## Approximate Reasoning by Parts

Contents

1 On Concepts. Aristotelian and Set–Theoretic Approaches . . . 1

1.1 An Aristotelian View on Concepts . . . 1

1.2 From Local to Global: Set Theory . . . 6

1.2.1 Naive Set Theory . . . 7

1.2.2 Algebra of Sets . . . 7

1.2.3 A Formal Approach . . . 12

1.3 Relations and Functions. . . 15

1.3.1 Algebra of Relations. . . 16

1.4 Ordering Relations . . . 18

1.5 Lattices and Boolean Algebras . . . 21

1.6 Infinite Sets . . . 22

1.7 Well–Ordered Sets. . . 23

1.8 Finite versus Infinite Sets . . . 26

1.9 Equipotency . . . 28

1.10 Countable Sets . . . 30

1.11 Filters and Ideals. . . 31

1.12 Equivalence Relations . . . 33

1.13 Tolerance Relations. . . 35

1.14 A Deeper Insight into Lattices and Algebras . . . 37

References . . . 42

2 Topology of Concepts . . . 45

2.1 Metric Spaces . . . 45

2.2 Products of Metric Spaces . . . 50

2.3 Compact Metric Spaces . . . 51

2.4 Complete Metric Spaces . . . 53

2.5 General Topological Spaces . . . 56

2.6 Regular Open and Regular Closed Sets . . . 58

2.7 Compactness in General Spaces . . . 61

2.8 Continuity . . . 64

2.9 Topologies on Subsets. . . 66

2.10 Quotient Spaces . . . 67

2.11 Hyperspaces . . . 68

2.11.1 Topologies on Closed Sets . . . 68

2.12 Cech Topologies . . . 73

References . . . 77

3 Reasoning Patterns of Deductive Reasoning . . . 79

3.1 The Nature of Exact Reasoning . . . 79

3.2 Propositional Calculus . . . 80

3.3 Many–Valued Calculi: 3–Valued Logic of .Lukasiewicz . . . 88

3.4 Many–Valued Calculi: n–Valued Logic . . . 96

3.5 Many–Valued Calculi: [0,1]–Valued Logics . . . 99

3.5.1 MV–Algebras. . . 114

3.6 Many–Valued Calculi: Logics of Residual Implications . . . 116

3.7 Automated Reasoning . . . 119

3.8 Predicate Logic . . . 121

3.9 Modal Logics . . . 133

3.9.1 Modal Logic K . . . 133

3.9.2 Modal Logic T . . . 137

3.9.3 Modal Logic S4 . . . 138

3.9.4 Modal Logic S5 . . . 138

References . . . 141

4 Reductive Reasoning Rough and Fuzzy Sets as Frameworks . . . 145

4.1 Rough Set Approach Main Lines . . . 145

4.2 Decision Systems . . . 148

4.3 Decision Rules . . . 150

4.4 Dependencies . . . 152

4.5 Topology of Rough Sets . . . 153

4.6 A Rough Set Reasoning Scheme: The Approximate Collage Theorem . . . 156

4.7 A Rough Set Scheme for Reasoning about Knowledge . . . 158

4.8 Fuzzy Set Approach: Main Lines . . . 161

4.9 Residual Implications . . . 168

4.10 Topological Properties of Residual Implications . . . 169

4.11 Equivalence and Similarity in Fuzzy Universe . . . 175

4.12 Inductive Reasoning: Fuzzy Decision Rules . . . 181

4.13 On the Nature of Reductive Reasoning . . . 183

References . . . 187

5 Mereology . . . 191

5.1 Mereology: The Theory of Lesniewski . . . 191

5.2 A Modern Structural Analysis of Mereology . . . 199

5.3 Mereotopology . . . 201

5.4 Timed Mereology . . . 203

5.5 Spatio–temporal Reasoning: Cells . . . 207

5.6 Mereology Based on Connection. . . 209

5.7 Classes in Connection Mereology . . . 215

5.8 C–Quasi–Boolean Algebra . . . 216

5.9 C–Mereotopology . . . 219

5.10 Spatial Reasoning: Mereological Calculi . . . 220

5.10.1 On Region Connection Calculus . . . 223

References . . . 225

6 Rough Mereology . . . 229

6.1 Rough Inclusions . . . 230

6.2 Rough Inclusions: Residual Models . . . 231

6.3 Rough Inclusions: Archimedean Models . . . 234

6.4 Rough Inclusions: Set Models . . . 236

6.5 Rough Inclusions: Geometric Models . . . 236

6.6 Rough Inclusions: Information Models . . . 237

6.7 Rough Inclusions: Metric Models . . . 240

6.8 Rough Inclusions: A 3–Valued Rough Inclusion on Finite Sets . . . 240

6.9 Symmetrization of Rough Inclusions . . . 241

6.10 Mereogeometry . . . 241

6.11 Rough Mereotopology . . . 246

6.11.1 The Case of Transitive and Symmetric Rough Inclusions . . . 246

6.11.2 The Case of Transitive Non–symmetric Rough Inclusions . . . 250

6.12 Connections from Rough Inclusions . . . 252

6.12.1 The Case of Transitive and Symmetric Rough Inclusions . . . 252

6.12.2 The Case of Symmetric Non–transitive Rough Inclusions and the General Case . . . 252

6.13 Rough Inclusions as Many–Valued Fuzzy Equivalences . . . 254

References . . . 256

7 Reasoning with Rough Inclusions . . . 259

7.1 On Granular Reasoning . . . 259

7.2 On Methods for Granulation of Knowledge . . . 261

7.2.1 Granules from Binary Relations. . . 261

7.2.2 Granules in Information Systems from Indiscernibility . . . 262

7.2.3 Granules from Generalized Descriptors . . . 263

7.3 Granules from Rough Inclusions . . . 263

7.4 Rough Inclusions on Granules. . . 264

7.5 General Properties of Rough Mereological Granules . . . 265

7.6 Reasoning by Granular Rough Mereological Logics . . . 266

7.7 A Logic for Information Systems . . . 269

7.7.1 Relations to Many–Valued Logics . . . 271

7.8 A Graded Notion of Truth. . . 272

7.9 Dependencies and Decision Rules . . . 277

7.10 An Application to Calculus of Perceptions . . . 278

7.11 Modal Aspects of Rough Mereological Logics . . . 280

7.11.1 A Modal Logic with Ingredient Accessibility . . . 281

7.11.2 Modal Logic of Rough Set Approximations . . . 282

7.12 Reasoning in Multi–agent and Distributed Systems . . . 283

7.13 Reasoning in Cognitive Schemes. . . 288

7.13.1 Rough Mereological Perceptron . . . 290

References . . . 292

8 Reasoning by Rough Mereology in Problems of Behavioral Robotics . . . 297

8.1 Planning of Robot Motion. . . 297

8.2 Potential Fields from Rough Inclusions . . . 300

8.3 Planning for Teams of Robots . . . 303

8.4 Rough Mereological Approach to Robot Formations . . . 308

References . . . 315

9 Rough Mereological Calculus of Granules . . . 319

9.1 On Decision Rules . . . 319

9.2 The Idea of Granular Rough Mereological Classifiers . . . 322

9.3 Classification by Granules of Training Objects . . . 324

9.4 A Treatment of Missing Values. . . 326

9.5 Granular Rough Mereological Classifiers Using Residuals . . 329

9.6 Granular Rough Mereological Classifiers with Modified

Voting Parameters . . . 331

References . . . 333

Author Index . . . 335

Term Index . . . 341

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