Rough concepts

The present paper proposes a novel way to unify Rough Set Theory and Formal Concept Analysis. Our method stems from results and insights developed in the algebraic theory of modal logic, and is based on the idea that Pawlak's original approximation spaces can be seen as special instances of enriched formal contexts, i.e. relational structures based on formal contexts from Formal Concept Analysis.

[1]  Dimiter Vakarelov,et al.  A model logic for similarity relations in pawlak knowledge representation systems , 1991, Fundam. Informaticae.

[2]  Rodolphe Durand,et al.  Category Spanning, Evaluation, and Performance: Revised Theory and Test on the Corporate Law Market , 2016 .

[3]  Petr Hájek,et al.  A Fuzzy Modal Logic for Belief Functions , 2001, Fundam. Informaticae.

[4]  Jesús Medina,et al.  Unifying Reducts in Formal Concept Analysis and Rough Set Theory , 2018, Trends in Mathematics and Computational Intelligence.

[5]  Willem Conradie,et al.  Modelling Informational Entropy , 2019, WoLLIC.

[6]  Yiyu Yao,et al.  Structured approximations as a basis for three-way decisions in rough set theory , 2019, Knowl. Based Syst..

[7]  Yiyu Yao,et al.  A Comparative Study of Formal Concept Analysis and Rough Set Theory in Data Analysis , 2004, Rough Sets and Current Trends in Computing.

[8]  R. van Rooij,et al.  Vagueness, tolerance and non-transitive entailment , 2011 .

[9]  Yiyu Yao,et al.  Min-max attribute-object bireducts: On unifying models of reducts in rough set theory , 2019, Inf. Sci..

[10]  Mihir K. Chakraborty,et al.  Algebraic structures in the vicinity of pre-rough algebra and their logics , 2014, Inf. Sci..

[11]  Alessandra Palmigiano,et al.  Toward a Dempster-Shafer theory of concepts , 2019, Int. J. Approx. Reason..

[12]  Willem Conradie,et al.  Modelling socio-political competition , 2019, Fuzzy Sets Syst..

[13]  Dexue Zhang,et al.  Concept lattices of fuzzy contexts: Formal concept analysis vs. rough set theory , 2009, Int. J. Approx. Reason..

[14]  On Acquiring Social Categories : Cognitive Development and Anthropological Wisdom , 1988 .

[15]  Melvin Fitting,et al.  Many-valued modal logics II , 1992 .

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

[17]  Yiyu Yao,et al.  Generalization of Rough Sets using Modal Logics , 1996, Intell. Autom. Soft Comput..

[18]  Cecelia Britz.  Correspondence theory in many-valued modal logic , 2016 .

[19]  Robert E. Kent,et al.  Rough Concept Analysis: A Synthesis of Rough Sets and Formal Concept Analysis , 1996, Fundam. Informaticae.

[20]  Alessandra Palmigiano,et al.  Algebraic semantics and model completeness for Intuitionistic Public Announcement Logic , 2011, Ann. Pure Appl. Log..

[21]  Ewa Orlowska,et al.  Rough Set Semantics for Non-classical Logics , 1993, RSKD.

[22]  Jon M. Kleinberg,et al.  Clustering categorical data: an approach based on dynamical systems , 2000, The VLDB Journal.

[23]  Robert E. Kent.  Soft Concept Analysis , 2018, ArXiv.

[24]  Xiangping Kang,et al.  Rough set model based on formal concept analysis , 2013, Inf. Sci..

[25]  Nachoem M. Wijnberg,et al.  Classification systems and selection systems: The risks of radical innovation and category spanning , 2011 .

[26]  Gwo-Hshiung Tzeng,et al.  An integration method combining Rough Set Theory with formal concept analysis for personal investment portfolios , 2010, Knowl. Based Syst..

[27]  Miroslav Haviar,et al.  TiRS graphs and TiRS frames: a new setting for duals of canonical extensions , 2015 .

[28]  Richmond Campbell.  The sorites paradox , 1974 .

[29]  Miroslav Haviar,et al.  A Fresh Perspective on Canonical Extensions for Bounded Lattices , 2012, Applied Categorical Structures.

[30]  Anirban Saha,et al.  Algebraic structures in the vicinity of pre-rough algebra and their logics II , 2016, Inf. Sci..

[31]  Willem Conradie,et al.  Probabilistic Epistemic Updates on Algebras , 2019, ACM Trans. Comput. Log..

[32]  Marcin Wolski.  Formal Concept Analysis and Rough Set Theory from the Perspective of Finite Topological Approximations , 2005, Trans. Rough Sets.

[33]  Willem Conradie,et al.  Unified Correspondence , 2014, Johan van Benthem on Logic and Information Dynamics.

[34]  Brian A. Davey,et al.  An Introduction to Lattices and Order , 1989 .

[35]  Gerda Gemser,et al.  The producer-consumer classification gap and its effects on music festival success , 2016 .

[36]  Melvin Fitting,et al.  Many-valued modal logics , 1991, Fundam. Informaticae.

[37]  Ronald Fagin,et al.  Uncertainty, belief, and probability 1 , 1991, IJCAI.

[38]  Alessandra Palmigiano,et al.  Logics for Rough Concept Analysis , 2018, ICLA.

[39]  U. Wybraniec-Skardowska.  On a generalization of approximation space , 1989 .

[40]  Gianpiero Cattaneo,et al.  On the connection of hypergraph theory with formal concept analysis and rough set theory , 2016, Inf. Sci..

[41]  Michael T. Hannan,et al.  Multiple Category Memberships in Markets: An Integrative Theory and Two Empirical Tests , 2009 .

[42]  Zdzislaw Pawlak,et al.  Rough Set Theory and its Applications to Data Analysis , 1998, Cybern. Syst..

[43]  Szymon Frankowski,et al.  DEFINABLE CLASSES OF MANY VALUED KRIPKE FRAMES , 2006 .

[44]  Melvin Fitting.  Many-Valued Model Logics II , 1992, Fundam. Informaticae.

[45]  Willem Conradie,et al.  Categories: How I Learned to Stop Worrying and Love Two Sorts , 2016, WoLLIC.

[46]  Gerda Gemser,et al.  Categorization and Willingness to Pay for New Products: The Role of Category Cues as Value Anchors , 2017 .

[47]  Radim Belohlávek,et al.  Fuzzy Galois Connections , 1999, Math. Log. Q..

[48]  Lluis Godo,et al.  On the Minimum Many-Valued Modal Logic over a Finite Residuated Lattice , 2008, J. Log. Comput..

[49]  Willem Conradie,et al.  Toward an Epistemic-Logical Theory of Categorization , 2017, TARK.

[50]  Richard K. Belew,et al.  Lexical dynamics and conceptual change: Analyses and implications for information retrieval , 2003 .

[51]  Didier Dubois,et al.  The Structure of Oppositions in Rough Set Theory and Formal Concept Analysis - Toward a New Bridge between the Two Settings , 2014, FoIKS.

[52]  E. Ruspini.  The Logical Foundations of Evidential Reasoning (revised) , 1987 .

[53]  Willem Conradie,et al.  Modelling competing theories , 2019, EUSFLAT Conf..

[54]  Willem Conradie,et al.  Algorithmic correspondence and canonicity for non-distributive logics , 2016, Ann. Pure Appl. Log..

[55]  Alessandra Palmigiano,et al.  Epistemic Updates on Algebras , 2013, Log. Methods Comput. Sci..

[56]  Minghui Ma,et al.  Sequent Calculi for Varieties of Topological Quasi-Boolean Algebras , 2018, IJCSR.

[57]  Ming-Wen Shao,et al.  Connections between two-universe rough sets and formal concepts , 2018, Int. J. Mach. Learn. Cybern..

[58]  S. D. Comer,et al.  Perfect extensions of regular double Stone algebras , 1995 .

[59]  T. Iwiński.  Algebraic approach to rough sets , 1987 .

[60]  Willem Conradie,et al.  Constructive Canonicity of Inductive Inequalities , 2016, Log. Methods Comput. Sci..

[61]  Willem Conradie,et al.  Algebraic modal correspondence: Sahlqvist and beyond , 2016, J. Log. Algebraic Methods Program..

[62]  Ewa Orlowska,et al.  Introduction: What You Always Wanted to Know about Rough Sets , 1998 .

[63]  Arthur P. Dempster,et al.  A Generalization of Bayesian Inference , 1968, Classic Works of the Dempster-Shafer Theory of Belief Functions.

[64]  Giacomo Negro,et al.  "Actual" and Perceptual Effects of Category Spanning , 2013, Organ. Sci..

[65]  Yiyu Yao,et al.  Interpretation of Belief Functions in The Theory of Rough Sets , 1998, Inf. Sci..

[66]  N. Wijnberg,et al.  Product proliferation, complexity, and deterrence to imitation in differentiated‐product oligopolies , 2019, Strategic Management Journal.

[67]  Alessandra Palmigiano,et al.  Proper Multi-Type Display Calculi for Rough Algebras , 2018, LSFA.

[68]  Dimiter Vakarelov,et al.  A Modal Characterization of Indiscernibility and Similarity Relations in Pawlak's Information Systems , 2005, RSFDGrC.

[69]  Fei Liang,et al.  Algebraic proof theory for LE-logics , 2018 .

[70]  Mohua Banerjee,et al.  Rough Sets Through Algebraic Logic , 1996, Fundam. Informaticae.

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

[72]  David Ripley,et al.  Tolerant, Classical, Strict , 2010, Journal of Philosophical Logic.

[73]  Willem Conradie,et al.  Constructive Canonicity for Lattice-Based Fixed Point Logics , 2016, WoLLIC.

[74]  B. K. Tripathy,et al.  A Framework for Intelligent Medical Diagnosis using Rough Set with Formal Concept Analysis , 2011, ArXiv.

[75]  Andrzej Skowron.  The evidence theory and decision tables , 1989, Bull. EATCS.