# On query languages for linear queries definable with polynomial constraints

## Abstract

It has been argued that the linear database model, in which semi-linear sets are the only geometric objects, is very suitable for most spatial database applications. For querying linear databases, the language FO + linear has been proposed. We present both negative and positive results regarding the expressiveness of FO+linear. First, we show that the dimension query is definable in FO + linear, which allows us to solve several interesting queries. Next, we show the non-definability of a whole class of queries that are related to sets not definable in FO+linear. This result both sharpens and generalizes earlier results independently found by Afrati et al. and the present authors, and demonstrates the need for more expressive linear query languages if we want to sustain the desirability of the linear database model. In this paper, we show how FO + linear can be strictly extended within FO + poly in a safe way. Whether any of the proposed extensions is complete for the linear queries definable in FO + poly remains open. We do show, however, that it is undecidable whether an expression in FO + poly induces a linear query.

## Preview

Unable to display preview. Download preview PDF.

### References

- 1.F. Afrati, T. Andronikos, T.G. Kavalieros, “On the Expressiveness of First-Order Constraint Languages,” in Proceedings
*ESPRIT WG CONTESSA Workshop*, (Friedrichshafen, Germany), G. Kuper and M. Wallace, eds.,*Lecture Notes in Computer Science*, vol. 1034, Springer-Verlag, Berlin, 1996, pp. 22–39.Google Scholar - 2.F. Afrati, S. Cosmadakis, S. Grumbach, and G. Kuper, “Linear Versus Polynomial Constraints in Database Query Languages,” in Proceedings
*2nd Int'l Workshop on Principles and Practice of Constraint Programming*(Rosario, WA), A. Borning, ed.,*Lecture Notes in Computer Science*, vol. 874, Springer-Verlag, Berlin, 1994, pp. 181–192.Google Scholar - 3.A. Brodsky and Y. Kornatzky, “The LyriC Language: Querying Constraint Objects,” in Proceedings
*Post-ILPS'94 Workshop on Constraints and Databases*(Ithaca, NY), 1994.Google Scholar - 4.I. Carlbom, “An Algorithm for Geometric Set Operations Using Cellular Subdivision Techniques,”
*IEEE Computer Graphics and Applications*, 7:5, 1987, pp. 44–55.Google Scholar - 5.E. Clementini, P. Di Felice, and P. van Oosterom, “A Small Set of Formal Topological Relationships Suitable for End-User Interaction,” in Proceedings
*3nd Symposium on Advances in Spatial Databases*,*Lecture Notes in Computer Science*, vol. 692. Springer-Verlag, Berlin, 1993, pp. 277–295.Google Scholar - 6.J. Nievergelt and M. Freeston, eds., Special issue on spatial data,
*Computer Journal*, 37:1, 1994.Google Scholar - 7.M.J. Egenhofer, “A Formal Definition of Binary Topological Relationships,” in Proceedings
*Foundations of Data Organization and Algorithms*, W. Litwin and H.-J. Schek, eds.,*Lecture Notes in Computer Science*, vol. 367, Springer-Verlag, Berlin, 1989, pp. 457–472.Google Scholar - 8.M.J. Egenhofer, and J. Herring, “A Mathematical Framework for the Definition of Topological Relationships,” in Proceedings
*Fourth International Symposium on Spatial Data Handling*, K. Brassel and H. Kishimoto, eds., Zurich, Switzerland, 1990, pp. 803–813.Google Scholar - 9.M.J. Egenhofer, “Reasoning about Binary Topological Relations,” in Proceedings
*Advances in Spatial Databases*, O. Günther and H.-J. Schek, eds.,*Lecture Notes in Computer Science*, vol. 525, Springer-Verlag, Berlin, 1991, pp. 143–160.Google Scholar - 10.M.J. Egenhofer, “What's Special about Spatial? Database Requirements for Vehicle Navigation in Geographic Space,”
*SIGMOD Records*, 22:2, 1993, pp. 398–402.Google Scholar - 11.O. Günther, ed.,
*Efficient Structures for Geometric Data Management*, in*Lecture Notes in Computer Science*, vol. 337, Springer-Verlag, Berlin, 1988.Google Scholar - 12.O. Günther, and A. Buchmann, “Research Issues in Spatial Databases,”
*SIGMOD Records*, 19:4, 1990, pp. 61–68.Google Scholar - 13.R.H. Güting, “Geo-Relational Algebra: A Model and Query Language for Geometric Database Systems,” in
*Advances in Database Technology-EDBT '88*, Proceedings*Int'l Conf. on Extending Database Technology*(Venice, Italy), J.W. Schmidt, S. Ceri, and M. Missikoff, eds.,*Lecture Notes in Computer Science*, vol. 303, Springer-Verlag, Berlin, 1988, pp. 506–527.Google Scholar - 14.R.H. Güting, “Gral: An Extensible Relational Database System for Geometric Applications,” in Proceedings
*15th Int'l Conf. on Very Large Databases*(Amsterdam, the Netherlands), 1989, pp. 33–34.Google Scholar - 15.R.H. Güting, “An Introduction to Spatial Database Systems,”
*VLDB-Journal*, 3:4, 1994, pp. 357–399.Google Scholar - 16.R.H. Güting, “Implementations of the ROSE Algebra: Efficient Algorithms for Real-Based Spatial Data Types,” in Proceedings
*Advances in Spatial Databases*, M. Egenhofer and J. Herring, eds.,*Lecture Notes in Computer Science*, vol. 951, Springer-Verlag, Berlin, 1995, pp. 216–239.Google Scholar - 17.T. Huynh, C. Lassez, and J.-L. Lassez. Fourier Algorithm Revisited. In Proceedings
*2nd Int'l Conf. on Algebraic an Logic Programming*, H. Kirchner and W. Wechler, eds.*Lecture Notes in Computer Science*, vol. 463. Springer Verlag, Berlin, 1990, pp. 117–131.Google Scholar - 18.P.C. Kanellakis and D.Q. Goldin, “Constraint Programming and Database Query Languages,” in Proceedings
*2nd Conf. on Theoretical Aspects of Computer Software*, M. Hagiya and J.C. Mitchell, eds.,*Lecture Notes in Computer Science*, vol. 789, Springer-Verlag, Berlin, 1994.Google Scholar - 19.A. Kemper, and M. Wallrath, “An Analysis of Geometric Modeling in Database Systems,”
*Computing Surveys*, 19:1, 1987, pp. 47–91.Google Scholar - 20.P.C. Kanellakis, G.M. Kuper and P.Z. Revesz, “Constraint Query Languages,”
*Journal of Computer and System Sciences*, to appear, also in Proceedings*9th ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems*(Nashville, TN), 1990, pp. 299–313.Google Scholar - 21.A. Tarski, ”A Decision Method for Elementary Algebra and Geometry,” University of California Press, Berkeley, California, 1951.Google Scholar
- 22.J. Paredaens, J. Van den Bussche, and D. Van Gucht, “Towards a Theory of Spatial Database Queries,” in Proceedings
*13th ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems*(Minneapolis, MN), 1994. pp. 279–288.Google Scholar - 23.N. Pissinou, R. Snodgrass, R. Elmasri, I. Mumick, T. Özsu, B. Pernici, A. Segef. B. Theodoulidis, and U. Dayal, “Towards an Infrastructure for Temporal Databases,”
*SIGMOD Records*, 23:1, 1994, pp. 35–51.Google Scholar - 24.L.K. Putnam and P.A. Subrahmanyam, “Boolean Operations on
*n*-Dimensional Objects,”*IEEE Computer Graphics and Applications*, 6:6, 1986, pp. 43–51.Google Scholar - 25.J. D. Ullman, “Principles of Database and Knowledge-base Systems,” Computer Science Press, 1988.Google Scholar
- 26.L. Vandeurzen, M. Gyssens, and D. Van Gucht, “On the Desirability and Limitations of Linear Spatial Query Languages,” in Proceedings
*4th Symposium on Advances in Spatial Databases*, M. J. Egenhofer and J.R. Herring, eds.*Lecture Notes in Computer Science*, vol. 951, Springer Verlag, Berlin, 1995, pp. 14–28.Google Scholar