Handbook of Spatial Logics

Marco Aiello, Ian Pratt-Hartmann, Johan van Benthem

Springer Science & Business Media, Sep 4, 2007 - Science - 1058 pages

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.

Contents

• 3: 99
• 4: 160
• Computational complexity: 177
• Identifying tractable subsets of spatial CSPs: 184
• Practical efficiency of reasoning methods: 190
• Combination of spatial calculi: 197
• Conclusions: 207
• Modal Logics of Space: 217
• basic results: 231
• further directions: 256
• Modal logic and geometry: 276
• Modal logic and linear algebra: 285
• Conclusions: 291
• Topology and Epistemic Logic: 299
• Perspectives: 300
• Tarski and McKinseys Theorem: 301
• Topologic: 308
• the subset space axioms: 312
• Further examples: 316
• Completeness of the subset space axioms: 319
• Decidability of the subset space logic: 322
• Heinemanns extensions to topologic: 325
• Common knowledge in topological settings: 327
• The topology of belief: 329
• Other work connected to this chapter: 339
• Logical Theories for Fragments of Elementary Geometry 343: 342
• Preliminaries: 348
• Structures and theories of parallelism: 352
• Structures and theories of orthogonality: 355
• Twosorted pointline incidence spaces: 358
• Coordinatization: 367
• On the firstorder theories of affine and projective spaces: 380
• Betweenness structures and ordered affine planes: 386
• Rich languages and structures for elementary geometry: 394
• Modal logic and spatial logic: 400
• Pointbased spatial logics: 404
• Linebased spatial logics: 406
• Tip spatial logics: 411
• Pointline spatial logics: 416
• Locales and Toposes as Spaces: 429
• Opens as propositions: 431
• Predicate geometric logic: 445
• Categorical logic: 457
• Sheaves as predicates: 474
• Summary of toposes: 488
• Other directions: 489
• Conclusions: 492
• Spatial Logic + Temporal Logic ?: 497
• Static and changing spatial models: 501
• Spatial logics: 506
• Temporal logics: 527
• Combination principles: 531
• Combining topologics with temporal logics: 533
• Combining distance logics with temporal logics: 543
• Logics for dynamical systems: 546
• Related temporalised formalisms: 557
• Dynamic Topological Logic: 565
• Basic definitions: 569
• Recurrence and the DTL of measurepreserving continuous functions on the closed unit interval: 573
• Purely topological and purely temporal fragments of DTLs: 576
• S4 is topologically complete for 0 1: 579
• The logic of homeomorphisms: 586
• The logic of continuous functions: 592
• Conclusion: 604
• Logic of SpaceTime and Relativity Theory: 607
• Special relativity: 608
• General relativistic spacetime: 660
• Black holes wormholes timewarp Distinguished general relativis tic spacetimes: 683
• Connections with the literature: 705
• Discrete Spatial Models Michael B Smyth Julian Webster 1 Introduction: 713
• Preliminaries correspondence principle: 717
• ˇCech closure spaces 4 Closure systems: 727
• Extended examples: 730
• Boundary and dimension: 742
• Discrete region geometry: 749
• Matroids: 764
• Spherical oriented matroids Flat oriented matroids: 768
• Algebraic spatial models: 787
• 10: 795
• 11: 796
• Real Algebraic Geometry and Constraint Databases Floris Geerts Bart Kuijpers From the relational database model to the constraint database model: 799
• Constraint data models and query languages: 805
• Introduction to real algebraic geometry: 812
• Query evaluation through quantifier elimination: 822
• Expressiveness results: 829
• Extensions of logical query languages 1: 841
• 2: 842
• 4: 845
• 5: 853
• 6: 854
• Mathematical Morphology Isabelle Bloch Henk Heijmans Christian Ronse 1 Introduction: 857
• Algebra: 884
• Related approaches: 897
• Logics: 919
• Conclusion: 936
• Parts Wholes and Locations Achille C Varzi: 945
• Philosophical issues in mereology: 947
• Philosophical issues in topology: 975
• Location theories: 1012
• 705 713 713: 1039
• 725: 1041
• 829: 1043
• 841: 1044
• 857: 1049
• 975: 1052
• 1012: 1053

Bibliographic information

• Title: Handbook of Spatial Logics
• Editors: Marco Aiello, Ian Pratt-Hartmann, Johan van Benthem
• Edition: illustrated
• Publisher: Springer Science & Business Media, 2007
• ISBN: 1402055870, 9781402055874
• Length: 1058 pages
• Subjects: Computers / Artificial Intelligence / General; Mathematics / Discrete Mathematics; Mathematics / Geometry / General; Mathematics / History & Philosophy; Mathematics / Logic; Mathematics / Topology; Philosophy / General; Science / Philosophy & Social Aspects