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An Extended Algebra for Constraint Databases
September/October 1998 (vol. 10 no. 5)
pp. 686-705

Abstract—Constraint relational databases use constraints to both model and query data. A constraint relation contains a finite set of generalized tuples. Each generalized tuple is represented by a conjunction of constraints on a given logical theory and, depending on the logical theory and the specific conjunction of constraints, it may possibly represent an infinite set of relational tuples. For their characteristics, constraint databases are well suited to model multidimensional and structured data, like spatial and temporal data. The definition of an algebra for constraint relational databases is important in order to make constraint databases a practical technology. In this paper, we extend the previously defined constraint algebra (called generalized relational algebra). First, we show that the relational model is not the only possible semantic reference model for constraint relational databases and we show how constraint relations can be interpreted under the nested relational model. Then, we introduce two distinct classes of constraint algebras, one based on the relational algebra, and one based on the nested relational algebra, and we present an algebra of the latter type. The algebra is proved equivalent to the generalized relational algebra when input relations are modified by introducing generalized tuple identifiers. However, from a user point of view, it is more suitable. Thus, the difference existing between such algebras is similar to the difference existing between the relational algebra and the nested relational algebra, dealing with only one level of nesting. We also show how external functions can be added to the proposed algebra.

[1] S. Abiteboul, R. Hull, and V. Vianu, Foundations of Databases. Addison-Wesley, 1995.
[2] S. Abiteboul and P. Kanellakis, "Query Languages for Complex Object Databases," SIGACT News, vol. 21, no. 3, pp. 9-18, 1990.
[3] J.L. Balcazar, J. Diaz, and J. Gabarro, Structural Complexity II, Springer-Verlag, 1989.
[4] A. Belussi, E. Bertino, and B. Catania, "Manipulating Spatial Data in Constraint Databases," Proc. Fifth Int'l Symp. Spatial Databases, Lecture Notes in Computer Science 1,262, pp. 115-141, 1997.
[5] A. Belussi, E. Bertino, and B. Catania, "Introducing External Functions in Constraint Query Languages," Proc. Fourth Int'l Conf. Principles and Practice of Constraint Programming, to appear, 1998.
[6] A. Belussi, E. Bertino, and B. Catania, "New Algebras for Constraint Relational Databases," Technical Report 211-298, Univ. of Milan, Italy, 1998.
[7] E. Bertino, B. Catania, and B. Shidlovsky, "Towards Optimal Two-Dimensional Indexing for Constraint Databases," Information Processing Letters, vol. 64, no. 1, pp. 1-8, 1997.
[8] A. Brodsky and Y. Kornatzky, The LyriC Language: Constraining Objects Proc. ACM SIGMOD Int'l Conf. Management of Data, 1995.
[9] J. Byon and P.Z. Revesz, "DISCO: A Constraint Database System with Sets," Proc. First Int'l CONTESSA Database Workshop: Constraint Databases and their Applications, Lecture Notes in Computer Science 1,034, pp. 68-83, 1995.
[10] A.K. Chandra and D. Harel, "Computable Queries for Relational Data Bases," J. Computer and System Sciences, vol. 21, no. 2, pp. 156-178, 1980.
[11] C.C. Chang and H.J. Keisler, Model Theory, North-Holland, 1973.
[12] J. Chomicki, D. Goldin, and G. Kuper, Variable Independence and Aggregation Closure Proc. ACM Symp. Principles of Database Systems, June 1996.
[13] J. Chomicki and G. Kuper, Measuring Infinite Relations Proc. ACM Symp. Principles of Database Systems, 1995.
[14] L. De Floriani, P. Marzano, and E. Puppo, "Spatial Queries and Data Models," Spatial Information Theory: A Theoretical Basis for GIS, Lecture Notes in Computer Science 716, pp. 123-138, 1993.
[15] M. Gargano, E. Nardelli, and M. Talamo, "Abstract Data Types for the Logical Modeling of Complex Data," Information Systems, vol. 16, no. 6, pp. 565-583, 1991.
[16] D.G. Goldin and P.C. Kanellakis, "Constraint Query Algebras," Constraints J., vol. 1, nos. 1-2, pp. 45-83, 1996.
[17] S. Grumbach and J. Su, "Dense-Order Constraint Databases," Proc. 14th ACM SIGACT-SIGMOD-SIGART Symp. Principles of Database Systems, pp. 66-77, 1995.
[18] R.H. Gueting and M. Schneider, "Realm-Based Spatial Data Types: The ROSE Algebra," VLDB J., vol. 4, no. 2, pp. 243-286, 1995.
[19] A. Guttman, “R-Trees: A Dynamic Index Structure for Spatial Searching,” Proc. ACM SIGMOD Conf. Management of Data, 1984.
[20] M.R. Hansen, B.S. Hansen, P. Lucas, and P. van Emde Boas, "Integrating Relational Databases and Constraint Languages," Computer Languages, vol. 14, no. 2, pp. 63-82, 1989.
[21] L. Hermosilla and G. Kuper, "Towards the Definition of a Spa-tial Object-Oriented Data Model with Constraints," Proc. First Int'l CONTESSA Database Workshop: Constraint Databases and Their Applications, Lecture Notes in Computer Science 1,034, pp. 120-131, 1995.
[22] P.C. Kanellakis, "Elements of Relational Database Theory," Handbook of Theoretical Computer Science, J. van Leeuwen, ed., chapter 17, Elsevier Science, 1990.
[23] P.C. Kanellakis and D. Goldin, "Constraint Programming and Database Query Languages," Proc. Int'l Symp. Theoretical Aspects of Computer Software, Lecture Notes in Computer Science 789, pp. 96-120, 1994.
[24] P.C. Kanellakis, G.M. Kuper, and P.Z. Revesz, Constraint Query Languages J. Computer and System Sciences, vol. 51, no. 1, pp. 26-52, 1995.
[25] M. Koubarakis, "Representation and Querying in Temporal Databases: The Power of Temporal Constraints,'' Proc. Int'l Conf. Data Eng., pp. 327-334, 1993.
[26] G.M. Kuper, "Aggregation in Constraint Databases," Proc. First Int'l Workshop Principles and Practice of Constraint Programming, pp. 166-175, 1993.
[27] J.L. Lassez, Querying Constraints Proc. Ninth ACM Symp. Principles of Database Systems, pp. 288-298, 1990.
[28] L. Libkin and L. Wong, "Conservativity of Nested Relational Calculi with Internal Generic Functions," Information Processing Letters, vol. 49, no. 6, pp. 272-280, 1994.
[29] S. Marcus and V.S. Subrahmanian, “Foundations of Multimedia Database Systems,” J. ACM, vol. 43, no. 3, pp. 474-523, 1996.
[30] J. Paredaens, "Spatial Databases, The Final Frontier," Proc. Fifth Int'l Conf. Database Theory, Lecture Notes in Computer Science 893, pp. 14-31, 1995.
[31] J. Paredaens, J. Van den Bussche, and D. Van Gucht, Towards a Theory of Spatial Database Queries Proc. ACM Symp. Principles of Database Systems, pp. 279-288, 1994.
[32] J. Paredaens and D. Van Gucht, "Possibilities and Limitations of Using Flat Operators in Nested Algebra Expressions," Proc. Seventh ACM SIGACT-SIGMOD-SIGART Symp. Principles of Database Systems, pp. 29-38, 1988.
[33] P. Revesz, "Datalog Queries of Set Constraint Databases," Proc. Fifth Int'l Conf. Database Theory, Lecture Notes in Computer Science 893, pp. 424-438, 1995.
[34] M. Scholl and A. Voisard, "Thematic Map Modeling," Proc. Symp. Design and Implementation of Large Spatial Databases, pp. 167-190, 1989.
[35] T. Sellis, N. Roussopoulos, and C. Faloutsos, “The R+-Tree: A Dynamic Index for Multidimensional Objects,” Proc. 13th Int'l Conf. Very Large Data Bases (VLDB), 1987.
[36] D. Suciu, "Bounded Fixpoints for Complex Objects," Proc. Int'l Workshop Database Programming Languages, pp. 263-281, 1994.
[37] D. Suciu, "Domain-Independent Queries on Databases with External Functions," Proc. Fifth Int'l Conf. Database Theory, Lecture Notes in Computer Science 893, pp. 177-190, 1995.
[38] P. Svensson, "GEO-SAL: A Query Language for Spatial Data Analysis," Proc. Int'l Symp. Advances in Spatial Databases, Lecture Notes in Computer Science 525, pp. 119-140, 1991.
[39] J. Van den Bussche, "Complex Object Manipulation Through Identifiers—An Algebraic Perspective," Technical Report 92-41, Univ. of Antwerp, Belgium, 1992.
[40] L. Vandeurzen, M. Gyssens, and D. Van Gucht, On the Desirability and Limitations of Linear Spatial Database Models Proc. Int'l Symp. Large Spatial Databases, pp. 14-28, 1995.
[41] L. Wong, "Normal Forms and Conservative Properties for Query Languages Over Collection Types," J. Computer and System Sciences, vol. 523, no. 3, pp. 495-505, 1996.
[42] A. Brodsky, J. Jaffar, and M.J. Maher, Towards Practical Constraint Databases Proc. Int'l Conf. Very Large Data Bases, 1993.

Index Terms:
Constraints, generalized relations, relational algebra, nested relational algebra, external functions.
Citation:
Alberto Belussi, Elisa Bertino, Barbara Catania, "An Extended Algebra for Constraint Databases," IEEE Transactions on Knowledge and Data Engineering, vol. 10, no. 5, pp. 686-705, Sept.-Oct. 1998, doi:10.1109/69.729722
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