Reasoning in Fuzzy Description Logics with General Concept Inclusion Axioms

描述逻辑(DLS)是一系列基于逻辑的知识表示语言，通过允许知识工程师在此逻辑中定义应用领域的相关概念，然后使用终止推理算法对这些定义进行推理，这些语言专为表示术语知识而定制。为了处理概念的成员和非成员之间的边界模糊的应用(例如，“高个子”、“高血压”或“沉重的网络负载”)，DLS已经与模糊逻辑相结合，导致模糊描述逻辑(FUZZY DLS)。考虑到过去20年关于模糊描述逻辑的文献，人们可以得到这样的印象：从算法的角度来看，模糊描述逻辑的行为与它们的精确对应的模糊描述逻辑非常相似：对于基于诸如G？odel的简单t-规范的模糊描述逻辑，可以使用为相应的精确描述逻辑调用推理器的黑盒过程，而基于更复杂的t-范数(例如乘积和Lukasiewicz)的模糊描述逻辑可以通过适当修改用于清晰描述逻辑的基于表的推理器来处理。然而，最近事实证明，在所谓的一般概念包含公理(GCIS)的存在下，已发表的基于表的推理器到模糊DLS的扩展并不能正确地工作。事实上，我们能够证明，对于某些基于乘积t-范数的模糊DLS，GCIS可以导致不可判定性。然而，对于大多数模糊DLS来说，推理的可决定性状态不是很好。GCIS仍在营业。本课题的目的是研究具有地理信息系统的模糊决策支持系统的可判定性和不可判断性之间的界限。一方面，我们将尝试对特定的模糊DLS给出更多的不可判定性结果，然后试图从这些结果中推导出隐含不可判断性的一般准则。另一方面，我们将试图确定可判定的特殊情况，通过将DLS的表式和自动机决策过程扩展到模糊情况，并寻找其他对于不可表达的DLS的推理方法。

Description logics (DLs) are a family of logic-based knowledge representation languages that are tailored towards representing terminological knowledge, by allowing the knowledge engineer to define the relevant concepts of an application domain within this logic and then reason about these definitions using terminating inference algorithms. In order to deal with applications where the boundaries between members and non-members of concepts (e.g., “tall man,” “high blood pressure,” or “heavy network load”) are blurred, DLs have been combined with fuzzy logics, resulting in fuzzy description logics (fuzzy DLs). Considering the literature on fuzzy description logics of the last 20 years, one could get the impression that, from an algorithmic point of view, fuzzy DLs behave very similarly to their crisp counterparts: for fuzzy DLs based on simple t-norms such as G¨odel, black-box procedures that call reasoners for the corresponding crisp DLs can be used, whereas fuzzy DLs based on more complicated t-norms (such as product and Lukasiewicz) can be dealt with by appropriately modifying the tableau-based reasoners for the crisp DLs. However, it has recently turned out that, in the presence of so-called general concept inclusion axioms (GCIs), the published extensions of tableaubased reasoners to fuzzy DLs do not work correctly. In fact, we were able to show that GCIs can cause undecidability for certain fuzzy DLs based on product t-norm. However, for most fuzzy DLs, the decidability status of reasoning w.r.t. GCIs is still open. The purpose of this project is to investigate the border between decidability and undecidability for fuzzy DLs with GCIs. On the one hand, we will try to show more undecidability results for specific fuzzy DLs, and then attempt to derive from these results general criteria that imply undecidability. On the other hand, we will try to determine decidable special cases, by extending tableau- and automatabased decision procedures for DLs to the fuzzy case, and also looking at other reasoning approaches for inexpressive DLs.