A geometry of approximation. Rough set theory: logic, algebra and topology of conceptual patterns

Description

'A Geometry of Approximation' addresses Rough Set Theory, a field of interdisciplinary research first proposed by Zdzislaw Pawlak in 1982, and focuses mainly on its logic-algebraic interpretation. The theory is embedded in a broader perspective that includes logical and mathematical methodologies pertaining to the theory, as well as related epistemological issues. Any mathematical technique that is introduced in the book is preceded by logical and epistemological explanations. Intuitive justifications are also provided, insofar as possible, so that the general perspective is not lost. Such an approach endows the present treatise with a unique character. Due to this uniqueness in the treatment of the subject, the book will be useful to researchers, graduate and pre-graduate students from various disciplines, such as computer science, mathematics and philosophy. It features an impressive number of examples supported by about 40 tables and 230 figures. The comprehensive index of concepts turns the book into a sort of encyclopaedia for researchers from a number of fields. 'A Geometry of Approximation' links many areas of academic pursuit without losing track of its focal point, Rough Sets. Contents......Page 14 Preface......Page 6 List of Figures......Page 26 Notation......Page 29 Abbreviations......Page 31 1. Perception and Concepts: A Phenomenological Approach......Page 32 1.1 Monological Approach and Dialogical Approach......Page 35 2. Monological Approach to Perception and Concepts......Page 37 3.1 Semantics vs Syntax......Page 49 3.2 Information and Interpretation: Correspondence Theory of Truth vs Pragmatism......Page 55 3.2.1 Meaning-conditions vs Truth-Conditions......Page 56 3.2.2 Logic, Meaning and Rough Set Theory......Page 61 4. The Logico-Algebraic Interpretation of Rough Set Systems......Page 63 5.1 Types, Tokens and Abstract Points......Page 67 5.3 Abstract Points and Rough Sets......Page 75 6. Rough Sets and Logic......Page 76 7. Concluding Remarks......Page 80 I. A Mathematics of Perception......Page 82 1.1 Foreword......Page 83 1.2 Formal Relationships Between "Noumena" and "Phenomena"......Page 86 1.3 Functional P-Systems and Conceptualisation......Page 94 1.4 Categorizing Through Relational P-Systems......Page 100 2.1 Concrete and Formal Observation Spaces......Page 123 2.2 The Basic Phenomenological Constructors......Page 130 2.3 Formal Operators on Points and on Observables......Page 138 3.1 Information, Concepts and Formal Operators......Page 152 3.2 Comparing Perception Systems......Page 164 3.3 Higher Level Operators......Page 169 3.4 Transforming Perception Systems......Page 176 3.5 Topological Approximation Operators......Page 179 3.6 Topological Approximation Systems......Page 182 4.1 Frame – Approximation......Page 185 4.2 Frame – Classification......Page 186 4.3 Frame – Categorizing Through Pointless Topology......Page 188 4.4 Frame – Observable Properties......Page 191 4.5 Frame – Finite Observations: The Binary Machine Example......Page 194 4.6 Frame – Quanta of Information......Page 198 4.7 Frame – Information Systems......Page 204 4.8 Frame – Dichotomic, Complementary and Functional Systems......Page 207 4.9 Frame – Concept Lattices......Page 211 4.10 Frame – Neighborhood Systems......Page 220 4.11 Frame – Basic Pairs and Point-Free Topology......Page 221 4.12 Frame – Chu Spaces......Page 222 4.13 Frame – Intuitionism, Modalities and Relational Semantics......Page 223 4.14 Frame – Galois Adjunctions......Page 229 4.15 Frame – Categories and Adjoint Functors......Page 236 4.16 Solutions......Page 239 II. The Logico-Algebraic Theory of Rough Sets......Page 245 5.1 Foreword......Page 246 5.2 Rough Set Systems and Three-Valued Logics......Page 249 5.3 Exact and Inexact Local Behaviours in Rough Set Systems......Page 251 5.4 Representing Rough Sets......Page 254 5.5 Rough Set Systems, Local Validity, and Logico-Algebraic Structures......Page 258 6. Basic Logico-Algebraic Structures......Page 269 6.1 Heyting Algebras......Page 270 6.2 Nelson Algebras......Page 274 6.3 N-Valued Łukasiewicz Algebras......Page 279 6.4 Chain-Based Lattices......Page 280 6.5 Relationships, Analogies and Differences Between Structures......Page 283 7.1 Representing Rough Sets......Page 287 7.2 Some Duality of Distributive Lattices......Page 292 7.3 Grothendieck Topologies......Page 295 7.4 Lawvere-Tierney Operators and Rough Set Systems......Page 299 8.1 Approximation Operators......Page 313 8.2 Adjointness, Approximations and the Center of a Rough Set System......Page 314 8.3 Multi-Valued Logics: A Knowledge-Oriented Interpretation......Page 318 9.1 Truth-Oriented and Knowledge-Oriented Approaches in Logic......Page 331 9.2 Understanding the Knowledge-Oriented Point of View......Page 332 9.3 Some Problems Arising From the Knowledge-Oriented Point of View......Page 335 9.4 A "Mixed-Radix" Attitude in Logic......Page 338 9.5 A Maximal Intermediate Constructive Logic......Page 342 9.6 Mixed-Radix Information Systems......Page 345 9.7 Conclusions......Page 354 10.1 Frame – Rough Set Systems and Chain-Based Lattices......Page 356 10.2 Frame – Rough Set Systems as Regular Double Stone Algebras......Page 358 10.3 Frame – Information-Oriented Duality Theorems......Page 359 10.4 Frame – Representation of Three-Valued Łukasiewicz Algebras as Rough Set System......Page 371 10.5 Frame – Proof of the Facts Stated in Window 7.1......Page 374 10.6 Frame – Proof of Proposition 8.3.1......Page 375 10.7 Frame – Grothendieck Topologies and Lawvere-Tierney Operators......Page 377 10.8 Frame – Representation of Rough Sets......Page 378 10.9 Frame – Rough Sets and Non Classical Logico-Algebraic Systems......Page 379 10.10 Frame – Representation Theorems and Decomposition of Distributive Lattices......Page 382 10.11 Frame – Representation of Logical Values by Ordered Pairs......Page 386 10.12 Frame – Negation......Page 387 10.13 Frame – Intuitionistic Logic: Natural Deduction System INT......Page 398 10.15 Frame – Nelson Logic: Natural Deduction System CLSN......Page 399 10.16 Frame – The System ε[sub(0)]......Page 401 10.17 Frame – The Logic F[sub(CL)]......Page 407 10.18 Frame – Medvedev's Logic of Finite Problems......Page 412 10.19 Frame – Atomic Decidability and Non-Standard Systems......Page 413 10.20 Frame – An Applications of the Algebraic Approach to Partial Information Systems......Page 416 10.21 Frame – Logical Operations in a Pure Algebraic Setting......Page 428 10.22 Solutions......Page 430 III. The Modal Logic of Rough Sets......Page 435 11.1 Foreword......Page 436 11.2 Modalities and Assertions......Page 438 11.3 Internal Modalities vs External Modalities......Page 440 11.4 Knowledge and Information......Page 444 11.5 Knowledge and Modal Systems......Page 448 12.1 Modal Systems and Binary Relations......Page 461 12.2 From Loosely Structured Spaces to Structured Spaces: A Variety of Modal Properties......Page 474 12.3 Relations, Pre-Topologies and Topologies......Page 478 12.4 Pre-Topological Spaces......Page 479 12.5 Towards Topology 1......Page 498 12.6 Towards Topology 2......Page 503 12.7 Pre-Topological Spaces and Binary Relations......Page 513 12.8 Topological Spaces and Binary Relations......Page 530 13.1 Topological Boolean Algebras......Page 542 13.2 Monadic Topological Boolean Algebras......Page 543 14.1 Introduction......Page 549 14.2 From Syntax to Semantics......Page 550 14.3 Rough Algebras......Page 558 14.4 The Systems L[sub(1)],L[sub(2)]......Page 562 14.5 Algebraic Interpretation and Modal Interpretation of Rough Set Systems......Page 572 15.1 Frame – Proof of the Duality Between L[sub(R)](X) = ∪{Z : R(Z) ⊆ X} and M[sub(R)](X) = ∩{–Z : R(Z) ⊆ – X}......Page 574 15.2 Frame – Relational Properties and Logical Characteristics......Page 575 15.3 Frame – Proof of Proposition 12.7.10......Page 576 15.5 Frame – Alternative Proofs of Corollary 12.8.2.(1)......Page 577 15.7 Frame – Transforming a Pre-Topological Spaceof Type [ ][sub(S)] into a Topological Space......Page 578 15.8 Frame – Modal Interpretations of Approximation Spaces and Rough Sets......Page 580 15.9 Frame – Kripke-Joyal Models......Page 581 15.10 Frame – Quantum Logic and Internal Modalities......Page 582 15.11 Frame – Persistence of Modalised Formulas......Page 584 15.12 Frame – Coherence Between Information and Knowledge......Page 587 15.13 Frame – Neighborhood Systems......Page 591 15.14 Frame – Pre-Topologies and Intuitionistic Formal Spaces......Page 599 15.15 Frame – Modal Structures and Pre-Topological Spaces......Page 633 15.16 Frame – Duality of Operations and Algebraic Structures......Page 639 15.17 Frame – Computing Dependency Relations in a Fragment of Intuitionistic Logic......Page 640 15.18 Frame – Approximation, Formal Concepts, Modalities and Relation Algebras......Page 643 15.19 Frame – Relational Proof Theory......Page 664 15.20 Frame – Some History of the Algebraic Concepts used in this Part......Page 669 15.21 Solutions......Page 670 16.1 A Mathematical Toolkit: Orders......Page 683 16.2 A Mathematical Toolkit: Functions......Page 684 16.3 A Mathematical Toolkit: Lattices......Page 686 16.4 A Mathematical Toolkit: Topology......Page 689 16.5 A Mathematical Toolkit: Relations......Page 691 Bibliography......Page 698 Index......Page 748 A......Page 752 C......Page 755 E......Page 756 F......Page 757 I......Page 759 K......Page 760 L......Page 761 M......Page 762 N......Page 764 O......Page 765 P......Page 766 R......Page 767 S......Page 769 T......Page 770 Z......Page 771