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

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Abstract

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.

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Data Availability

No data were used for the research described in the article.

Notes

  1. For \(A\rightarrow B\), A comes from the condition part of \(\mathbb {\mathbb {K}}^{\uparrow \downarrow }\) or \(\mathbb {\mathbb {K}}^{\downarrow \uparrow }\) and B comes from the decision part.

  2. Augmentation and Combination are applicable to the implications within formal contexts.

  3. Clearly, Reflexivity applies to both necessary and acceptable implications.

  4. Using the method in [19], as referenced in our paper, we can calculate a superset of necessary implications and that of acceptable implications by employing Augmentation and Combination. Then, we traverse the sets to remove those redundant ones and finally obtain Tables 10 and 12.

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Acknowledgements

This work is supported by the Fundamental Research Program of Shanxi Province (202103021223303), the Open Project Foundation of Intelligent Information Processing Key Laboratory of Shanxi Province (CICIP2022006), the National Natural Science Foundation of China (62072294, 61972238), Shanxi University of Finance and Economics Talent Introduction Research Startup Fund (Z18368), Shanxi Province Doctoral Graduates Research Funding (Z24223), NNSFC (62272284), the Special Fund for Science and Technology Innovation Teams of Shanxi (202204051001015), and the Natural Scientific Research Projects in Shanxi Province, China (202203021221218).

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  1. School of Information, Shanxi University of Finance and Economics, Taiyuan, 030006, Shanxi, China

    Shaoxia Zhang

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Zhang, S. Decision Implication-Based Knowledge Representation and Reasoning Within Incomplete Fuzzy Formal Context. Int. J. Fuzzy Syst. 26, 2058–2073 (2024). https://doi.org/10.1007/s40815-024-01707-1

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  • DOI: https://doi.org/10.1007/s40815-024-01707-1

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