Probabilistic Description Logic as a Fragment of Probabilistic First-Order Logic

描述逻辑是形式化知识表示的核心形式。他们一方面关注术语知识，即概念之间的相互关系，另一方面关注断言知识，即关于具体个体之间的相互关系以及抽象概念对其描述的知识。知识常常受到各种形式的不确定性的影响；例如，知识可以是统计上的（“90%的鸟会飞”），也可以由于对世界的实际状态缺乏完全的了解而在主观上不确定（“病人X表现出红斑，因此有80%的可能性患有莱姆病”）。概率描述逻辑的目的是用概率方法将这种不确定知识形式化。从技术上讲，概率描述逻辑是多维模态逻辑；模型基于一组世界的概率分布，每个世界都与描述逻辑维度的标准解释相关联。后者代表个体的集合，可能具有统计分布。借助这个模型的概念，概率描述逻辑作为一个片段嵌入到Halpern的概率一阶逻辑中，类似于将经典模态逻辑嵌入到经典一阶逻辑中。这为概率描述逻辑的语义提供了坚实的基础，并为估计和比较它们的表达能力提供了一个框架。在经典案例中，中心问题涉及不同表达水平的概率描述逻辑中核心推理问题的可决性和复杂性。这包括特别轻量级的方言，如概率EL，它在很大程度上限制了表达性，但反过来允许易于处理的推理。第二个项目阶段的目标包括元理论的扩展，以涵盖表达性的公理化和形式界限问题，以及可用表达手段的扩展；特别地，我们将针对不动点扩展和具有模糊真值的概率逻辑，例如“probably”的模糊逻辑。此外，主观概率和统计概率之间的关系仍然是人们关注的焦点。我们的目标是设计逻辑，以支持直观预期的推理模式，例如从统计概率中直接推断主观，同时具有可决定的推理问题。

Description logics are core formalisms in formal knowledge representation. They are concerned on the one hand with terminological knowledge, i.e. with interrelations among concepts, and on the other hand with assertional knowledge, i.e. knowledge about interrelations between concrete individuals and their description by abstract concepts. Knowledge is often subject to various forms of uncertainty; e.g. knowledge can be statistical ("90% of all birds fly") or subjectively uncertain due to lack of complete knowledge about the actual state of the world ("Patient X exhibits erythema, and therefore suffers from Lyme borreliosis with 80% probability"). Probabilistic description logics aim to formalize such uncertain knowledge using probabilistic methods. Technically, probabilistic description logics are many-dimensional modal logics; models are based on probability distributions on a set of worlds, each of which is associated with a standard interpretation of the description logical dimension. The latter represents the set of individuals, possibly equipped with a statistical distribution. By dint of this notion of model, probabilistic description logic embeds as a fragment into Halpern's probabilistic first-order logic, in analogy to the embedding of classical modal logic into classical first-order logic. This puts the semantics of probabilistic description logics on a firm footing and provides a framework for the estimation and comparison of their expressive power. As in the classical case, central questions concern the decidability and complexity of the core reasoning problems in probabilistic description logics of various levels of expressivity. This includes in particular lightweight dialects such probabilistic EL, which limits expressivity rather substantially but in return allows for tractable reasoning.The aims of the second project phase include a broadening of the metatheory to cover also questions of axiomatization and formal bounds on expressivity, as well as an extension of the expressive means made available; in particular, we will target fixpoint extensions and probabilistic logics featuring fuzzy truth values, such as the fuzzy logic of `probably'. Moreover, the relationship between subjective and statistical probabilities remains in the focus of attention. We will aim to design logics that support intuitively expected reasoning patterns such as direct inference of subjective from statistical probabilities while at the same time having decidable reasoning problems.