Probabilistic Description Logics Based on the Aggregating Semantics and the Principle of Maximum Entropy

描述逻辑（dl）是一种基于逻辑的知识表示语言，专门用于表示术语知识。这些术语知识在TBox中使用一般概念包含（gci）表示，而关于个人的知识（断言知识）在ABox中表示。在许多应用领域中，知识并不总是确定的，这一事实激发了dl的概率扩展。在这种扩展中，有必要区别对待断言性知识和术语性知识。原则上，概率术语知识具有统计色彩，而概率断言知识具有主观色彩。然而，为了对包含这两种知识的知识库进行推理，需要一个涵盖这两方面的通用语义框架。这一领域以前的工作并没有以完全令人满意的方式解决这一双重需要。这个项目的主要思想是通过将ABox断言和gci分别推广到封闭和开放概率条件，来适应和扩展最近开发的聚合语义，从受限的一阶情况扩展到dl。这种语义在可能世界语义的基础上结合了主观概率和基于人口的陈述，从而为主观概率和统计概率提供了一个共同的语义框架。作为第二个主要特征，我们在聚合语义之上应用了最大熵原理。这克服了为推断概率获得大而无信息区间的缺陷，这是许多关于概率分布集的推理方法的共同特征。虽然该方法的语义特性已经对一阶逻辑片段进行了一些详细的研究，但在算法和计算特性方面只做了初步的工作。为了在实践中有用，应用这种方法得到的概率深度学习需要配备有效的推理程序。因此，这个项目的主要重点将是研究概率逻辑的计算性质（可判定性和复杂性），通过实例化方法获得，特别是使用不同表达能力的dl。除了显示可判定性和复杂性结果外，我们还将为一些已研究的dl开发实用算法，并提供原型实现。另一个主要的挑战是将方法从固定的有限大小的宇宙扩展到无限的情况，要么考虑不断增长的宇宙的极限概率，要么考虑可数无限的宇宙。此外，除了基本方法之外，我们还将研究扩展，例如在概念中使用概率，允许知识库中的附加约束和独立性假设。

Description Logics (DLs) are a well-investigated family of logic-based knowledge representation languages which are tailored towards representing terminological knowledge. This terminological knowledge is represented in the TBox using general concept inclusions (GCIs), whereas knowledge about individuals (assertional knowledge) is stated in the ABox. Probabilistic extensions of DLs are motivated by the fact that, in many application domains, knowledge is not always certain. In such extensions, there is a need for treating assertional knowledge differently from terminological knowledge. In principle, probabilistic terminological knowledge has a statistical flavour whereas probabilistic assertional knowledge has a subjective flavour. However, in order to reason with respect to a knowledge base containing both kinds of knowledge, one needs a common semantic framework covering both aspects. Previous work in this area has not addressed this dual need in a completely satisfactory way.The main idea underlying this project is to adapt and extend the recently developed aggregating semantics from a restricted first-order case to DLs by respectively generalizing ABox assertions and GCIs to closed and open probabilistic conditionals. This semantics combines subjective probabilities with population-based statements on the basis of a possible-worlds semantics, thus providing a common semantic framework for both subjective and statistical probabilities. As a second main feature, we apply the principle of maximum entropy on top of aggregating semantics. This overcomes the pitfall of obtaining large and uninformative intervals for inferred probabilities which is a common feature of many of the approaches that reason with respect to sets of probability distributions. Whereas the semantic properties of the approach have been investigated in some detail for a fragment of first-order logic, only preliminary work has been done on algorithmic and computational properties. To be useful in practice, the probabilistic DL obtained by applying this approach need to be equipped with effective reasoning procedures. Thus, the main emphasis of this project will be on investigating computational properties (decidability and complexity) of the probabilistic logics obtained by instantiating the approach in particular with DLs of different expressive power. In addition to showing decidability and complexity results, we will develop practical algorithms for some of the investigated DLs and provide prototypical implementations. Another major challenge will be to extend the approach from universes of fixed finite size to the infinite case by either considering limit probabilities for universes of growing size or considering a countably infinite universe. Furthermore, in addition to the basic approach, we will also investigate extensions, such as using probabilities also within concepts, allowing for additional constraints in the knowledge base and for independence assumptions.