Abstract
This paper presents a calculus for mereotopological reasoning in which two-dimensional spatial regions are treated as primitive entities. A first order predicate language \(\mathcal{L}\)with a distinguished unary predicate c(x), function-symbols , · and − and constants 0 and 1 is defined. An interpretation \(\Re \)for \({\mathcal{L}}\)is provided in which polygonal open subsets of the real plane serve as elements of the domain. Under this interpretation the predicate c(x) is read as ‘region x is connected’ and the function-symbols and constants are given their meaning in terms of a Boolean algebra of polygons. We give an alternative interpretation \({\mathfrak{S}}\) based on the real closed plane which turns out to be isomorphic to \(\Re \)A set of axioms and a rule of inference are introduced. We prove the soundness and completeness of the calculus with respect to the given interpretation.
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REFERENCES
Asher, N. and Vieu, L.: Toward a geometry of common sense – A semantics and acomplete axiomatization of mereotopology, in Proceedings of the Fourteenth International Joint Conference on Artificial Intelligence (IJCAI-95), 1995, pp. 846–852.
Biacino, L. and Gerla, G.: Connectionstructures: Grzegorczyk's and Whitehead's definition of point, Notre Dame J. Formal Logic 37(3) (1996), 431–439.
Bollobás, Béla: Graph Theory: An Introductory Course, Spinger-Verlag, New York, 1979.
Borgo, S., Guarino, N. and Masolo, C.: A pointless theory of space basedon 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), Morgan Kaufmann, San Francisco, CA, 1996, pp. 220–229.
Chang, C. C. and Keisler, H. J.: ModelTheory, 3rd edition, North-Holland, Amsterdam, 1990.
Clarke, B. L.:A calculus ofindividuals 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.
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.
Johnstone, P. T.: Stone Spaces,Cambridge University Press, Cambridge, 1982.
Newman, M. H. A.: Elements of the Topology ofPlane Sets of Points, Cambridge University Press, Cambridge, 1964.
Pratt, Ian and Lemon, Oliver: Ontologies for plane, polygonal mereotopology, Notre Dame J. Formal Logic 38(2) (1997), 225–245.
Randell, D. A., Cui, Z. and Cohn, A. G.: A spatial logic based on regions andconnection, in B. Nebel, C. Rich, and W. Swartout (eds), Principles of Knowledge Representation and Reasoning: Proceedings of the Third International Conference (KR' 92), Morgan Kaufmann, Los Altos, CA, 1992, pp. 165–176.
Tarski, Alfred: Foundations of the geometry of solids, in Logic, Semantics,and Metamathematics, Clarendon Press, Oxford, 1956, pp. 24–29.
Whitehead, A. N.:Process and Reality, The MacMillan Company, New York, 1929.
Wilson, Robin J.:Introduction to Graph Theory, Longman, London, 1979.