Model-Theoretic Characterization of Asher and Vieu's Ontology of Mereotopology

We characterize the models of Asher and Vieu's first-order mereotopology RT0 in terms of mathematical structures with well-defined properties: topological spaces, lattices, and graphs. We give a full representation theorem for the models of the subtheory RT- (RT0 without existential axioms) as p-ortholattices (pseudocomplemented, orthocomplemented). We further prove that the finite models of RT-EC, an extension of RT-, are isomorphic to a graph representation of p-ortholattices extended by additional edges and we show how to construct finite models of the full mereotopology. The results are compared to representations of Clarke's mereotopology and known models of the Region Connection Calculus (RCC). Although soundness and completeness of the theory RT0 has been proved with respect to a topological translation of the axioms, our characterization provides more insight into the structural properties of the mereotopological models.

[1]  Achille C. Varzi.  Spatial Reasoning and Ontology: Parts, Wholes, and Locations , 2007, Handbook of Spatial Logics.

[2]  Laure Vieu,et al.  Toward a Geometry of Common Sense: A Semantics and a Complete Axiomatization of Mereotopology , 1995, IJCAI.

[3]  P. Simons.  Parts: A Study in Ontology , 1991 .

[4]  Giangiacomo Gerla,et al.  Connection Structures , 1991, Notre Dame J. Formal Log..

[5]  Barry Smith,et al.  Mereotopology: A Theory of Parts and Boundaries , 1996, Data Knowl. Eng..

[6]  Bowman L. Clarke,et al.  A calculus of individuals based on "connection" , 1981, Notre Dame J. Formal Log..

[7]  Feodor F. Dragan,et al.  Dually Chordal Graphs , 1998, SIAM J. Discret. Math..

[8]  Johan van Benthem,et al.  Handbook of Spatial Logics , 2007 .

[9]  Roberto Casati,et al.  Parts And Places , 1999 .

[10]  Anthony G. Cohn,et al.  A Spatial Logic based on Regions and Connection , 1992, KR.

[11]  Brandon Bennett,et al.  A Categorical Axiomatisation of Region-Based Geometry , 2001, Fundam. Informaticae.

[12]  O. Stock.  Spatial and Temporal Reasoning , 1899 .

[13]  John G. Stell,et al.  Boolean connection algebras: A new approach to the Region-Connection Calculus , 2000, Artif. Intell..

[14]  Manfred Stern,et al.  Semimodular Lattices: Theory and Applications , 1999 .

[15]  T. Hahmann.  Model-Theoretic Analysis of Asher and Vieu's Mereotopology , 2008 .

[16]  Anthony G. Cohn,et al.  Representing and Reasoning with Qualitative Spatial Relations About Regions , 1997 .

[17]  F. J. Pelletier,et al.  316 Notre Dame Journal of Formal Logic , 1982 .

[18]  Michael Winter,et al.  A representation theorem for Boolean contact algebras , 2005, Theor. Comput. Sci..

[19]  Nelson Goodman,et al.  The calculus of individuals and its uses , 1940, Journal of Symbolic Logic.

[20]  Michael F. Worboys,et al.  The Algebraic Structure of Sets of Regions , 1997, COSIT.

[21]  Carola Eschenbach,et al.  Classical mereology and restricted domains , 1995, Int. J. Hum. Comput. Stud..

[22]  Ian Pratt-Hartmann,et al.  A Complete Axiom System for Polygonal Mereotopology of the Real Plane , 1998, J. Philos. Log..

[23]  Bowman L. Clarke,et al.  Individuals and points , 1985, Notre Dame J. Formal Log..

[24]  Anthony G. Cohn,et al.  Qualitative Spatial Representation and Reasoning with the Region Connection Calculus , 1997, GeoInformatica.

[25]  G. Bruns.  ORTHOMODULAR LATTICES (London Mathematical Society Monographs, 18) , 1984 .

[26]  Michael Winter,et al.  Weak Contact Structures , 2005, RelMiCS.

[27]  B. L. Clark.  Individuals and points. , 1985 .

[28]  G. Grätzer.  General Lattice Theory , 1978 .

[29]  Giangiacomo Gerla,et al.  Connection Structures: Grzegorczyk's and Whitehead's Definitions of Point , 1996, Notre Dame J. Formal Log..

[30]  Ivo Düntsch,et al.  Relations Algebras in Qualitative Spatial Reasoning , 1999, Fundam. Informaticae.

[31]  G. Kalmbach.  On Orthomodular Lattices , 1990 .

[32]  Eugene C. Luschei,et al.  The logical systems of Lesniewski , 1962 .

[33]  Nicola Guarino,et al.  A Pointless Theory of Space Based on Strong Connection and Congruence , 1996, KR.