Axioms, Algebras and Topology

Axiomatic theories provide a very general means for specifying the logical properties of formal concepts. From the axiomatic point of view, it is symbolic formulae and the logical relations between them — especially the entailment relation — that form the primary subject of interest. The vocabulary of concepts of any theory can be interpreted in terms of a domain of entities, which exemplify properties, relations and functional mappings corresponding to the formal symbols of the theory. Moreover, by interpreting logical operations as functions of these seman-

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

[2]  F. Wolter,et al.  Qualitative spatiotemporal representation and reasoning: a computational perspective , 2003 .

[3]  James F. Allen.  Maintaining knowledge about temporal intervals , 1983, CACM.

[4]  Christoph Dornheim.  Undecidability of Plane Polygonal Mereotopology , 1998, KR.

[5]  Antony Galton.  Taking Dimension Seriously in Qualitative Spatial Reasoning , 1996, ECAI.

[6]  David Robertson,et al.  Proceedings of the 12th European Conference on Artificial Intelligence , 1996 .

[7]  A. Tarski,et al.  Boolean Algebras with Operators. Part I , 1951 .

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

[9]  Alfred Tarski,et al.  Distributive and Modular Laws in the Arithmetic of Relation Algebras , 1953 .

[10]  Yde Venema,et al.  Points, Lines and Diamonds: A two-sorted Modal Logic for Projective Planes , 1999, J. Log. Comput..

[11]  Anthony G. Cohn.  Modal and Non Modal Qualitative Spatial Logics , 1993 .

[12]  Anthony G. Cohn,et al.  Multi-Dimensional Modal Logic as a Framework for Spatio-Temporal Reasoning , 2002, Applied Intelligence.

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

[14]  Frank Wolter,et al.  Spatio-temporal representation and reasoning based on RCC-8 , 2000, International Conference on Principles of Knowledge Representation and Reasoning.

[15]  Peter Roeper,et al.  Region-Based Topology , 1997, J. Philos. Log..

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

[17]  Bjarni Jónsson,et al.  On the canonicity of Sahlqvist identities , 1994, Stud Logica.

[18]  M. de Rijke,et al.  Modal Logic , 2001, Cambridge Tracts in Theoretical Computer Science.

[19]  Krister Segerberg,et al.  Two-dimensional modal logic , 1973, J. Philos. Log..

[20]  Algebraic logic , 1985, Problem books in mathematics.

[21]  Max J. Cresswell,et al.  A New Introduction to Modal Logic , 1998 .

[22]  Anthony G. Cohn,et al.  A more expressive formulation of many sorted logic , 1987, Journal of Automated Reasoning.

[23]  M. de Rijke,et al.  Sahlqvist's theorem for boolean algebras with operators with an application to cylindric algebras , 1995, Stud Logica.

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

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

[26]  Alfred Tarski,et al.  Tarski's System of Geometry , 1999, Bulletin of Symbolic Logic.

[27]  R. Sikorski,et al.  The mathematics of metamathematics , 1963 .

[28]  Anthony G. Cohn,et al.  A Comparison Of Structures In Spatial And Temporal Logics , 1994 .

[29]  B. Nebel.  Introduction to Modal Logic Introduction , 2009 .

[30]  Alan Bundy,et al.  Proceedings of the Fourteenth International Joint Conference on Artificial Intelligence - IJCAI-95 , 1995 .

[31]  Thomas Mormann.  Continuous lattices and Whiteheadian theory of space , 2004 .

[32]  N. Goodman,et al.  The Structure of Appearance. , 1953 .

[33]  Max J. Egenhofer,et al.  Reasoning about Binary Topological Relations , 1991, SSD.

[34]  Ernest Davis,et al.  The Expressivity of Quantifying over Regions , 2006, J. Log. Comput..

[35]  Maarten Marx,et al.  Multi-dimensional modal logic , 1997, Applied logic series.

[36]  Jochen Renz,et al.  A Canonical Model of the Region Connection Calculus , 1997, J. Appl. Non Class. Logics.

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

[38]  Laure Vieu,et al.  Atomicity vs. Infinite Divisibility of Space , 1999, COSIT.

[39]  A. Tarski,et al.  A Formalization Of Set Theory Without Variables , 1987 .

[40]  A. Tarski,et al.  Boolean Algebras with Operators , 1952 .

[41]  Werner Nutt.  On the Translation of Qualitative Spatial Reasoning Problems into Modal Logics , 1999, KI.

[42]  Hajnal Andréka,et al.  Free algebras in discriminator varieties , 1991 .

[43]  E. Orlowska.  Relational interpretation of modal logics , 1988 .

[44]  Henry A. Kautz,et al.  Constraint Propagation Algorithms for Temporal Reasoning , 1986, AAAI.

[45]  Ivo Düntsch,et al.  A relation - algebraic approach to the region connection calculus , 2001, Theor. Comput. Sci..

[46]  M. Egenhofer.  Categorizing Binary Topological Relations Between Regions, Lines, and Points in Geographic Databases , 1998 .

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

[48]  Nicholas Mark GottsDivision.  Formalizing Commonsense Topology : The INCH Calculus , 1996 .

[49]  Brandon Bennett,et al.  Spatial Reasoning with Propositional Logics , 1994, KR.

[50]  Sanjiang Li,et al.  On countable RCC models , 2004, Fundam. Informaticae.

[51]  Brandon Bennett,et al.  Modal Logics for Qualitative Spatial Reasoning , 1996, Log. J. IGPL.

[52]  Ralf Röhrig,et al.  A Theory for Qualitative Spatial Reasoning Based on Order Relations , 1994, AAAI.

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

[54]  István Németi,et al.  Algebraization of quantifier logics, an introductory overview , 1991, Stud Logica.

[55]  Brandon Bennett,et al.  Logical representations for automated reasoning about spatial relationships , 1997 .

[56]  Heinrich Wansing,et al.  Proof Theory of Modal Logic , 1996 .

[57]  Ian Pratt-Hartmann,et al.  Expressivity in Polygonal, Plane Mereotopology , 2000, J. Symb. Log..

[58]  Luis Fariñas del Cerro,et al.  Modal Logics for Incidence Geometries , 1997, J. Log. Comput..

[59]  Gert Sabidussi,et al.  Algebras and orders , 1993 .

[60]  A. Tarski,et al.  The Algebra of Topology , 1944 .

[61]  Yuan-Ko Huang,et al.  Continuous K-Nearest Neighbor Query for Moving Objects with Uncertain Velocity , 2009, GeoInformatica.

[62]  Anthony G. Cohn,et al.  Computing Transivity Tables: A Challenge For Automated Theorem Provers , 1992, CADE.

[63]  Dimitris Papadias,et al.  Topological Inference , 1995, IJCAI.

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

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

[66]  Anthony G. Cohn,et al.  Constraint Networks of Topological Relations and Convexity , 1999, Constraints.

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

[68]  Patrick J. Hayes,et al.  The second naive physics manifesto , 1995 .

[69]  A. Tarski,et al.  Cylindric Algebras. Part II , 1988 .

[70]  A. R. Turquette,et al.  Logic, Semantics, Metamathematics , 1957 .

[71]  Jan Kratochvíl,et al.  String graphs requiring exponential representations , 1991, J. Comb. Theory, Ser. B.

[72]  Sanjiang Li,et al.  Extensionality of the RCC8 Composition Table , 2002, Fundam. Informaticae.

[73]  D. Gabbay,et al.  Many-Dimensional Modal Logics: Theory and Applications , 2003 .

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

[75]  Ivo Düntsch,et al.  A note on proximity spaces and connection based mereology , 2001, FOIS.

[76]  Steven Givant,et al.  The Calculus of Relations as a Foundation for Mathematics , 2006, Journal of Automated Reasoning.

[77]  Judit X. Madarász,et al.  Interpolation in Algebraizable Logics; Semantics for Non-normal Multi-modal Logic , 1998, J. Appl. Non Class. Logics.

[78]  I. Hodkinson,et al.  Relation Algebras by Games , 2002 .

[79]  Hajnal Andréka,et al.  Decision problems for equational theories of relation algebras , 1997, Memoirs of the American Mathematical Society.

[80]  Roberto Casati,et al.  Philosophy and the Cognitive Sciences: Proceedings of the 16th International Wittgenstein Symposium, 15-22 August 1993, Kirchberg Am Wechsel (Austria) , 1994 .

[81]  Bjarni Jónsson,et al.  A Survey of Boolean Algebras with Operators , 1993 .

[82]  Carsten Lutz,et al.  Modal Logics of Topological Relations , 2006, Log. Methods Comput. Sci..

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

[84]  Ivo Düntsch,et al.  A Proof System for Contact Relation Algebras , 2000, J. Philos. Log..

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

[86]  Ian Pratt-Hartmann.  Empiricism and Rationalism in Region-based Theories of Space , 2001, Fundam. Informaticae.

[87]  M. Egenhofer,et al.  Point-Set Topological Spatial Relations , 2001 .

[88]  Kirby A. Baker.  Lattice theory and its applications , 1995 .

[89]  Michael Winter,et al.  Construction of Boolean contact algebras , 2004, AI Commun..

[90]  L. Birch,et al.  The concept of nature , 1992 .

[91]  Valentin Goranko,et al.  Using the Universal Modality: Gains and Questions , 1992, J. Log. Comput..

[92]  Oliver Lemon,et al.  Ontologies for Plane, Polygonal Mereotopology , 1997, Notre Dame J. Formal Log..

[93]  Alfred Tarski,et al.  Relational selves as self-affirmational resources , 2008 .

[94]  Ivo Düntsch,et al.  A Proximity Approach to Some Region-Based Theories of Space , 2002, J. Appl. Non Class. Logics.

[95]  Ivo Düntsch,et al.  Relation Algebras and their Application in Temporal and Spatial Reasoning , 2005, Artificial Intelligence Review.

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

[97]  Ewa Orlowska,et al.  Relational Proof Systems for Modal Logics , 1996 .

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

[99]  Carus Paul.  The Foundation of Mathematics. , 1920 .

[100]  E. V. Shchepin,et al.  Real functions and near-normal spaces , 1972 .

[101]  Christian Freksa,et al.  Spatial Information Theory. Cognitive and Computational Foundations of Geographic Information Science , 1999, Lecture Notes in Computer Science.

[102]  P. Johnstone,et al.  The point of pointless topology , 1983 .

[103]  Jan Kratochvíl,et al.  String graphs. II. recognizing string graphs is NP-hard , 1991, J. Comb. Theory, Ser. B.

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

[105]  Christian Freksa,et al.  Temporal Reasoning Based on Semi-Intervals , 1992, Artif. Intell..

[106]  A. Tarski.  What is Elementary Geometry , 1959 .

[107]  Robert C. Moore,et al.  Formal Theories of the Commonsense World , 1985 .

[108]  Antony Galton.  Multidimensional Mereotopology , 2004, KR.

[109]  Gunther Schmidt,et al.  A Necessary Relation Algebra for Mereotopology , 2001, Stud Logica.

[110]  Michael Winter,et al.  Remarks on lattices of contact relations Preliminary report , 2006 .

[111]  Frank Wolter,et al.  Exploring Artificial Intelligence in the New Millenium , 2002 .