Interval sets and three-way concept analysis in incomplete contexts

In situations with incomplete information, we may have only partial knowledge about a concept. This motivates the introduction of the notion of a partially-known concept represented by a set of known instances, a set of known non-instances, and a set of objects with unknown states. We present a common conceptual framework of the notions of interval sets and incomplete formal contexts for representing partially-known concepts. An interval set is interpreted as a family of sets bounded by a pair of sets, and any one in the family may possibly be the actual set of instances of the concept when the information or knowledge becomes complete. An incomplete formal context is interpreted as a family of complete formal contexts, and any one in the family may possibly be the actual formal context when the information or knowledge becomes complete. While a complete formal context is induced by a binary relation, an incomplete formal context is induced by an interval binary relation that is interpreted as a family of binary relations. Within the proposed framework, we identify four possible forms for representing partially-known concepts. We examine, interpret, and extend existing studies on concept analysis in complete formal contexts.

[1]  Ling Wei,et al.  The attribute reductions of three-way concept lattices , 2016, Knowledge-Based Systems.

[2]  Yiyu Yao,et al.  A Note on Definability and Approximations , 2007, Trans. Rough Sets.

[3]  YuHong,et al.  A tree-based incremental overlapping clustering method using the three-way decision theory , 2016 .

[4]  Siegfried Gottwald,et al.  Applications of fuzzy sets to systems analysis , 1977, IEEE Transactions on Systems, Man, and Cybernetics.

[5]  Yiyu Yao,et al.  Interval sets and interval-set algebras , 2009, 2009 8th IEEE International Conference on Cognitive Informatics.

[6]  Yumin Chen,et al.  Three-way decision reduction in neighborhood systems , 2016, Appl. Soft Comput..

[7]  Yiyu Yao,et al.  An Outline of a Theory of Three-Way Decisions , 2012, RSCTC.

[8]  Yiyu Yao,et al.  Rough-set concept analysis: Interpreting RS-definable concepts based on ideas from formal concept analysis , 2016, Inf. Sci..

[9]  Yiyu Yao,et al.  Three-way decisions with probabilistic rough sets , 2010, Inf. Sci..

[10]  Fan Min,et al.  Three-way recommender systems based on random forests , 2016, Knowl. Based Syst..

[11]  Qing Li,et al.  Three-way decisions based software defect prediction , 2016, Knowl. Based Syst..

[12]  Max A. Sobel,et al.  Introduction to mathematics , 1984 .

[13]  Theresa Beaubouef,et al.  Rough Sets , 2019, Lecture Notes in Computer Science.

[14]  Melvin Fitting,et al.  Kleene's Logic, Generalized , 1991, J. Log. Comput..

[15]  Yiyu Yao,et al.  The two sides of the theory of rough sets , 2015, Knowl. Based Syst..

[16]  Lotfi A. Zadeh,et al.  Fuzzy Sets , 1996, Inf. Control..

[17]  Victor W. Marek,et al.  Contributions to the Theory of Rough Sets , 1999, Fundam. Informaticae.

[18]  Witold Lipski,et al.  On semantic issues connected with incomplete information databases , 1979, ACM Trans. Database Syst..

[19]  Bao Qing Hu,et al.  Fuzzy probabilistic rough sets and their corresponding three-way decisions , 2016, Knowl. Based Syst..

[20]  D. Ciucci.  Orthopairs and granular computing , 2016 .

[21]  Shuli Liu,et al.  An extended three-way decision and its application in member selection , 2015, J. Intell. Fuzzy Syst..

[22]  Bao Qing Hu,et al.  Three-way decisions space and three-way decisions , 2014, Inf. Sci..

[23]  Richard Holzer,et al.  On the Treatment of Incomplete Knowledge in Formal Concept Analysis , 2000, ICCS.

[24]  A. V. Savchenko,et al.  Fast multi-class recognition of piecewise regular objects based on sequential three-way decisions and granular computing , 2016, Knowl. Based Syst..

[25]  Hong-Ying Zhang,et al.  Ranking interval sets based on inclusion measures and applications to three-way decisions , 2016, Knowl. Based Syst..

[26]  Janusz Zalewski,et al.  Rough sets: Theoretical aspects of reasoning about data , 1996 .

[27]  Decui Liang,et al.  Three-way group decisions with decision-theoretic rough sets , 2016, Inf. Sci..

[28]  Witold Pedrycz,et al.  Determining Three-Way Decisions With Decision-Theoretic Rough Sets Using a Relative Value Approach , 2017, IEEE Transactions on Systems, Man, and Cybernetics: Systems.

[29]  Bernhard Ganter,et al.  Formal Concept Analysis: Mathematical Foundations , 1998 .

[30]  Weihua Xu,et al.  Cognitive concept learning from incomplete information , 2016, International Journal of Machine Learning and Cybernetics.

[31]  Decui Liang,et al.  Deriving three-way decisions from intuitionistic fuzzy decision-theoretic rough sets , 2015, Inf. Sci..

[32]  Bing Huang,et al.  Sequential three-way decision and granulation for cost-sensitive face recognition , 2016, Knowl. Based Syst..

[33]  Yiyu Yao,et al.  Three-Way Decision: An Interpretation of Rules in Rough Set Theory , 2009, RSKT.

[34]  Yiyu Yao,et al.  Three-Way Formal Concept Analysis , 2014, RSKT.

[35]  Guoyin Wang,et al.  A tree-based incremental overlapping clustering method using the three-way decision theory , 2016, Knowl. Based Syst..

[36]  Bao Qing Hu,et al.  Three-way decision spaces based on partially ordered sets and three-way decisions based on hesitant fuzzy sets , 2016, Knowl. Based Syst..

[37]  Decui Liang,et al.  A novel three-way decision model based on incomplete information system , 2016, Knowl. Based Syst..

[38]  Yuhua Qian,et al.  Three-way cognitive concept learning via multi-granularity , 2017, Inf. Sci..

[39]  LiangDecui,et al.  Three-way group decisions with decision-theoretic rough sets , 2016 .

[40]  Rudolf Wille,et al.  Restructuring Lattice Theory: An Approach Based on Hierarchies of Concepts , 2009, ICFCA.

[41]  Bing Shi,et al.  Regression-based three-way recommendation , 2017, Inf. Sci..

[42]  Brian F. Chellas.  Modal Logic: Normal systems of modal logic , 1980 .

[43]  Michal Krupka,et al.  Concept Lattices of Incomplete Data , 2012, ICFCA.

[44]  Heung Wong,et al.  The aggregation of multiple three-way decision spaces , 2016, Knowl. Based Syst..

[45]  Yiyu Yao,et al.  Three-Way Decisions and Cognitive Computing , 2016, Cognitive Computation.

[46]  Jiye Liang,et al.  Decision-Oriented Rough Set Methods , 2015, RSFDGrC.

[47]  Nouman Azam,et al.  Web-Based Medical Decision Support Systems for Three-Way Medical Decision Making With Game-Theoretic Rough Sets , 2015, IEEE Transactions on Fuzzy Systems.

[48]  Yiyu Yao,et al.  Advances in three-way decisions and granular computing , 2016, Knowl. Based Syst..

[49]  Didier Dubois,et al.  Borderline vs. unknown: comparing three-valued representations of imperfect information , 2014, Int. J. Approx. Reason..

[50]  Ali Shakiba,et al.  S-approximation Spaces: A Three-way Decision Approach , 2015, Fundam. Informaticae.

[51]  Tianrui Li,et al.  PICKT: A Solution for Big Data Analysis , 2015, RSKT.

[52]  Jinhai Li,et al.  Incomplete decision contexts: Approximate concept construction, rule acquisition and knowledge reduction , 2013, Int. J. Approx. Reason..

[53]  Robert LIN,et al.  NOTE ON FUZZY SETS , 2014 .

[54]  Yao Li,et al.  TDUP: an approach to incremental mining of frequent itemsets with three-way-decision pattern updating , 2015, International Journal of Machine Learning and Cybernetics.

[55]  Yanping Zhang,et al.  Multi-granular mining for boundary regions in three-way decision theory , 2016, Knowl. Based Syst..

[56]  Yiyu Yao,et al.  Interval-set algebra for qualitative knowledge representation , 1993, Proceedings of ICCI'93: 5th International Conference on Computing and Information.

[57]  Yiyu Yao,et al.  Decision-theoretic three-way approximations of fuzzy sets , 2014, Inf. Sci..

[58]  Guoyin Wang,et al.  Approximate concept construction with three-way decisions and attribute reduction in incomplete contexts , 2016, Knowl. Based Syst..

[59]  Didier Dubois,et al.  Graduality, Uncertainty and Typicality in Formal Concept Analysis , 2010, 35 Years of Fuzzy Set Theory.

[60]  Sergei A. Obiedkov,et al.  Modal Logic for Evaluating Formulas in Incomplete Contexts , 2002, ICCS.

[61]  Urszula Wybraniec-Skardowska,et al.  Extensions and Intentions in the Ruogh Set Theory , 1998, Inf. Sci..

[62]  Pierre Nicole,et al.  Logic or the Art of Thinking , 1996 .

[63]  Ling Wei,et al.  The connections between three-way and classical concept lattices , 2016, Knowl. Based Syst..

[64]  W.,et al.  Categories and Concepts · Theoretical Views and Inductive Data Analysis , 2009 .