Table of Contents
Introduction. Qualitative Reasoning xvii
Chapter 1. Allen’s Calculus 1
1.1. Introduction 1
1.2. Allen’s interval relations 6
1.3. Constraint networks 8
1.4. Constraint propagation 17
1.5. Consistency tests 26
Chapter 2. Polynomial Subclasses of Allen’s Algebra 29
2.1. “Show me a tractable relation!” 29
2.2. Subclasses of Allen’s algebra 30
2.3. Maximal tractable subclasses of Allen’s algebra 52
2.4. Using polynomial subclasses 57
2.5. Models of Allen’s language 60
2.6. Historical note 61
Chapter 3. Generalized Intervals 63
3.1. “When they built the bridge . “ 63
3.2. Entities and relations 65
3.3. The lattice of basic (p, q)-relations 68
3.4. Regions associated with basic (p, q)-relations 69
3.5. Inversion and composition 73
3.6. Subclasses of relations: convex and pre-convex relations 79
3.7. Constraint networks 82
3.8. Tractability of strongly pre-convex relations 83
3.9. Conclusions 84
3.10. Historical note 85
Chapter 4. Binary Qualitative Formalisms 87
4.1. “Night driving” 87
4.2. Directed points in dimension 1 92
4.3. Directed intervals 97
4.4. The OPRA direction calculi 99
4.5. Dipole calculi 100
4.6. The Cardinal direction calculus 101
4.7. The Rectangle calculus 104
4.8. The n-point calculus 106
4.9. The n-block calculus 108
4.10. Cardinal directions between regions 109
4.11. The INDU calculus 123
4.12. The 2n-star calculi 126
4.13. The Cyclic interval calculus 128
4.14. The RCC–8 formalism 131
4.15. A discrete RCC theory 137
Chapter 5. Qualitative Formalisms of Arity Greater than 2 145
5.1. “The sushi bar” 145
5.2. Ternary spatial and temporal formalisms 146
5.3. Alignment relations between regions 155
5.4. Conclusions 158
Chapter 6. Quantitative Formalisms, Hybrids, and Granularity 159
6.1. “Did John meet Fred this morning?”159
6.2. TCSP metric networks 160
6.3. Hybrid networks 164
6.4. Meiri’s formalism 168
6.5. Disjunctive linear relations (DLR) 174
6.6. Generalized temporal networks 175
6.7. Networks with granularity 179
Chapter 7. Fuzzy Reasoning 187
7.1. “Picasso’s Blue period” 187
7.2. Fuzzy relations between classical intervals 188
7.3. Events and fuzzy intervals 195
7.4. Fuzzy spatial reasoning: a fuzzy RCC 208
7.5. Historical note 222
Chapter 8. The Geometrical Approach and Conceptual Spaces 223
8.1. “What color is the chameleon?” 223
8.2. Qualitative semantics 224
8.3. Why introduce topology and geometry? 225
8.4. Conceptual spaces 226
8.5. Polynomial relations of INDU 237
8.6. Historical note 258
Chapter 9. Weak Representations 259
9.1. “Find the hidden similarity” 259
9.2. Weak representations 261
9.3. Classifying the weak representations of An 275
9.4. Extension to the calculi based on linear orders 283
9.5. Weak representations and configurations 290
9.6. Historical note 304
Chapter 10. Models of RCC−8 305
10.1. “Disks in the plane” 305
10.2. Models of a composition table 307
10.3. The RCC theory and its models 312
10.4. Extensional entries of the composition table 319
10.5. The generalized RCC theory 329
10.6. A countable connection algebra 337
10.7. Conclusions 341
Chapter 11. A Categorical Approach of Qualitative Reasoning 343
11.1. “Waiting in line” 343
11.2. A general construction of qualitative formalisms 346
11.3. Examples of partition schemes 349
11.4. Algebras associated with qualitative formalisms 350
11.5. Partition schemes and weak representations 352
11.6. A general definition of qualitative formalisms 353
11.7. Interpretating consistency 355
11.8. The category of weak representations 357
11.9. Conclusions 360
Chapter 12. Complexity of Constraint Languages 363
12.1. “Sudoku puzzles” 363
12.2. Structure of the chapter 365
12.3. Constraint languages 366
12.4. An algebraic approach of complexity 367
12.5. CSPs and morphisms of relational structures 368
12.6. Clones of operations 373
12.7. From local consistency to global consistency 375
12.8. The infinite case 376
12.9. Disjunctive constraints and refinements 382
12.10. Refinements and independence 389
12.11. Historical note 390
Chapter 13. Spatial Reasoning and Modal Logic 391
13.1. “The blind men and the elephant” 391
13.2. Space and modal logics 393
13.3. The modal logic S4 393
13.4. Topological models 396
13.5. Translating the RCC−8 predicates 408
13.6. An alternative modal translation of RCC−8 409
13.7. Generalized frames 410
13.8. Complexity 411
13.9. Complements 412
Chapter 14. Applications and Software Tools 413
14.1. Applications 413
14.2. Software tools 416
Chapter 15. Conclusion and Prospects 423
15.1. Introduction 423
15.2. Combining qualitative formalisms 423
15.3. Spatio-temporal reasoning 426
15.4. Alternatives to qualitative reasoning 430
15.5. To conclude — for good 434
Appendix A. Elements of Topology 435
A.1. Topological spaces 435
A.2. Metric spaces 445
A.3. Connectedness and convexity 447
Appendix B. Elements of Universal Algebra 451
B.1. Abstract algebras 451
B.2. Boolean algebras 452
B.3. Binary relations and relation algebras 454
B.4. Basic elements of the language of categories 457
Appendix C. Disjunctive Linear Relations 463
C.1. DLRs: definitions and satisfiability 463
C.2. Linear programming 464
C.3. Complexity of the satisfiability problem 466
Bibliography 471
Index 501
