Relation algebras and their application in temporal and spatial reasoning

Artificial Intelligence Review

Volumn 23, Issue 4, 2005, Pages 315-357

  Düntsch, Ivo ^a  

^a BROCK UNIVERSITY   (Canada)

Author keywords

Constraint satisfaction; Qualitative temporal and spatial reasoning; Relation algebra

Indexed keywords

CONSTRAINT THEORY; DATA HANDLING; MATHEMATICAL OPERATORS; RELATIONAL DATABASE SYSTEMS; THEOREM PROVING; THERMAL EFFECTS;

CONSTRAINT SATISFACTION; QUALITATIVE TEMPORAL AND SPATIAL REASONING (QTSR); QUALITATIVE TEMPORAL REASONING (QTR); RELATION ALGEBRA;

ALGEBRA;

EID: 17544372675     PISSN: 02692821     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10462-004-5899-8     Document Type: Article

References (96)

1. Allen, J. F. (1983). Maintaining Knowledge About Temporal Intervals. Communications of the ACM 26(11): 832-843.
2. Andréka, H., Düntsch, I. & Németi, I. (1995). Binary Relations and Permutation Groups. Mathematical Logic Quarterly 41: 197-216.
3. Andréka, H., Givant, S. & Németi, I. (1997). Decision Problems for Equational Theories of Relation Algebras, number 604 in Memoirs of the American Mathematical Society. American Mathematical Society: Providence, RI.
4. Andréka, H. & Maddux, R. (1994). Representations for Small Relation Algebras. Notre Dame Journal of Formal Logic 35(4): 550-562.
5. Andréka, H., Monk, J. D. & Németi, I. (eds.) (1991). Algebraic Logic, Vol. 54 of Colloquia Mathematica Societatis János Bolyai. North Holland: Amsterdam.
6. Anellis, I. & Houser, N. (1991). Nineteenth Century Roots of Algebraic Logic and Universal Algebra, In Andréka, H., Monk, J. D. Németi, I. (eds.), Algebraic Logic, Vol. 54 of Colloquia Mathematica Societatis János Bolyai. North Holland: Amsterdam, 1-36.
7. Asher, N. & Vieu, L. (1995). Toward a Geometry of Common Sense: A Semantics and A Complete Axiomatization of Mereotopology. In Mellish, C. (ed.), IJCAI 95, Proceedings of the 14th International Joint Conference on Artificial Intelligence.
8. Behnke, R., Berghammer, R., Meyer, E. & Schneider, P. (1998). RELVIEW - A System for Calculating with Relations and Relational Programming. Lecture Notes in Computer Science 1382: 318-321.
9. Bennett, B. (1994a). Some Observations and Puzzles About Composing Spatial and Temporal Relations, 11th European Conference on Artificial Intelligence, Workshop on Spatial Reasoning.
10. Bennett, B. (1994b). Spatial Reasoning with Prepositional Logics. In Jon Doyle, P. T. & Erik Sandewall (ed.), Proceedings of the 4th International Conference on Principles of Knowledge Representation and Reasoning, Morgan Kaufmann, Bonn, FRG, 51-62.
11. Bennett, B., Isli, A. & Cohn, A. (1997). When Does a Composition Table Provide A Complete and Tractable Proof Procedure For A Relational Constraint Language?. IJCAI 97, Proceedings of the Workshop of Spatial Reasoning.
12. Birkhoff, G. (1948). Lattice Theory, Vol. 25 of American Mathematical Society Colloquium Publications, 2 edn. AMS: Providence, MI.
13. Chin, L. & Tarski, A. (1951). Distributive and Modular Laws in the Arithmetic of Relation Algebras. University of California Publications in Mathematics 1: 341-384.
14. Clarke, B. L. (1981). A Calculus of Individuals Based on 'Connection'. Notre Dame Journal of Formal Logic 22: 204-218.
15. Comer, S. (1983). A Remark on Chromatic Polygroups. Congressus Numerantium 38: 85-95.
16. de Laguna, T. (1922). Point, Line and Surface as Sets of Solids. The Journal of Philosophy 19: 449-461.
17. de Morgan, A. (1864). On The Syllogism: IV, and on The Logic of Relations, Transactions of the Cambridge Philosophical Society 10: 331-358. (read April 23, 1860).
18. Düntsch, I. (1991). Small Integral Relation Algebras generated by A Partial Order. Period. Math. Hungar. 23: 129-138.
19. Düntsch, I. & Mikulás, S. (2001). Cylindric Structures and Dependencies in Relational Databases. Theoretical Computer Science 269: 451-468.
20. Düntsch, L, Orłowska, E. & Radzikowska, A. (2003). Lattice-Based Relation Algebras and Their Representability. Technical Report CS-03-03, Department of Computer Science, Brock University.
21. Düntsch, I., Schmidt, G. & Winter, M. (2001a). A Necessary Relation Algebra for Mereotopology. Studia Logica 69: 381-409.
22. Düntsch, I., Wang, H. & McCloskey, S. (2001 b). A Relation Algebraic Approach to the Region Connection Calculus. Theoretical Computer Science 255: 63-83.
23. Düntsch, I. & Winter, M. (2003). A Representation Theorem for Boolean Contact Algebras. Research Report CS-03-08, Department of Computer Science, Brock University.
24. Düntsch, I. & Winter, M. (2004). Construction of Boolean contact algebras, AI Communications. To appear.
25. Egenhofer, M. (1991). Reasoning About Binary Topological Relations. In Günther, O. & Schek, H. J. (eds.), Proceedings of the Second Symposium on Large Spatial Databases, SSD'91 (Zurich, Switzerland), 143-160, Vol. 525 of Lecture Notes in Computer Science.
26. Egenhofer, M. (1993). A Model for Detailed Binary Topological Relationships. Geomatica 47: 261-273.
27. Egenhofer, M. (1994a). Deriving the Composition of Binary Topological Relations. Journal of Visual Languages and Computing 5: 133-149.
28. Egenhofer, M. & Franzosa, R. (1991). Point-Set Topological Spatial Relations. International Journal of Geographic Information Systems 5(2): 161-174.
29. Egenhofer, M. & Herring, J. (1991). Categorizing Binary Topological Relationships Between Regions, Lines and Points in Geographic Databases. Technical Report, Department of Surveying Engineering, University of Maine.
30. Egenhofer, M. J. (1994b). Deriving the Composition of Binary Topological Relations. Journal of Visual Languages and Computing 5(1): 133-149.
31. Egenhofer, M. J., Clementini, E. & Felice, P. D. (1994). Topological Relations between Regions With Holes. International Journal of Geographical Information Systems 8(2): 129-144.
32. Egenhofer, M. J. & Franzosa, R. D. (1995). On The Equivalence of Topological Relations. International Journal of Geographical Information Systems 9(2): 133-152.
33. Egenhofer, M. & Rodriguez, A. (1999). Relation Algebras Over Containers and Surfaces: An Ontological Study of a Room Space. Journal of Spatial Cognition and Computation 1: 155-180.
34. Egenhofer, M. & Sharma, J. (1992). Topological Consistency. Fifth International Symposium on Spatial Data Handling. Charleston, SC.
35. Egenhofer, M. & Sharma, J. (1993). Assessing the Consistency of Complete and Incomplete Topological Information. Geographical Systems 1: 47-68.
36. Frank, A. U. (1996). Qualitative Spatial Reasoning: Cardinal Directions as an Example. International Journal of Geographical Information Science 10(3): 269-290.
37. Gerla, G. (1995). Pointless Geometries. In Buekenhout, F. (ed.) Handbook of Incidence Geometry, chapter 18, pp. 1015-1031. Elsevier Science B.V.
38. Grzegorczyk, A. (1960). Axiomatization of Geometry Without Points. Synthese 12: 228-235.
39. Henkin, L., Monk, J. D. & Tarski, A. (1985). Cylindric Algebras, Part II, North Holland: Amsterdam.
40. Hirsch, R. (1996). Relation Algebras of Intervals. Artificial Intelligence 83(2): 267-295.
41. Hirsch, R.: (1997). Expressive Power and Complexity in Algebraic Logic. Journal of Logic and Computation 7(3): 309-351.
42. Hirsch, R. (1999). A Finite Relation Algebra With An Undecidable Network Satisfaction Problem. Journal of the IGPL 7: 547-554.
43. Hirsch, R. (2000). Tractable Approximations for Temporal Constraint Handling. Artificial Intelligence 116(1-2): 287-295.
44. Hirsch, R. & Hodkinson, I. (2002). Relation Algebras by Games, Vol. 147 of Studies in Logic and the Foundations of Mathematics. Elsevier: Amsterdam.
45. Jónsson, B. (1959). Representation of Modular Lattices and of Relation Algebras. Transaction of the American Mathematical Society: Providence, RI. 92: 449-464.
46. Jónsson, B. (1982). Varieties of Relation Algebras. Algebra Universalis 15: 273-298.
47. Jónsson, B. (1984). The Theory of Binary Relations. Lecture notes, University of Montreal.
48. Jónsson, P. & Drakengren, T. (1997). A Complete Classification of Tractability in RCC-5. Journal of Artificial Intelligence Research 6: 211-222.
49. Kahl, W. & Schmidt, G. (2000). Exploring (finite) Relation Algebras using Tools written in Haskell. Technical Report 2000-02, Fakultät für Informatik, Universität der Bundeswehr München. Avalaible from http://ist.unibw-muenchen.de/Publications/TR/2000-02/ (December 31, 2001).
50. Köhler, C. (2002). The Occlusion Calculus. Proceedings of the "Cognitive Vision" Workshop, Zürich.
51. Koppelberg, S. (1989). General Theory of Boolean Algebras, Vol. 1 of Handbook on Boolean Algebras. North Holland: Amsterdam.
52. Kurucz, Á. (1997). Decision Problems in Algebraic Logic. PhD Thesis, Hungarian Academy of Sciences, Budapest.
53. Ladkin, P. B. & Maddux, R. (1988). The Algebra of Binary Constraint Networks. Technical Report KES.U.88.9, Kestrel Institute, http://www.math. iastate.edu/maddux/papers/tabcn.ps.
54. Ladkin, P. B. & Maddux, R. (1994). On Binary Constraint Problems. Journal of the ACM 41(3): 435-469.
55. Ladkin, P. B. & Maddux, R. D. (1989). On Binary Constraint Networks. Technical report, Kestrel Institute, Palo Alto, CA, USA.
56. Ladkin, P. B. & Reinefeld, A. (1992). Effective Solution of Qualitative Interval Constraint Problems. Artificial Intelligence 57(1): 105-124.
57. Ladkin, P. B. & Reinefeld, A. (1997). Fast Algebraic Methods for Interval Constraint Problems. Annals of Mathematics and Artificial Intelligence 19(3-4): 383-411. URL: citeseer.nj.nec.com/article/ladkin97fast.html
58. Leśniewski, S. (1927-1931). O Podstawach Matematyki. Przeglad Filozoficzny 30-34.
59. Leśniewski, S. (1983). On The Foundation of Mathematics. Topoi 2: 7-52.
60. Li, S. & Ying, M. (2003a). Extensionality of the RCC8 Composition Table. Fundamenta Informaticae 55: 363-385.
61. Li, S. & Ying, M. (2003b). Region Connection Calculus: Its Models and Composition Table. Artificial Intelligence 145: 121-145.
62. Li, Y., Li, S. & Ying, M. (2003). Relational Reasoning in The Region Connection Calculus. (Preprint)
63. Lobachevskij, N. I. (1835). New Principles of Geometry With A Complete Theory of Parallels. Polnoe Sobranie Socinenij 2. (In Russian)
64. Luschei, E. C. (1962). The Logical Systems of Leśniewski. North Holland, Amsterdam.
65. Lyndon, R. C. (1950). The Representation of Relational Algebras. Annals of Mathematics 51(2): 707-729.
66. Lyndon, R. C. (1961). Relation Algebras and Projective Geometries. Michigan Mathematical Journal 8: 21-28.
67. Mackworth, A. K. (1977). Consistency In Networks of Relations. Artificial Intelligence 8(1): 99-118.
68. Maddux, R. (1982). Some Varieties Containing Relation Algebras. Transactions of the American Mathematical Society 272: 501-526.
69. Maddux, R. (1990). Some Algebras and Algorithms for Reasoning About Time and Space, ftp://ftp.math.iastate.edu/pub/maddux/cmps4.ps.
70. Maddux, R. (1991a). Introductory Course on Relation Algebras, Finite-Dimensional Cylindric Algebras, and Their Interconnections. In Andréka, H., Monk, J.D. & Nérneti, I. (eds.). Algebraic Logic, vol. 54 of Colloquia Mathematica Societatis János Bolyai. North Holland: Amsterdam, pp. 361-392.
71. Maddux, R. (1991b). The Origin of Relation Algebras in The Development and Axiomatization of The Calculus of Relations. Studia Logica 50: 421-455.
72. Maddux, R. (1993). Relation Algebras for Reasoning About Time and Space. In Nivat, M., Rattray, C., Rus, T. and Scollo, G. (eds.) Algebraic Methodology and Software Technology (AMAST '93) 27-44. Workshops in Computing, Springer-Verlag, New York, NY.
73. McKenzie, R. (1970). Representations of Integral Relation Algebras. Michigan Mathematical Journal 17: 279-287.
74. Monk, D. (1964). On Representable Relation Algebras. Michigan Mathematical Journal 11: 207-210.
75. Montanari, U. (1974). Networks of Constraints: Fundamental Properties and Applications to Picture Processing. Information Sciences 7: 95-132.
76. Mormann, T. (2001). Holes in The Region Connection Calculus. Preprint, Presented at RelMiCS 6, Oisterwijk, October 2001.
77. Nebel, B. (1995). Computational Properties of Qualitative Spatial Reasoning: First Results, In Wachsmuth, I. Rollinger, C.-R. and Brauer, W. (eds.) KI-95: Advances in Artificial Intelligence, 233-244, Vol. 981 of Lecture Notes in Computer Science, Springer, Heidelberg.
78. Nebel, B. & Bürckert, H.-J. (1995). Reasoning About Temporal Relations: A Maximal Tractable Subclass of Allen's Interval Algebra. Journal of the ACM 42(1): 43-66.
79. Nérneti, I. (1987). Decidability of Relation Algebras With Weakened Associativity. Proceedings of the American Mathematical Society 100: 340-344.
80. Peirce, C. S. (1870). Description of A Notation for The Logic of Relatives, Resulting from An Amplification of The Conceptions of Boole's Calculus of Logic. Memoirs of the American Academy of Sciences 9, 317-378. Reprint by Welch, Bigelow and Co., Cambridge, Mass., 1870, pp. 1-62.
81. Pratt, I. & Schoop, D. (1998). A Complete Axiom System for Polygonal Mereotopology of The Real Plane. Journal of Philosophical Logic 27(6): 621-658.
82. Pratt, I. & Schoop, D. (2000). Expressivity in Polygonal, Plane Mereotopology. Journal of Symbolic Logic 65(2): 822-838.
83. Pratt, V. (1992). Origins of The Calculus of Binary Relations, 7th Annual Symposium on Logic in Computer Science, 248-254. IEEE: Santa Cruz, CA.
84. Randell, D. A., Cohn, A. G. & Cui, Z. (1992). Computing Transitivity tables: A Challenge for Automated Theorem Provers. In Kapur, D. (ed.), Proceedings of the 11th International Conference on Automated Deduction (CADE-11), Vol. 607 of LNAI, 786-790. Springer: Saratoga Springs, NY.
85. Randell, D., Witkowski, M. & Shanahan, M. (2001). From Images to Bodies: Modelling and Exploiting Spatial Occlusion and Motion Parallax. In Nebel, B. (ed.), Proceedings of the seventeenth International Conference on Artificial Intelligence (IJCAI-01), 57-66. Morgan Kaufmann Publishers, Inc.: San Francisco, CA.
86. Renz, J. (1999). Maximal Tractable Fragments of The Region Connection Calculus: A Complete Analysis. In Thomas, D. (ed.), Proceedings of the 16th International Joint Conference on Artificial Intelligence (IJCAI-99- Vol1), 448-455. Morgan Kaufmann Publishers, Inc.: San Francisco, CA.
87. Renz, J. & Nebel, B. (1997). On the Complexity of Qualitative Spatial Reasoning: A Maximal Tractable Fragment of The Region Connection Calculus. IJCAI 97, Proceedings of the 15th International Joint Conference on Artificial Intelligence.
88. Renz, J. & Nebel, B. (1999). On the Complexity of Qualitative Spatial Reasoning: A Maximal Tractable Fragment of The Region Connection Calculus, Artificial Intelligence 108(1-2): 69-123.
89. Schröder, E. (1890-1905). Vorlesungen über die Algebra der Logik, Volumes 1 to 3, Teubner, Leipzig. Reprinted by Chelsea, New York, 1966.
90. Smith, T. P. & Park, K. K. (1992). An Algebraic Approach to Spatial Reasoning, International Journal of Geographical Information Systems 6: 177-192.
91. Surma, S. J., Srzednicki, J. T, Barnett, D. I. and Ricky, V. F. (eds.) (1992). Stanislaw Leśniewski: Collected works.
92. Tarski, A. (1941). On the Calculus of Relations. Journal Symbolic Logic 6: 73-89.
93. Tarski, A. (1955). Contributions to the Theory of Models. III. Indag. Math 17: 56-64.
94. Tarski, A. & Givant, S. (1987). A Formalization of Set Theory Without Variables, Vol. 41 of Colloquium Publications, American Mathematical Society: Providence, RI.
95. Vilain, M., Kautz, H. & van Beek, P. (1986). Constraint Propagation Algorithms For Temporal Reasoning: A Revised Report. Proceedings of the Fifth National Conference on Artificial Intelligence, 377-382. American Association for Artificial Intelligence. AAAI Press: Menlo Park.
96. Whitehead, A. N. (1929). Process and Reality, MacMillan: New York.