### Ontologies for Plane, Polygonal Mereotopology

Oliver Lemon and Ian Pratt
Source: Notre Dame J. Formal Logic Volume 38, Number 2 (1997), 225-245.

#### Abstract

Several authors have suggested that a more parsimonious and conceptually elegant treatment of everyday mereological and topological reasoning can be obtained by adopting a spatial ontology in which regions, not points, are the primitive entities. This paper challenges this suggestion for mereotopological reasoning in two-dimensional space. Our strategy is to define a mereotopological language together with a familiar, point-based interpretation. It is proposed that, to be practically useful, any alternative region-based spatial ontology must support the same sentences in our language as this familiar interpretation. This proposal has the merit of transforming a vague, open-ended question about ontologies for practical mereotopological reasoning into a precise question in model theory. We show that (a version of) the familiar interpretation is countable and atomic, and therefore prime. We conclude that useful alternative ontologies of the plane are, if anything, less parsimonious than the one which they are supposed to replace.

First Page:
Primary Subjects: 03B30
Secondary Subjects: 03C65
Full-text: Open access

Permanent link to this document: http://projecteuclid.org/euclid.ndjfl/1039724888
Mathematical Reviews number (MathSciNet): MR1489411
Digital Object Identifier: doi:10.1305/ndjfl/1039724888
Zentralblatt MATH identifier: 0897.03014

### References

[1] Allen, J. F., An interval-based representation of temporal knowledge,'' pp. 221--26 in The Seventh International Joint Conference on Artificial Intelligence (IJCAI), 1981.
[2] Asher, N., and L. Vieu, Toward a geometry of common sense: a semantics and a complete axiomatization of mereotopology,'' pp. 846--52 in International Joint Conference on Artificial Intelligence (IJCAI), 1995.
Mathematical Reviews (MathSciNet): MR1464505
[3] Balbiani, P., L. F. del Cerro, T. Tinchev, and D. Vakarelov, Geometrical structures and modal logic,'' pp. 43--57 in Practical Reasoning. Lecture Notes in Artificial Intelligence, 1085, edited by D. Gabbay and H.-J. Ohlbach, Springer-Verlag, Berlin, 1996.
Mathematical Reviews (MathSciNet): MR1457027
Digital Object Identifier: doi:10.1007/3-540-61313-7_62
[4] Basri, S. A., A Deductive Theory of Space and Time. Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1966.
Mathematical Reviews (MathSciNet): MR35:6431
Zentralblatt MATH: 0161.00208
[5] Biacino, L., and Gerla, G., Connection structures,'' Notre Dame Journal of Formal Logic, vol. 32 (1991), pp. 242--47.
Mathematical Reviews (MathSciNet): MR92i:03067
Zentralblatt MATH: 0749.06004
Project Euclid: euclid.ndjfl/1093635748
Digital Object Identifier: doi:10.1305/ndjfl/1093635748
[6] Borgo, S., N. Guarino, and C. Masolo, A pointless theory of space based on strong connection and congruence,'' pp. 220--29 in Principles of Knowledge Representation and Reasoning. Proceedings of the Fifth International Conference (KR), edited by L. C. Aiello, J. Doyle, and S. C. Shapiro, Morgan Kaufmann, San Francisco, 1996.
[7] Carnap, R., Introduction to Symbolic Logic and its Applications, Dover, New York, 1958.
Mathematical Reviews (MathSciNet): MR21:2578
Zentralblatt MATH: 0083.00104
[8] Casati, R., and A. Varzi, Holes and Other Superficialities, The MIT Press, Cambridge, 1994.
[9] Casati, R., and A. Varzi, The structure of spatial localization,'' Philosophical Studies, vol. 82 (1996), pp. 205--39.
[10] Chang, C. C., and H. J. Keisler, Model Theory, 3d edition, North-Holland, Amsterdam, 1990.
Mathematical Reviews (MathSciNet): MR91c:03026
Zentralblatt MATH: 0697.03022
[11] Clarke, B. L., A calculus of individuals based on connection','' Notre Dame Journal of Formal Logic, vol. 23 (1981), pp. 204--18.
Mathematical Reviews (MathSciNet): MR82i:03036
Zentralblatt MATH: 0476.03035
Project Euclid: euclid.ndjfl/1093883455
Digital Object Identifier: doi:10.1305/ndjfl/1093883455
[12] Clarke, B. L., Individuals and points,'' Notre Dame Journal of Formal Logic, vol. 26 (1985), pp. 61--75.
Mathematical Reviews (MathSciNet): MR86h:03047
Zentralblatt MATH: 0597.03005
Project Euclid: euclid.ndjfl/1093870761
Digital Object Identifier: doi:10.1305/ndjfl/1093870761
[13] Davis, E., Representing and Acquiring Geographic Knowledge. Research Notes in Artificial Intelligence, Morgan Kaufmann, Los Altos, 1986.
[14] Davis, E., Representations of Commonsense Knowledge, Morgan Kaufmann, San Mateo, 1990.
[15] Dudek, G., P. Freedman, and S. Hadjres, Using local information in a non-local way for mapping graph-like worlds,'' pp. 1639--45 in Thirteenth International Joint Conference on Artificial Intelligence (IJCAI), Morgan Kaufmann, San Mateo, 1993.
[16] Goldblatt, R., Mathematics of Modality. CSLI Lecture Notes, 43, CSLI, Stanford, 1993.
Mathematical Reviews (MathSciNet): MR95k:03022
Zentralblatt MATH: 0942.03516
[17] Gotts, N., J. Gooday, and A. Cohn, A connection based approach to commonsense topological description and reasoning,'' Monist, vol. 79 (1996), pp. 51--75.
[18] Haarslev, V., Formal semantics of visual languages using spatial reasoning,'' pp. 156--63 in IEEE Symposium on Visual Languages, IEEE Computer Society Press, Los Alamitos, 1995.
[19] Henkin, L., P. Suppes, and A. Tarski, editors, The Axiomatic Method, with Special Reference to Geometry and Physics, North-Holland, Amsterdam, 1959.
Zentralblatt MATH: 0088.24414
[20] Hodges, W., Model Theory. Encyclopedia of Mathematics and its Applications, vol. 42, Cambridge University Press, Cambridge, 1993.
Mathematical Reviews (MathSciNet): MR94e:03002
Zentralblatt MATH: 0789.03031
[21] Knight, J. F., A. Pillay, and C. Steinhorn, Definable sets in ordered structures II,'' Transactions of the American Mathematical Society, vol. 295 (1986), pp. 593--605.
Mathematical Reviews (MathSciNet): MR88b:03050b
Zentralblatt MATH: 0662.03024
Digital Object Identifier: doi:10.2307/2000053
[22] Koppelberg, S., Handbook of Boolean Algebras, vol. 1, North-Holland, Amsterdam, 1989.
Mathematical Reviews (MathSciNet): MR90k:06002
Zentralblatt MATH: 0671.06001
[23] Kuipers, B., Modeling spatial knowledge,'' Cognitive Science, vol. 2 (1978), pp. 129--53.
Zentralblatt MATH: 0332.68007
[24] Lemon, O., Review of Logic and Visual Information by E. M. Hammer,'' Journal of Logic, Language, and Information, vol. 6 (1997), pp. 213--16.
Zentralblatt MATH: 0941.03538
[25] Lemon, O., and I. Pratt, Spatial logic and the complexity of diagrammatic reasoning,'' Machine Graphics and Vision. Special Issue on Diagrammatic Representation and Reasoning, vol. 6 (1997), pp. 89--108.
[26] Pillay, A., and C. Steinhorn, Definable sets in ordered structures I,'' Transactions of the American Mathematical Society, vol. 295 (1986), pp. 565--92.
Mathematical Reviews (MathSciNet): MR88b:03050a
Zentralblatt MATH: 0662.03023
Digital Object Identifier: doi:10.2307/2000052
[27] Pratt, I., Map semantics,'' pp. 77--91 in Spatial Information Theory: A Theoretical Basis for GIS, edited by A. Frank and I. Campari, vol. 716 of Lecture Notes in Computer Science, Springer-Verlag, Berlin, 1993.
[28] Pratt, I., and D. Schoop, A complete axiom system for polygonal mereotopology of the real plane,'' Technical Report UMCS97-2-2, University of Manchester, Manchester, 1997.
Mathematical Reviews (MathSciNet): MR1663396
Digital Object Identifier: doi:10.1023/A:1004361501703
Zentralblatt MATH: 0921.03009
[29] Rescher, N., and J. Garson, Topological logic,'' The Journal of Symbolic Logic, vol. 33 (1968), pp. 537--48.
Mathematical Reviews (MathSciNet): MR38:5595
Zentralblatt MATH: 0187.26702
Digital Object Identifier: doi:10.2307/2271360
[30] Rescher, N., and A. Urquhart, Temporal Logic. Library of Exact Philosophy, vol. 3, Springer-Verlag, New York, 1971.
Mathematical Reviews (MathSciNet): MR49:2267
Zentralblatt MATH: 0229.02027
[31] Roeper, P., Region-based topology,'' Journal of Philosophical Logic, vol. 26 (1997) pp. 251--309.
Mathematical Reviews (MathSciNet): MR99f:54005
Zentralblatt MATH: 0873.54001
Digital Object Identifier: doi:10.1023/A:1017904631349
[32] Segerberg, K., Two dimensional modal logic,'' Journal of Philosophical Logic, vol. 2 (1973) pp. 77--96.
Mathematical Reviews (MathSciNet): MR54:12488
Zentralblatt MATH: 0259.02013
Digital Object Identifier: doi:10.1007/BF02115610
[33] Shanahan, M., Default reasoning about spatial occupancy,'' Artificial Intelligence, vol. 74 (1995), pp. 147--63.
[34] Shehtman, V. B., Modal logics of domains on the real plane,'' Studia Logica, vol. 42 (1983) pp. 63--80.
Mathematical Reviews (MathSciNet): MR86d:03020
Digital Object Identifier: doi:10.1007/BF01418760
[35] Tarski, A., Foundations of the geometry of solids,'' pp. 24--29 in Logic, Semantics, Metamathematics, Clarendon Press, Oxford, 1956.
[36] Varzi, A., Spatial reasoning in a holey world,'' pp. 326--36 in Advances in Artificial Intelligence, edited by P. Torasso, vol. 728 of Lecture Notes in Artificial Intelligence, Springer-Verlag, Berlin, 1993.
[37] Vieu, L., A logical framework for reasoning about space,'' pp. 25--35 in Spatial Information Theory: A Theoretical Basis for GIS, edited by A. Frank and I. Campari, vol. 716 of Lecture Notes in Computer Science, Springer-Verlag, Berlin, 1993.
Mathematical Reviews (MathSciNet): MR1290134
Digital Object Identifier: doi:10.1007/BFb0028190
[38] von Wright, G. H., `A modal logic of place,'' pp. 65--73 in The Philosophy of Nicholas Rescher: Discussion and Replies, edited by E. Sosa, Reidel, Dordrecht, 1979.
[39] Whitehead, A. N., Process and Reality, MacMillan, New York, 1929.
Zentralblatt MATH: 55.0035.03