Handbook of Spatial Logics

Description

A spatial logic is a formal language interpreted over any class of structures featuring geometrical entities and relations, broadly construed. In the past decade, spatial logics have attracted much attention in response to developments in such diverse fields as Artificial Intelligence, Database Theory, Physics, and Philosophy. The aim of this handbook is to create, for the first time, a systematic account of the field of spatial logic. The book comprises a general introduction, followed by fourteen chapters by invited authors. Each chapter provides a self-contained overview of its topic, describing the principal results obtained to date, explaining the methods used to obtain them, and listing the most important open problems. Jointly, these contributions constitute a comprehensive survey of this rapidly expanding subject. Preface......Page 6 Contents......Page 8 1 – What is Spatial Logic? Marco Aiello, Ian Pratt-Hartmann, Johan van Benthem......Page 24 1. Introduction......Page 36 2. Mereotopologies......Page 37 3. Defining topological relations......Page 49 4. Expressiveness of first-order languages in plane mereotopologies......Page 61 5. Axiomatization......Page 81 6. Spatial mereotopology......Page 92 7. Model theory......Page 105 8. Philosophical considerations......Page 114 1. Introduction......Page 122 2. Preliminary definitions and notation......Page 127 3. Contact relations......Page 142 4. Boolean contact algebras......Page 145 5. Other theories of topological relations......Page 157 6. Reasoning about topological relations......Page 160 7. Conclusion......Page 172 1. Introduction......Page 184 2. Constraint-based methods for qualitative spatial representation and reasoning......Page 186 3. Spatial constraint calculi......Page 192 4. Computational complexity......Page 200 5. Identifying tractable subsets of spatial CSPs......Page 207 6. Practical efficiency of reasoning methods......Page 213 7. Combination of spatial calculi......Page 220 8. Conclusions......Page 230 1. Modal logics and spatial structures......Page 240 2. Modal logic and topology: basic results......Page 254 3. Modal logic and topology: further directions......Page 279 4. Modal logic and geometry......Page 299 5. Modal logic and linear algebra......Page 308 6. Conclusions......Page 314 1. Introduction......Page 322 2. Perspectives......Page 323 3. The original topological interpretation of modal logic: Tarski and McKinsey’s Theorem......Page 324 4. Topologic......Page 331 5. A logical system: the subset space axioms......Page 335 6. Further examples......Page 339 7. Completeness of the subset space axioms......Page 342 8. Decidability of the subset space logic......Page 345 9. Heinemann’s extensions to topologic......Page 348 10. Common knowledge in topological settings......Page 350 11. The topology of belief......Page 352 12. Other work connected to this chapter......Page 362 1. Introduction and historical overview......Page 366 2. Preliminaries......Page 371 3. Structures and theories of parallelism......Page 375 4. Structures and theories of orthogonality......Page 378 5. Two-sorted point-line incidence spaces......Page 381 6. Coordinatization......Page 390 7. On the first-order theories of affine and projective spaces......Page 403 8. Betweenness structures and ordered affine planes......Page 409 9. Rich languages and structures for elementary geometry......Page 417 10. Modal logic and spatial logic......Page 423 11. Point-based spatial logics......Page 427 12. Line-based spatial logics......Page 429 13. Tip spatial logics......Page 434 14. Point-line spatial logics......Page 439 1. Introduction......Page 452 2. Opens as propositions......Page 454 3. Predicate geometric logic......Page 468 4. Categorical logic......Page 480 5. Sheaves as predicates......Page 497 6. Summary of toposes......Page 511 7. Other directions......Page 512 8. Conclusions......Page 515 1. Introduction......Page 520 2. Static and changing spatial models......Page 524 3. Spatial logics......Page 529 4. Temporal logics......Page 550 5. Combination principles......Page 554 6. Combining topo-logics with temporal logics......Page 556 7. Combining distance logics with temporal logics......Page 566 8. Logics for dynamical systems......Page 569 9. Related ‘temporalised’ formalisms......Page 580 1. Introduction......Page 588 2. Basic definitions......Page 592 3. Recurrence and the DTL of measure-preserving continuous functions on the closed unit interval......Page 596 4. Purely topological and purely temporal fragments of DTLs......Page 599 5. S4 is topologically complete for (0, 1)......Page 602 6. The logic of homeomorphisms......Page 609 7. The logic of continuous functions......Page 615 8. Conclusion......Page 627 1. Introduction......Page 630 2. Special relativity......Page 631 3. General relativistic space-time......Page 683 4. Black holes, wormholes, timewarp. Distinguished general relativistic space-times......Page 706 5. Connections with the literature......Page 728 1. Introduction......Page 736 2. Preliminaries; correspondence principle......Page 740 3. Čech closure spaces......Page 746 4. Closure systems......Page 748 5. Extended examples......Page 753 6. (Boundary and) dimension......Page 766 7. Discrete region geometry......Page 772 8. Matroids......Page 784 9. Spherical oriented matroids......Page 791 10. Flat oriented matroids......Page 806 11. Algebraic spatial models......Page 810 1. From the relational database model to the constraint database model......Page 822 2. Constraint data models and query languages......Page 828 3. Introduction to real algebraic geometry......Page 835 4. Query evaluation through quantifier elimination......Page 845 5. Expressiveness results......Page 852 6. Extensions of logical query languages......Page 864 1. Introduction......Page 880 2. Algebra......Page 899 3. Related approaches......Page 920 4. Logics......Page 941 5. Conclusion......Page 959 15 – Spatial Reasoning and Ontology: Parts, Wholes, and Locations, Achille C. Varzi......Page 968 1. Philosophical issues in mereology......Page 970 2. Philosophical issues in topology......Page 998 3. Location theories......Page 1035