ISBN: 3790815101
TITLE: Rough sets
AUTHOR: Polkowski
TOC:

Preface vii
Part 1: Rough Sets
1 Rough Set Theory: An Introduction 3
1.1 Knowledge representation 3
1.2 Information systems 5
1.3 Exact sets, rough sets, approximations 6
1.4 Set-algebraic structures 9
1.5 Topological structures 9
1.6 Logical aspects of rough sets 11
1.6.1 Dependencies 11
1.6.2 Modal aspects of rough sets 15
1.6.3 Many-valued logics for rough sets 16
1.7 Decision systems 16
1.8 Knowledge reduction 18
1.8.1 Reducts via boolean reasoning: discernibility approach 19
1.8.2 Reducts in decision systems 20
1.8.3 Rough membership functions 21
1.9 Similarity based techniques 22
1.9.1 General approximation spaces 23
1.9.2 Rough mereology 24
1.9.3 Rough mereology in complex information systems: in search of features in distributed systems 26
1.10 Generalized and approximate reducts 31
1.10.1 Frequency based reducts 31
1.10.2 Local reducts 34
1.10.3 Dynamic reducts 35
1.10.4 Generalized dynamic reducts 35
1.10.5 Genetic and hybrid algorithms in reduct computation 36
1.11 Template techniques 38
1.11.1 Templates 39
1.11.2 Template goodness measures 40
1.11.3 Searching for optimal descriptors 42
1.12 Discretization 42
1.12.1 Value partition via cuts 43
1.12.2 Heuristics 44
1.13 Selected Bibliography on Rough Sets 50
Part 2: Prerequisites
2 The Sentential Calculus 95
2.1 Introduction 95
2.2 Functors 96
2.3 Meaningful expressions 97
2.4 The Sentential Calculus 98
2.5 Exemplary derivations in (L) 101
2.6 Completeness of (L) 104
2.7 The sequent approach 107
2.7.1 Axioms and inference rules for the sequent calculus of the sentential calculus 107
3 Logical Theory of Approximations 121
3.1 Introduction 121
3.2 Figures of syllogisms 121
3.3 Syllogistic as a deductive system 125
3.4 Selected syllogisms 129
4 Many-valued Sentential Calculi 137
4.1 Introduction 137
4.2 A formal development 138
4.3 Consistent sets of meaningful expressions 140
4.4 Completeness 141
4.5 n-valued logics 145
4.6 4-valued logic: Modalities 148
4.7 Modalities 151
5 Propositional Modal Logic 159
5.1 Introduction 159
5.2 The system K 160
5.3 The system T 166
5.4 The system S4 166
5.5 The system S5 167
6 Set Theory 173
6.1 Introduction 173
6.2 Naive set theory 174
6.2.1 Algebra of sets 175
6.3 A formal approach 180
6.4 Relations and functions 183
6.4.1 Algebra of relations 184
6.5 Orderings 186
6.6 Lattices and Boolean algebras 189
6.7 Infinite sets 190
6.8 Well-ordered sets 192
6.9 Finite versus infinite sets 194
6.10 Equipotency 196
6.11 Countable Sets 198
6.12 Filters and ideals 200
6.13 Equivalence, tolerance 201
6.13.1 Tolerance relations 204
7 Topological Structures 213
7.1 Introduction 213
7.2 Metric spaces 213
7.3 Topological Cartesian products 218
7.4 Compactness in metric spaces 220
7.5 Completeness in metric spaces 221
7.6 General topological spaces 225
7.7 Regular sets 227
7.8 Compactness in general spaces 229
7.9 Continuity 233
7.10 Topologies on subsets 235
7.11 Quotient spaces 236
7.12 Hyperspaces 236
7.12.1 Topologies on closed sets 237
7.13 Cech topologies 242
8 Algebraic Structures 251
8.1 Introduction 251
8.2 Lattices 251
8.3 Distributive lattices 254
8.4 Pseudo-complement 256
8.5 Stone lattices 258
8.6 Complement 258
8.7 Boolean algebras 260
8.8 Filters on lattices 261
8.9 Filters on Boolean algebras 264
8.10 Pseudo-Boolean algebras 266
9 Predicate Calculus 273
9.1 Introduction 273
9.2 A formal predicate calculus 274
9.3 The Lindenbaum-Tarski algebra 278
9.4 Completeness 282
9.4.1 Calculus of open expressions 286
9.5 Calculus of unary predicates 286
9.6 Fractional truth values 287
9.7 Intuitionistic propositional logic 289
9.7.1 Gentzen-type formalization of predicate and intuitionistic calculi 291
Part 3: Mathematics of Rough Sets
10 Independence, Approximation 299
10.1 Introduction 299
10.2 Independence 300
10.2.1 Functional dependence 302
10.2.2 An abstract view on independence 306
10.2.3 Dependence spaces 308
10.2.4 Independence 309
10.2.5 Dependence 312
10.2.6 Interpretation in information systems 314
10.3 Classification/approximation spaces 315
10.3.1 Approximation spaces of an information system 319
10.4 Partial dependence 321
11 Topology of Rough Sets 331
11.1 Introduction 331
11.2 Pi_0-rough sets 332
11.3 Metrics on rough sets 334
11.3.1 Some examples 339
11.4 Almost rough sets 341
11.5 Fractals, Approximate Collage 345
11.5.1 Fractals 345
11.5.2 The Approximate Collage Theorem 353
12 Algebra and Logic of Rough Sets 361
12.1 Introduction 361
12.2 Algebraic structures via rough sets 362
12.2.1 Nelson algebras of rough sets 362
12.2.2 Heyting algebras of rough sets 365
12.2.3 Stone algebras of rough sets 367
12.3 Lukasiewicz algebras of rough sets 372
12.3.1 Wajsberg algebras 375
12.3.2 Post algebra representation of rough sets 387
12.4 A logic of indiscernibility 389
12.4.1 The syntax of FD-logic 389
12.4.2 Semantics of FD-logic 391
12.5 Information logics 394
12.5.1 The logic IL 396
12.5.2 A canonical model 399
Part 4: Rough vs. Fuzzy
13 Infinite-valued Logical Calculi 413
13.1 Introduction 413
13.2 Syntax of L_\infty 415
13.3 Semantics of L_\infty 421
13.3.1 Polynomials, polynomial formulae 422
13.4 Fuzzy logics of sentences 427
13.4.1 Basic ingredients of a fuzzy logic 429
13.4.2 The Lukasiewicz residuated lattice 431
13.4.3 Filters on residuated lattices 434
13.4.4 Syntax and semantics of a fuzzy sentential calculus 436
13.4.5 Syntax and semantics at work 439
13.4.6 Completeness of the fuzzy sentential calculus 441
13.4.7 Discrete Lukasiewicz residuated lattices 449
13.4.8 3 - Lukasiewicz algebras vs. Lindenbaum-Tarski algebras of the fuzzy sentential calculus 450
14 From Rough to Fuzzy 465
14.1 Introduction 465
14.2 Triangular norms 466
14.3 Rough fuzzy and fuzzy rough sets 479
14.3.1 Rough fuzzy sets 481
14.3.2 Fuzzy rough sets 483
14.4 Brouwer-Zadeh lattices 491
Bibliography 501
Index 521
List of Symbols 531
END
