Differential Dependencies: Reasoning and Discovery

The importance of difference semantics (e. g., "similar" or "dissimilar") has been recently recognized for declaring dependencies among various types of data, such as numerical values or text values. We propose a novel form of Differential Dependencies (DDs), which specifies constraints on difference, called differential functions, instead of identification functions in traditional dependency notations like functional dependencies. Informally, a differential dependency states that if two tuples have distances on attributes X agreeing with a certain differential function, then their distances on attributes Y should also agree with the corresponding differential function on Y. For example, [date(<= 7)] -> [price(< 100)] states that the price difference of any two days within a week length should be no greater than 100 dollars. Such differential dependencies are useful in various applications, for example, violation detection, data partition, query optimization, record linkage, etc. In this article, we first address several theoretical issues of differential dependencies, including formal definitions of DDs and differential keys, subsumption order relation of differential functions, implication of DDs, closure of a differential function, a sound and complete inference system, and minimal cover for DDs. Then, we investigate a practical problem, that is, how to discover DDs and differential keys from a given dataset. Due to the intrinsic hardness, we develop several pruning methods to improve the discovery efficiency in practice. Finally, through an extensive experimental evaluation on real datasets, we demonstrate the discovery performance and the effectiveness of DDs in several real applications.