An Investigation About Rough Set Theory: Some Foundational and Mathematical Aspects

The approach to rough set theory is investigated from the foundational point of view, starting from the formal analysis of the mathematical structures generated by equivalence relations (the standard Pawlak approach to complete information systems) and then by tolerance or similarity relations (the approach to roughness by incomplete information systems). Making use of these approaches the “minimal” meta-theoretical principles which allow one to treat approximations in rough set theory are discussed and formulated in explicit way, in particular the conditions which can assure what can be rightly considered as lower and upper approximations of some approximable set. The hierarchy of approximation spaces generated in this way on a universe is then discussed.

[1]  Z. Pawlak.  Rough Sets: Theoretical Aspects of Reasoning about Data , 1991 .

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

[3]  C. J. Everett.  Closure operators and Galois theory in lattices , 1944 .

[4]  A. Tarski.  Fundamentale Begriffe der Methodologie der deduktiven Wissenschaften. I , 1930 .

[5]  Francesc Rosselló,et al.  On the coverings by tolerance classes , 2004, Inf. Sci..

[6]  R. Carnap.  Der logische Aufbau der Welt , 1998 .

[7]  W. Zakowski.  APPROXIMATIONS IN THE SPACE (U,π) , 1983 .

[8]  Zdzis?aw Pawlak,et al.  Rough sets , 2005, International Journal of Computer & Information Sciences.

[9]  Gianpiero Cattaneo,et al.  Abstract Approximation Spaces for Rough Theories , 2008 .

[10]  Jerzy W. Grzymala-Busse,et al.  Rough Sets , 1995, Commun. ACM.

[11]  Yiyu Yao,et al.  Relational Interpretations of Neigborhood Operators and Rough Set Approximation Operators , 1998, Inf. Sci..

[12]  Jerzy Stefanowski,et al.  Rough classification in incomplete information systems , 1989 .

[13]  Roman Slowinski,et al.  Handling Various Types of Uncertainty in the Rough Set Approach , 1993, RSKD.

[14]  Zdzisław Pawlak,et al.  Rough sets: a new approach to vagueness , 1992 .

[15]  Yiyu Yao,et al.  On Generalizing Rough Set Theory , 2003, RSFDGrC.

[16]  Alexis Tsoukiàs,et al.  On the Extension of Rough Sets under Incomplete Information , 1999, RSFDGrC.

[17]  Roman Slowinski,et al.  Rough-Set Reasoning about Uncertain Data , 1996, Fundam. Informaticae.

[18]  Oystein Ore,et al.  Combinations of Closure Relations , 1943 .

[19]  Marzena Kryszkiewicz,et al.  Generation of Rules from Incomplete Information Systems , 1997, PKDD.

[20]  Zdzislaw Pawlak,et al.  Information systems theoretical foundations , 1981, Inf. Syst..

[21]  Jerzy W. Grzymala-Busse,et al.  On the Unknown Attribute Values in Learning from Examples , 1991, ISMIS.

[22]  M. de Rijke,et al.  Modal Logic , 2001, Cambridge Tracts in Theoretical Computer Science.

[23]  H. Poincaré.  La science et l'hypothèse , 1968 .

[24]  C. Kuratowski.  Sur l'opération Ā de l'Analysis Situs , 1922 .

[25]  Zdzislaw Pawlak,et al.  Rough sets and decision tables , 1984, Symposium on Computation Theory.

[26]  A. R. Turquette,et al.  Logic, Semantics, Metamathematics , 1957 .

[27]  Z. Pawlak.  Rough set elements , 1998 .

[28]  Zdzisław Pawlak.  On rough relations , 1986 .

[29]  Gianpiero Cattaneo,et al.  A Hierarchical Lattice Closure Approach to Abstract Rough Approximation Spaces , 2008, RSKT.

[30]  Lech Polkowski,et al.  Rough Sets in Knowledge Discovery 2 , 1998 .

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

[32]  A. Skowron,et al.  On topology in information systems , 1988 .

[33]  Kiyoshi Iseki.  On Closure Operation in Lattice Theory , 1951 .

[34]  S. Marcus.  Tolerance rough sets, Čech topologies, learning processes , 1994 .

[35]  Marzena Kryszkiewicz,et al.  Rough Set Approach to Incomplete Information Systems , 1998, Inf. Sci..

[36]  Gianpiero Cattaneo,et al.  Lattices with Interior and Closure Operators and Abstract Approximation Spaces , 2009, Trans. Rough Sets.

[37]  Paul R. Halmos,et al.  The Basic Concepts of Algebraic Logic , 1956 .

[38]  William Zhu,et al.  Topological approaches to covering rough sets , 2007, Inf. Sci..