study the {\em relaxation problem}: when does an exact implication relax to an approximate implication? We use information theory to define the degree of satisfaction, and prove several results. First, we show that any implication from a set of data dependencies (MVDs+FDs) can be relaxed to a simple linear inequality with a factor at most quadratic in the number of variables; when the consequent is an FD, the factor can be reduced to 1. Second, we prove that there exists an implication between CIs that does not admit any relaxation; however, we prove that every implication between CIs relaxes "in the limit". Then, we show that the implication problem for differential constraints in market basket analysis also admits a relaxation with a factor equal to 1. Finally, we show how some of the results in the paper can be derived using the {\em I-measure} theory, which relates between information theoretic measures and set theory. Our results recover, and sometimes extend, previously known results about the implication problem: the implication of MVDs and FDs can be checked by considering only 2-tuple relations.

翻译：功能依赖(FD) 和多值依赖(MVD) 等完整性限制,例如功能依赖(FD) 和多值依赖(MVD) 等功能依赖(MVD) 在数据库系统设计中至关重要。同样, 概率性有条件独立(CI) 对多变概率分布的推理至关重要。 隐含的问题研究是,一组约束(保证) 是否意味着另一个制约(后果), 并在数据库和AI文献中进行了调查, 假设所有制约都确切存在 。 但是, 许多应用今天都考虑限制, 仅能维持大约 。 在本文中, 我们定义一种近似隐含的隐含性是, 预示性前位和后位的满足程度之间的直线性不平等。 当结果为: 当结果为缓度和后期的满意度之间的线性不平等时, 我们使用后期的隐含的隐含的隐含性, 后期的隐含性, 我们证明后期的隐含性。