Reasoning in Fuzzy Description Logics with General Concept Inclusion Axioms

描述逻辑（DL）是一个家庭的逻辑为基础的知识表示语言，是专门为表示术语知识，通过允许知识工程师定义的相关概念的应用领域内的这个逻辑，然后对这些定义使用终止推理算法的原因。为了处理概念的成员和非成员之间的边界（例如，“高个子”、“高血压”或“重网络负载”）是模糊的，DL已经与模糊逻辑相结合，产生模糊描述逻辑（模糊DL）。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DL的行为与它们的清晰对应物非常相似：对于基于简单t-范数的模糊DL，如哥德尔，可以使用为相应的清晰DL调用推理机的黑盒过程，而基于更复杂的t-范数（例如乘积和Lukasiewicz）的模糊DL可以通过适当地修改用于清晰DL的基于表格的推理器来处理。然而，它最近被证明，在所谓的一般概念包含公理（GCI）的存在下，基于表的推理机的模糊DL的已公布的扩展不正确的工作。事实上，我们能够证明GCI可以导致基于乘积t范数的某些模糊DL的不可判定性。然而，对于大多数模糊DL，推理w.r.t. GCI仍然开放。本计画的目的是探讨具有一般认知能力的模糊描述语言的可判定性与不可判定性之间的界限。一方面，我们将尝试为特定的模糊DL显示更多的不可判定性结果，然后尝试从这些结果中推导出隐含不可判定性的一般标准。另一方面，我们将试图确定可判定的特殊情况下，通过扩展的tableau和automatabased决策过程的DL的模糊情况下，也看其他推理方法的非表达DL。

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.