Relational Representation Theorems for Extended Contact Algebras

In topological spaces, the relation of extended contact is a ternary relation that holds between regular closed subsets A, B and D if the intersection of A and B is included in D. The algebraic counterpart of this mereotopological relation is the notion of extended contact algebra which is a Boolean algebra extended with a ternary relation. In this paper, we are interested in the relational representation theory for extended contact algebras. In this respect, we study the correspondences between point-free and point-based models of space in terms of extended contact. More precisely, we prove new representation theorems for extended contact algebras.

[1]  Dimiter Vakarelov,et al.  Contact Algebras and Region-based Theory of Space: A Proximity Approach - I , 2006, Fundam. Informaticae.

[2]  Yavor Nenov,et al.  Topological Logics with Connectedness over Euclidean Spaces , 2011, TOCL.

[3]  Sanjiang Li,et al.  Region Connection Calculus: Its models and composition table , 2003, Artif. Intell..

[4]  Dimiter Vakarelov,et al.  A mereotopology based on sequent algebras , 2017, J. Appl. Non Class. Logics.

[5]  Jochen Renz,et al.  Qualitative Spatial Reasoning with Topological Information , 2002, Lecture Notes in Computer Science.

[6]  Dimiter Vakarelov,et al.  Modal Logics for Region-based Theories of Space , 2007, Fundam. Informaticae.

[7]  Andrzej Grzegorczyk,et al.  Axiomatizability of geometry without points , 1960, Synthese.

[8]  Tatyana Ivanova,et al.  Extended Contact Algebras and Internal Connectedness , 2020, Stud Logica.

[9]  Michael Zakharyaschev,et al.  Spatial reasoning with RCC8 and connectedness constraints in Euclidean spaces , 2014, Artif. Intell..

[10]  Ian Pratt-Hartmann,et al.  First-Order Mereotopology , 2007, Handbook of Spatial Logics.

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

[12]  Majid Alizadeh,et al.  Boolean Algebras , 2022, Set Theory and Foundations of Mathematics: An Introduction to Mathematical Logic.

[13]  Ivo Düntsch,et al.  Region–based theory of discrete spaces: A proximity approach , 2007, Annals of Mathematics and Artificial Intelligence.

[14]  Bernhard Nebel,et al.  On the Complexity of Qualitative Spatial Reasoning: A Maximal Tractable Fragment of the Region Connection Calculus , 1999, Artif. Intell..

[15]  Antony Galton,et al.  The Mereotopology of Discrete Space , 1999, COSIT.

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

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

[18]  Frank Wolter,et al.  Spatial logics with connectedness predicates , 2010, Log. Methods Comput. Sci..

[19]  Dimiter Vakarelov,et al.  Region-Based Theory of Space: Algebras of Regions, Representation Theory, and Logics , 2007 .

[20]  T. D. Laguna.  Point, Line, and Surface, as Sets of Solids , 1922 .