Incomplete information allow to deal with data with errors, uncertainty or inconsistencies and have been studied in different application areas such as query answering or data integration. In this paper, we investigate classical functional dependencies in presence of incomplete information. To do so, we associate each attribute with a comparability function which maps every pair of domain values to abstract values, assumed to be organized in a lattice. Thus, every relation schema has an associated product lattice from which we define abstract functional dependencies over abstract tuples, leading to reasoning in a multi-valued logic. In this setting, we revisit classical notions like soundness and completeness of Armstrong axioms, attribute set closure, implication problem and give associated results. We also focus on the interpretations of abstract values in true/false logic to define the notion of reality which corresponds to a {0,1}-embedding of the product lattice. Based on this semantic, we introduce the notions of possible (there exists one reality in which the given FD holds) and certain (for every reality, the given FD holds) functional dependencies. We show that the problem of checking if a functional dependency is certain can be solved in polynomial time, whereas the problem of checking if a FD is possible is NP-Complete. We also identify tractable cases depending on lattices properties.

中文翻译：

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可能/某些功能依赖

不完整的信息允许处理带有错误、不确定性或不一致的数据，并且已经在不同的应用领域进行了研究，例如查询回答或数据集成。在本文中，我们研究了存在不完整信息的经典函数依赖关系。为此，我们将每个属性与一个可比性函数相关联，该函数将每对域值映射到抽象值，假设被组织在格子中。因此，每个关系模式都有一个关联的产品格，我们从中定义抽象元组上的抽象函数依赖关系，从而在多值逻辑中进行推理。在这种情况下，我们重新审视经典概念，如阿姆斯壮公理的健全性和完备性、属性集闭包、蕴涵问题并给出相关结果。我们还关注对真/假逻辑中抽象值的解释，以定义对应于乘积点阵的 {0,1} 嵌入的现实概念。基于这个语义，我们引入了可能（存在一个给定 FD 存在的现实）和某些（对于每个现实，给定 FD 持有）函数依赖的概念。我们表明检查函数依赖是否确定的问题可以在多项式时间内解决，而检查 FD 是否可能的问题是 NP-Complete。我们还根据格子属性确定易处理的情况。我们引入了可能（存在一个给定 FD 成立的现实）和某些（对于每个现实，给定 FD 成立）函数依赖的概念。我们表明检查函数依赖是否确定的问题可以在多项式时间内解决，而检查 FD 是否可能的问题是 NP-Complete。我们还根据格子属性确定易处理的情况。我们引入了可能（存在一个给定 FD 成立的现实）和某些（对于每个现实，给定 FD 成立）函数依赖的概念。我们表明检查函数依赖是否确定的问题可以在多项式时间内解决，而检查 FD 是否可能的问题是 NP-Complete。我们还根据格子属性确定易处理的情况。