Approximate Reasoning by Parts: An Introduction to Rough Mereology (2011) .. by Lech Polkowski
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