A representation of continuous domains via relationally approximable concepts in a generalized framework of formal concept analysis

Abstract In this paper, in order to realize a representation of continuous domains, the notions of relationally consistent F-augmented contexts and relationally approximable concepts are introduced, which provides a generalized framework of formal concept analysis. We also introduce the notion of F-approximable mappings which serves as the morphism between relationally consistent F-augmented contexts. The main result is that the category of relationally consistent F-augmented contexts is equivalent to that of continuous domains with Scott continuous maps being morphisms. This provides a new approach to concretely representing continuous domains and demonstrates the efficiency of formal concept analysis in representing some important partially ordered structures.

[1]  Guo-Qiang Zhang.  A Representation of SFP , 1994, Inf. Comput..

[2]  Gerd Stumme,et al.  Hierarchies of conceptual scales , 1999 .

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

[4]  Dieter Spreen,et al.  Information systems revisited - the general continuous case , 2008, Theor. Comput. Sci..

[5]  Qing Wan,et al.  Approximate concepts acquisition based on formal contexts , 2015, Knowl. Based Syst..

[6]  Dana S. Scott,et al.  Some Domain Theory and Denotational Semantics in Coq , 2009, TPHOLs.

[7]  Gaihua Fu,et al.  FCA based ontology development for data integration , 2016, Inf. Process. Manag..

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

[9]  Guo-Qiang Zhang.  dI-Domains as Prime Information Systems , 1992, Inf. Comput..

[10]  Jinhai Li,et al.  Knowledge representation using interval-valued fuzzy formal concept lattice , 2016, Soft Comput..

[11]  Ming-Wen Shao,et al.  Attribute reduction in generalized one-sided formal contexts , 2017, Inf. Sci..

[12]  Michael Barr,et al.  Category theory for computing science , 1995, Prentice Hall International Series in Computer Science.

[13]  Bernhard Ganter,et al.  Attribute Exploration with Background Knowledge , 1999, Theor. Comput. Sci..

[14]  Vilém Vychodil,et al.  Formal Concept Analysis With Background Knowledge: Attribute Priorities , 2009, IEEE Transactions on Systems, Man, and Cybernetics, Part C (Applications and Reviews).

[15]  Maokang Luo,et al.  Rough concept lattices and domains , 2009, Ann. Pure Appl. Log..

[16]  K. Hofmann,et al.  Continuous Lattices and Domains , 2003 .

[17]  Guo-Qiang Zhang,et al.  APPROXIMABLE CONCEPTS, CHU SPACES, AND INFORMATION SYSTEMS , 2006 .

[18]  M. Stone.  Topological representations of distributive lattices and Brouwerian logics , 1938 .

[19]  R. Hoofman.  Continuous Information Systems , 1993, Inf. Comput..

[20]  Alasdair Urquhart,et al.  A topological representation theory for lattices , 1978 .

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

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

[23]  Radim Belohlávek,et al.  Ordinally equivalent data: A measurement-theoretic look at formal concept analysis of fuzzy attributes , 2013, Int. J. Approx. Reason..

[24]  Gerd Stumme,et al.  Formal Concept Analysis: foundations and applications , 2005 .

[25]  Qingguo Li,et al.  Representation of algebraic domains by formal association rule systems , 2017, Math. Struct. Comput. Sci..

[26]  Gerd Hartung,et al.  A topological representation of lattices , 1992 .

[27]  Pascal Hitzler,et al.  A Categorical View on Algebraic Lattices in Formal Concept Analysis , 2004, Fundam. Informaticae.

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

[29]  Hilary A. Priestley,et al.  Representation of Distributive Lattices by means of ordered Stone Spaces , 1970 .

[30]  Stanislav Krajci,et al.  On stability of fuzzy formal concepts over randomized one-sided formal context , 2018, Fuzzy Sets Syst..

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

[32]  Andreas Hotho,et al.  Conceptual Knowledge Processing with Formal Concept Analysis and Ontologies , 2004, ICFCA.

[33]  Radim Belohlávek,et al.  Concept lattices and order in fuzzy logic , 2004, Ann. Pure Appl. Log..