Abstract
We consider two-dimensional spatial databases defined in terms of polynomial inequalities and investigate the expressibility of the topological connectivity query for these databases in spatial Datalog. In [10], a spatial Datalog program for piecewise linear connectivity was given and proved to correctly test the connectivity of linear spatial databases. In particular, the program was proved to terminate on these inputs. Here, we generalize this result and give a program that correctly tests connectivity of spatial databases definable by a quantifier-free formula in which at most quadratic polynomials appear. We also show that a further generalization of our approach to spatial databases that are only definable in terms of polynomials of higher degree is impossible. The class of spatial databases that can be defined by a quantifier-free formula in which at most quadratic polynomials appear is shown to be decidable. Finally, we give a number of possible other approaches to attack the problem of expressing the connectivity query for arbitrary two-dimensional spatial databases in spatial Datalog.
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References
D.S. Arnon. Geometric reasoning with logic and algebra. Artificial Intelligence, 37:37–60, 1988.
M. Benedikt, G. Dong, L. Libkin, and L. Wong. Relational expressive power of constraint query languages. In Proceedings 15th ACM Symposium on Principles of Database Systems. ACM Press, 1996.
J. Bochnak, M. Coste, and M.-F. Roy. Géométrie algébrique réelle. Springer-Verlag, 1987.
G.E. Collins. Quantifier elimination for real closed fields by cylindrical algebraic decomposition. Lecture Notes in Computer Science, 33:134–183, 1975.
M. Coste. Ensembles semi-algébriques. In Géometrie algébrique réelle et formes quadratiques, volume 959 of Lecture Notes in Mathematics, pages 109–138. Springer, 1982.
W. Fulton Algebraic curves: an introduction to algebraic geometry New York, 1969.
S. Grumbach and J. Su. First-order definability over constraint databases. In Montanari and Rossi [12], pages 121–136.
M. Gyssens, J. Van den Bussche, and D. Van Gucht. Complete geometrical query languages. Manuscript, 1996.
P.C. Kanellakis, G.M. Kuper, and P.Z. Revesz. Constraint query languages. Journal of Computer and System Sciences, 51(1):26–52, August 1995.
B. Kuijpers, J. Paredaens, M. Smits, and J. Van den Bussche Termination properties of spatial datalog programs. Proceedings of the International Workshop Logic in Databases (LID'96), (Eds. D. Pedreschi, C. Zaniolo) Lecture Notes in Computer Science, 1154, October 1996.
E.E. Moise. Geometric topology in dimensions 2 and 3, volume 47 of Graduate Texts in Mathematics. Springer, 1977.
U. Montanari and F. Rossi, editors. Principles and practice of constraint programming, volume 976 of Lecture Notes in Computer Science. Springer, 1995.
J. Paredaens, J. Van den Bussche, and D. Van Gucht. Towards a theory of spatial database queries. In Proceedings 13th ACM Symposium on Principles of Database Systems, pages 279–288. ACM Press, 1994.
J. Renegar. On the computational complexity and geometry of the first-order theory of the reals. Journal of Symbolic Computation, 13, 1992.
J.T. Schwartz and M. Sharir. On the piano movers' problem II. In J.T. Schwartz, M. Sharir, and J. Hopcroft, editors, Planning, Geometry, and Complexity of Robot Motion, pages 51–96. Ablex Publishing Corporation, Norwood, New Jersey, 1987.
A. Tarski. A Decision Method for Elementary Algebra and Geometry. University of California Press, 1951.
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© 1996 Springer-Verlag Berlin Heidelberg
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Kuijpers, B., Smits, M. (1996). On expressing topological connectivity in spatial Datalog. In: Gaede, V., Brodsky, A., Günther, O., Srivastava, D., Vianu, V., Wallace, M. (eds) Constraint Databases and Applications. CDB 1997. Lecture Notes in Computer Science, vol 1191. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-62501-1_29
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DOI: https://doi.org/10.1007/3-540-62501-1_29
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