Elementary Polyhedral Mereotopology
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
A region-based model of physical space is one in which the primitive spatial entities are regions, rather than points, and in which the primitive spatial relations take regions, rather than points, as their relata. Historically, the most intensively investigated region-based models are those whose primitive relations are topological in character; and the study of the topology of physical space from a region-based perspective has come to be called mereotopology. This paper concentrates on a mereotopological formalism originally introduced by Whitehead, which employs a single primitive binary relation C(x,y) (read: “x is in contact with y”). Thus, in this formalism, all topological facts supervene on facts about contact. Because of its potential application to theories of qualitative spatial reasoning, Whitehead's primitive has recently been the subject of scrutiny from within the Artificial Intelligence community. Various results regarding the mereotopology of the Euclidean plane have been obtained, settling such issues as expressive power, axiomatization and the existence of alternative models. The contribution of the present paper is to extend some of these results to the mereotopology of three-dimensional Euclidean space. Specifically, we show that, in a first-order setting where variables range over tame subsets of R 3, Whitehead's primitive is maximally expressive for topological relations; and we deduce a corollary constraining the possible region-based models of the space we inhabit.
- Borgo, S., Guarino, N. and Masolo, C.: A pointless theory of space based on strong connection and congruence, in L. C. Aiello, J. Doyle, and S. C. Shapiro (eds.), Principles of Knowledge Representation and Reasoning: Proceedings of the Fifth International Conference (KR '96), 1996, pp. 220-229.
- Chang, C. C. and Keisler, H. J.: Model Theory, 3rd edn, North Holland, Amsterdam, 1990.
- Clarke, B. L.: A calculus of individuals based on "connection", Notre Dame J. Formal Logic 22(3) (1981), 204-218.
- Clarke, B. L.: Individuals and points, Notre Dame J. Formal Logic 26(1) (1985), 61-75.
- de Laguna, T.: Point, line, and surface as sets of solids, J. Philos. 19 (1922), 449-461.
- Dornheim, C.: Undecidability of plane polygonal mereotopology, in A. G. Cohn, L. K. Schubert, and S. C. Schubert (eds.), Principles of Knowledge Representation and Reasoning: Proceedings of the Sixth International Conference (KR '98), 1998, pp. 342-353.
- Galton, A.: Qualitative Spatial Change, Oxford University Press, Oxford, 2000.
- Gotts, N. M., Gooday, J. M. and Cohn, A. G.: A connection based approach to commonsense topological description and reasoning, Monist 79(1) (1996), 51-75.
- Hodges, W.: Model Theory, Cambridge University Press, Cambridge, 1993.
- Koppelberg, S.: Handbook of Boolean Algebras, Vol. 1, North-Holland, Amsterdam, 1989.
- Massey, W. S.: Algebraic Topology: An Introduction, Harcourt, Brace & World, New York, 1967.
- Newman, M. H. A.: Elements of the Topology of Plane Sets of Points, Cambridge University Press, Cambridge, 1964.
- Papadimitriou, C. H., Suciu, D. and Vianu, V.: Topological queries in spatial databases, in Proceedings of PODS'96, 1996, pp. 81-92.
- Pratt, I. and Lemon, O.: Ontologies for plane, polygonal mereotopology, Notre Dame J. Formal Logic 38(2) (1997), 225-245.
- Pratt, I. and Schoop, D.: A complete axiom system for polygonal mereotopology of the real plane, J. Philos. Logic 27(6) (1998), 621-658.
- Pratt, I. and Schoop, D.: Expressivity in polygonal, plane mereotopology, J. Symbolic Logic 65(2) (2000), 822-838.
- Randell, D. A., Cui, Z. and Cohn, A. G.: A spatial logic based on regions and connection, in B. Nebel, C. Rich, and W. Swartout (eds.), Principles of Knowledge Representation and Reasoning: Proceedings of the Third International Conference (KR '92), 1992, pp. 165-176.
- Simons, P.: Parts: A Study in Ontology, Clarendon Press, Oxford, 1987.
- van den Dries, L.: O-minimal structures, in W. Hodges, M. Hyland, C. Steinhorn, and J. Truss (eds.), Logic: From Foundations to Applications, Oxford University Press, Oxford, 1996, pp. 137-186.
- Whitehead, A. N.: Process and Reality, Macmillan, New York, 1929.
- Wilson, R. J.: Introduction to Graph Theory, Longman, London, 1979.
- Elementary Polyhedral Mereotopology
Journal of Philosophical Logic
Volume 31, Issue 5 , pp 469-498
- Cover Date
- Print ISSN
- Online ISSN
- Kluwer Academic Publishers
- Additional Links
- ontology of space
- spatial reasoning