Approximate Keys and Functional Dependencies in Incomplete Databases With Limited Domains-Algorithmic Perspective (2402.05161v1)
Abstract: A possible world of an incomplete database table is obtained by imputing values from the attributes (infinite) domain to the place of \texttt{NULL} s. A table satisfies a possible key or possible functional dependency constraint if there exists a possible world of the table that satisfies the given key or functional dependency constraint. A certain key or functional dependency is satisfied by a table if all of its possible worlds satisfy the constraint. Recently, an intermediate concept was introduced. A strongly possible key or functional dependency is satisfied by a table if there exists a strongly possible world that satisfies the key or functional dependency. A strongly possible world is obtained by imputing values from the active domain of the attributes, that is from the values appearing in the table. In the present paper, we study approximation measures of strongly possible keys and FDs. Measure $g_3$ is the ratio of the minimum number of tuples to be removed in order that the remaining table satisfies the constraint. We introduce a new measure $g_5$, the ratio of the minimum number of tuples to be added to the table so the result satisfies the constraint. $g_5$ is meaningful because the addition of tuples may extend the active domains. We prove that if $g_5$ can be defined for a table and a constraint, then the $g_3$ value is always an upper bound of the $g_5$ value. However, the two measures are independent of each other in the sense that for any rational number $0\le\frac{p}{q}<1$ there are tables of an arbitrarily large number of rows and a constant number of columns that satisfy $g_3-g_5=\frac{p}{q}$. A possible world is obtained usually by adding many new values not occurring in the table before. The measure $g_5$ measures the smallest possible distortion of the active domains. We study complexity of determining these approximate measures.
- M. Al-Atar and A. Sali. Approximate keys and functional dependencies in incomplete databases with limited domains. In Foundations of Information and Knowledge Systems 12th International Symposium, FoIKS 2022 Helsinki, Finland, June 20–23, 2022 Proceedings, volume 13388 of LNCS, pages 147–167. Springer Nature Switzerland AG, 2022.
- M. Al-Atar and A. Sali. Strongly possible functional dependencies for sql. Acta Cybernetica, 2022.
- M. Alattar and A. Sali. Keys in relational databases with nulls and bounded domains. In European Conference on Advances in Databases and Information Systems, pages 33–50. Springer, 2019.
- M. Alattar and A. Sali. Functional dependencies in incomplete databases with limited domains. In International Symposium on Foundations of Information and Knowledge Systems, pages 1–21. Springer, 2020.
- M. Alattar and A. Sali. Strongly possible keys for sql. Journal on Data Semantics, 9(2):85–99, 2020.
- L. Bertossi. Database repairs and consistent query answering: Origins and further developments. In Proceedings of the 38th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems, pages 48–58, 2019.
- J. Biskup and L. Wiese. A sound and complete model-generation procedure for consistent and confidentiality-preserving databases. Theoretical Computer Science, 412(31):4044–4072, 2011.
- A novel framework for imputation of missing values in databases. IEEE Transactions on Systems, Man, and Cybernetics-Part A: Systems and Humans, 37(5):692–709, 2007.
- C. Giannella and E. Robertson. On approximation measures for functional dependencies. Information Systems, 29(6):483–507, 2004.
- Measures of association for cross classifications. Measures of association for cross classifications, pages 2–34, 1979.
- J. Kivinen and H. Mannila. Approximate inference of functional dependencies from relations. Theoretical Computer Science, 149(1):129–149, 1995.
- Possible and certain keys for sql. The VLDB Journal, 25(4):571–596, 2016.
- W. Lipski Jr. On databases with incomplete information. Journal of the ACM (JACM), 28(1):41–70, 1981.
- L. Lovász and M. Plummer. Matching theory, volume 367. American Mathematical Soc., 2009.
- J. Wijsen. Foundations of query answering on inconsistent databases. ACM SIGMOD Record, 48(3):6–16, 2019.