Reasoning and Query Answering Using Concept Similarity Measures and Graded Membership Functions

描述逻辑（DL）是一个基于逻辑的知识表示语言，例如，通常用于形式化应用领域的本体，例如生物学和医学。为了将这种应用领域的重要概念定义为形式概念，DL陈述了个体属于概念的充分必要条件。一旦应用程序域的相关概念以这种方式形式化，就可以在查询中使用它们，以便从数据中检索新信息。由于传统的DL是基于经典的一阶逻辑，它们的语义是严格的，在这个意义上，所有规定的属性需要满足一个人属于一个概念，同样是真实的查询的答案。在很难得到精确定义的应用中，放松这个严格的要求并允许概念的近似定义是有用的，其中大多数但不是所有的所述性质都需要保持。类似地，如果一个查询没有精确的答案，满足查询所寻找的大多数特征的近似答案可能是有用的。为了允许概念的近似定义，我们引入了分级隶属函数的概念，它代替布尔隶属值0或1，从区间[0，1]产生隶属度。例如，阈值概念可以要求个体属于度至少为0.8的概念C。一种不同的方法，其基于对概念的相似性度量的使用，被用于放松实例查询（即，由单个概念组成的查询）。给定查询概念C，我们正在寻找查询D的答案，其与C的相似性高于某个阈值。虽然这两种方法是由两位申请人与博士生一起独立开发的，但事实证明它们之间存在密切联系。相似性度量可用于定义分级隶属函数，阈值概念可用于定义分级隶属函数。这些函数为宽松的实例查询提供了更自然的语义。2这个项目的目标是1.详细探讨两种方法之间的联系，并开发和研究一种组合方法; 2.通过考虑a）比DL EL更有表达力的DL（该方法最初是针对DL EL开发的）; B）更一般的术语形式; c）更一般的查询，诸如合取查询; d）更一般的推理问题，诸如确定适当的阈值而不是要求用户提供它们，来扩展该组合方法的适用性。

Description Logics (DLs) are a well-investigated family of logic-based knowledge representation languages, which are, e.g., frequently used to formalize ontologies for application domains such as biology and medicine. To define the important notions of such an application domain as formal concepts, DLs state necessary and sufficient conditions for an individual to belong to a concept. Once the relevant concepts of an application domain are formalized this way, they can be used in queries in order to retrieve new information from data. Since traditional DLs are based on classical first-order logic, their semantics is strict in the sense that all the stated properties need to be satisfied for an individual to belong to a concept, and the same is true for answers to queries. In applications where exact definitions are hard to come by, it would be useful to relax this strict requirement and allow for approximate definitions of concepts, where most, but not all, of the stated properties are required to hold. Similarly, if a query has no exact answer, approximate answers that satisfy most of the features the query is looking for could be useful.In order to allow for approximate definitions of concepts, we have introduced the notion of a graded membership function, which instead of a Boolean membership value 0 or 1 yields a membership degree from the interval [0,1]. Threshold concepts can then, for example, require that an individual belongs to a concept C with degree at least 0.8. A different approach, which is based on the use of similarity measures on concepts, was used to relax instance queries (i.e., queries that consist of a single concept). Given a query concept C, we are looking for answers to queries D whose similarity to C is higher than a certain threshold. While these two approaches were developed independently by the two applicants together with doctoral students, it has turned out that there are close connections. Similarity measures can be used to define graded membership functions, and threshold concepts w.r.t. these functions provide a more natural semantics for relaxed instance queries.The goals of this project are to1. explore the connection between the two approaches in detail, and develop and investigate a combined approach; 2. extend the applicability of this combined approach by considering a) more expressive DLs than the DL EL for which the approaches were initially developed; b) more general terminological formalisms; c) more general queries such as conjunctive queries; d) more general inference problems such as determining appropriate threshold values rather than requiring the user to provide them.