提出一种用于模糊形式上下文的可能性推理逻辑，可表达概念分析中的三类扩展概念。
A Modal Logic for Possibilistic Reasoning with Fuzzy Formal Contexts
- 设计双类型加权模态算子，分别对应必要性与充分性，基于可能性理论解释公式
- 证明该逻辑及其必要性、充分性片段在所有模糊上下文模型下均完备
- 能表示模糊形式概念分析中的三类推广概念，适合逻辑与知识挖掘研究者
我们提出一种针对模糊形式上下文的可能性推理的双类型加权模态逻辑。该逻辑语法包含两类加权模态算子，分别对应经典必要性（$oxplus$）与充分性（$oxminus$）模态，其公式在基于可能性理论的模糊形式上下文中进行语义解释。我们给出了该逻辑的公理化体系，其对所有模糊上下文模型类是可靠的。此外，该逻辑的必要性与充分性片段也各自对所有模糊上下文模型类是完备的。通过若干示例展示了该逻辑的表达能力。由于形式上下文是形式概念分析（FCA）的基本构造，我们将其三个核心概念——形式概念、对象导向概念与属性导向概念——推广为模糊形式上下文中的对应$c$-截集概念。我们证明了该逻辑语言能够表示这三类推广概念。最后，我们展示了将该逻辑扩展至多关系模糊上下文的可行性，允许不同模糊关系的布尔组合。
原文摘要 · Abstract (English)
We introduce a two-sort weighted modal logic for possibilistic reasoning with fuzzy formal contexts. The syntax of the logic includes two types of weighted modal operators corresponding to classical necessity ($\Box$) and sufficiency ($\boxminus$) modalities and its formulas are interpreted in fuzzy formal contexts based on possibility theory. We present its axiomatization that is \emph{sound} with respect to the class of all fuzzy context models. In addition, both the necessity and sufficiency fragments of the logic are also individually complete with respect to the class of all fuzzy context models. We highlight the expressive power of the logic with some illustrative examples. As a formal context is the basic construct of formal concept analysis (FCA), we generalize three main notions in FCA, i.e., formal concepts, object oriented concepts, and property oriented concepts, to their corresponding $c$-cut concepts in fuzzy formal contexts. Then, we show that our logical language can represent all three of these generalized notions. Finally, we demonstrate the possibility of extending our logic to reasoning with multi-relational fuzzy contexts, in which the Boolean combinations of different fuzzy relations are allowed.
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