Decision Implication-Based Knowledge Representation and Reasoning Within Incomplete Fuzzy Formal Context

Formal Concept Analysis (FCA) is an order theory-based methodology employed for concept analysis and construction. Incomplete fuzzy formal context is employed to present the uncertainty or lack of memberships between individuals and attributes. Acceptable implications and necessary implications are two types of implications that assess the validity of knowledge within incomplete formal contexts. On the one hand, attribute exploration approaches within incomplete formal contexts rely on the prior knowledge of experts. On the other hand, in the existing reasoning mechanism for acceptable implications and necessary implications, the bases are inconvenient as they recursively involve the bases of all the completions of the incomplete formal context. Another critical issue is that the inference rules, originally apply to the implications in formal contexts, may yield invalid implications when they are applied to the two types of implications. In this paper, we firstly discretize incomplete fuzzy formal context into incomplete formal context by employing a dual-threshold filter function and then model the incomplete formal context by two specially constructed decision contexts. Next, we re-represent acceptable implications and necessary implications based on decision implications and demonstrate that the inference rules Augmentation and Combination, initially designed for decision implications, are practicable for necessary implications and acceptable implications. Furthermore, we utilize Augmentation, Combination, and another inference rule Reflexivity to jointly define the completeness and non-redundancy for sets of necessary implications and that of acceptable implications. Finally, we establish necessary implication basis and acceptable implication basis, which preserve all the information implied in the two types of implications while simultaneously minimizing the total number of implications.