# On the desirability of γ-acyclic BCNF database schemes

## Abstract

In this paper, we show that γ-acyclic Boyce-Codd Normal Form database schemes are highly desirable with respect to query processing and updates. We first prove that this class of schemes is bounded with respect to the set of functional dependencies embodied in the database scheme. This result enlarges the class of known bounded database schemes. We then show that this class of schemes is simple in semantics by proving that there is a simple and efficient way to compute the *X*-total projection of the representative instance. As a consequence, answers to many queries for this class of schemes can be computed easily and efficiently. We also show that if a γ-acyclic Boyce-Codd Normal Form database scheme is lossless, then it is connection-trap-free. Finally, we derive a simple and efficient algorithm that determines if an updated state is consistent. This allows the system to enforce the satisfaction of functional dependencies embodied in the database scheme incrementally and efficiently.

## Keywords

Functional Dependency Query Processing Consistent State Database Scheme Connection Point## Preview

Unable to display preview. Download preview PDF.

## References

- [ABU]Aho, A.V., Beeri, C., Ullman, J.D. “The Theory of Joins in Relational Databases.”
*ACM TODS 4*, 3, September 1979, pp. 297–314.Google Scholar - [AtC]Atzeni, P., Chan E.P.F. “Efficient Query Answering in the Representative Instance Approach.”
*Proc. ACM PODS 1985*, pp. 181–188.Google Scholar - [B]Berge, C.
*Graphs and Hypergraphs*. North-Holland, Amsterdam, The Netherlands, 1973.Google Scholar - [BB]Beeri, C., Bernstein, P.A. “Computational Problems Related to the Design of Normal Form Relational Schemas.”
*ACM TODS 4*, 1, March 1979, pp. 30–59.Google Scholar - [BBC]Bernstein, P.A., Blaustein, B.T., Clarke, E.M. “Fast Maintenance of Semantic Integrity Assertions using Redundant Aggregate Data.”
*Proc. VLDB 1980*, pp. 126–136.Google Scholar - [BBG]Beeri, C., Bernstein, P.A., and Goodman, N. “A Sophisticate's Introduction to Database Normalization Theory.”
*Proc. VLDB 1978*, pp. 113–124.Google Scholar - [BDB]Biskup, J., Dayal, U., and Bernstein, P.A. “Synthesizing Independent Database Schemas.”
*Proc. ACM-SIGMOD 1979*, Boston, Ma., pp. 143–151.Google Scholar - [BG]Bernstein, P.A., Goodman, N. “What Does Boyce-Codd Normal Form Do?”
*Proc. VLDB 1980*, pp. 245–259.Google Scholar - [BV]Beeri, C., Vardi, M.Y. “A Proof Procedure for Data Dependencies.”
*JACM 31*, 4, October 1984, pp. 718–741.Google Scholar - [C1]Chan, E.P.F. “Optimal Computation of Total Projections with Unions of Simple Chase Join Expressions.”
*Proc. ACM-SIGMOD 1984*, Boston, Ma., pp. 149–163.Google Scholar - [C2]Chan, E.P.F. “An Incremental Approach to Testing Satisfaction of Functional Dependencies.” Unpublished manuscript, University of Toronto, 1981.Google Scholar
- [CA]Chan, E.P.F., Atzeni, P. “On the Properties and Characterization of Connection-trap-free Schemes.”
*Proc. PODS 1986*, pp. 140–147.Google Scholar - [CH]Chan, E.P.F., Hernańdez, H.J. “On the Desirability of γ-acyclic BCNF Database Schemes.” Submitted to
*Theoretical Computer Science*.Google Scholar - [CM]Chan, E.P.F., Mendelzon, A.O. “Answering Queries on the Embedded-complete Database Schemes.” Unpublished manuscript, 1984.Google Scholar
- [Co]Codd, E.F. “A Relational Model for Large Shared Data Banks.”
*CACM 13*, 6 (June 1970), pp. 377–387.Google Scholar - [Da]Date, C.J.
*An Introduction to Database Systems*. 3rd. edition, Reading, Ma., Addison-Wesley, 1981.Google Scholar - [DM]D'Atri, A., Moscarini, M. “Acyclic Hypergraphs: Their Recognition and Top-down vs. Bottom-up Generation.”
*IASI-CNR, R.29*, Rome, Italy, 1982.Google Scholar - [F]Fagin, R. “Hypergraphs and Relational Database Schemes.”
*JACM 30*, 3, July 1983, pp. 514–550.Google Scholar - [GM]Graham, M.H., Mendelzon, A.O. “On Total Projections Computable by Relational Algebra.” Unpublished manuscript, Department of Computer Science, University of Toronto, August 1982.Google Scholar
- [GMV]Graham, M.H., Mendelzon, A.O., and Vardi, M.Y. “Notions of Dependency Satisfaction.”
*JACM 33*, 1, January 1986, pp. 105–129.Google Scholar - [GY]Graham, M.H., Yannakakis, M. "Independent Database Schemas."
*JCSS*28, pp. 121–141 (1984).Google Scholar - [H1]Honeyman, P. "Extension Joins."
*Proc. VLDB 1980*, pp. 239–244.Google Scholar - [H2]Honeyman, P. "Testing Satisfaction of Functional Dependencies."
*JACM 29*, 3, July 1982, pp. 668–677.Google Scholar - [IIK]Ito, M., Iwasaki, M., Kasami, T. "Some Results on the Representative Instance in Relational Databases."
*SIAM J. of Computing 14*, 2 (1985), pp. 334–354.Google Scholar - [LeP]LeDoux, C.H., Parker, D.S. "Reflections on Boyce-Codd Normal Form."
*Proc. VLDB 1982*, pp. 131–141.Google Scholar - [LO]Lucchesi, C.L., Osborn, S.L. "Candidate Keys for Relations."
*JCSS 17*, 2, October 1978, pp. 270–279.Google Scholar - [M]Mendelzon, A.O. "Database States and their Tableaux."
*ACM TODS 9*, 2, June 1984, pp. 264–282.Google Scholar - [Ma]Maier, D.
*The Theory of Relational Databases*. Computer Science Press, 1983.Google Scholar - [MMS]Maier, D., Mendelzon, A.O., and Sagiv, Y. "Testing Implications of Data Dependencies."
*ACM TODS 4*, 4, December 1979, pp. 455–469.Google Scholar - [MRW]Maier, D., Rozenshtein, D., Warren, D.S. "Windows On the World."
*Proc. ACM-SIGMOD 1983*, pp. 68–78.Google Scholar - [MUV]Maier, D., Ullman, J.D., and Vardi, M.Y. "On the Foundations of the Universal Relation Model."
*ACM TODS 9*, 2, June 1984, pp. 283–308.Google Scholar - [NG]Nicholas, J-M, Gallaire, H. "Data Bases: Theory vs Interpretation."
*Logic and Data Bases*, Plenum Press, pp. 33–54.Google Scholar - [O]Osborn, S.L. "Testing for Existence of a Covering Boyce-Codd Normal Form."
*IPL 8*, 1, January 1979, pp. 11–14.Google Scholar - [S1]Sagiv, Y. "Can We Use the Universal Instance Assumption Without Using Nulls?"
*Proc. ACM-SIGMOD 1981*, pp. 108–120.Google Scholar - [S2]Sagiv, Y. "A Characterization of Globally Consistent Databases and their Correct Access Paths."
*ACM TODS 8*, 2, June 1983, pp. 266–286.Google Scholar - [S3]Sagiv, Y. "Evaluation of Queries in Independent Database Schemes." Unpublished manuscript, 1984.Google Scholar
- [St]Stonebraker, M.R. "Implementation of Integrity Constraints and Views by Query Modification."
*Proc. ACM-SIGMOD 1975*, pp. 65–78.Google Scholar - [U]Ullman, J.D.
*Principles of Database Systems*. 2nd. edition, Computer Science Press, 1982.Google Scholar - [V]Vassiliou, Y. "A Formal Treatment of Imperfect Information in Data Management."
*CSRG TR-123*, University of Toronto, Nov. 1980.Google Scholar - [Y]Yannakakis, M. "Algorithms for Acyclic Database Schemes."
*Proc. VLDB 1981*, pp. 82–94.Google Scholar