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On Satisfiability, Equivalence, and Implication Problems Involving Conjunctive Queries in Database Systems
August 1996 (vol. 8 no. 4)
pp. 604-616

Abstract—Satisfiability, equivalence, and implication problems involving conjunctive queries are important and widely encountered problems in database management systems. These problems need to be efficiently and effectively solved. In this paper, we consider queries which are conjunctions of the inequalities of the form (XopC), (XopY), and/or (XopY + C), where X and Y are two attributes, C is a constant, and op∈ {< , ≤ , =, ≠, >, ≥}. These types of inequalities are widely used in database systems, since the first type is a selection, the second type is a θ-join, and the third type is a very popular clause in a deductive database system. The satisfiability, equivalence, and implication problems in the integer domain (for attributes and constants) have been shown to be NP-hard [20], [24]. However, we show that these problems can be solved efficiently in the real domain. The incorporation of the real domain is significant, because the real domain is practically and widely used in a database. Necessary and sufficient conditions and algorithms are presented. A novel concept of the "modulo closure" and a set of sound and complete axioms with respect to the "modulo closure" are also proposed to infer all correct and necessary inequalities from a given query. The proposed axioms generalize Ullman's axioms [27] where queries only consist of θ-joins.

[1] A. Aho, J. Hopcroft, and J. Ullman, Data Structures and Algorithms.Reading, Mass: Addison-Wesley, 1983.
[2] A.V. Aho, Y. Sagiv, and J.D. Ullman, Efficient Optimization of a Class of Relational Expressions ACM Trans. Database Systems, vol. 4, no. 4, pp. 435-454, Dec. 1979.
[3] J.A. Blakeley, N. Coburn, and P.A. Larson, "Updating Derived Relations: Detecting Irrelevant and Autonomously Computable Updates, Proc. Fifth VLDB, Morgan Kaufmann, 1986, pp. 457-466.
[4] J.A. Blakeley, P. Larson, and F.W. Tompa, "Efficiently Updating Materialized Views," Proc. ACM SIGMOD Conf.,Washington, D.C., pp. 61-71, May 1986.
[5] U.S. Chakravarthy, J. Grant, and J. Minker, “Logic-based Approach to Semantic Query Optimization,” ACM Trans. Database Systems, vol. 15, no. 2, pp. 162–207, 1990.
[6] T.H. Cormen,C.E. Leiserson, and R.L. Rivest,Introduction to Algorithms.Cambridge, Mass.: MIT Press/McGraw-Hill, 1990.
[7] R.W. Floyd, “Algorithm 97: Shortest Path,” Comm. ACM, vol. 5, no. 6, pp. 345, 1962.
[8] S. Guo, W. Sun, and M.A. Weiss, "On Satisfiability, Equivalience, and Implication Problems Involving Conjunctive Inequalities of the Form (x op c) and/or (x op y + c) in Database Systems," Technical Report, FIU, Miami, Fla., Oct. 1992.
[9] J. Jaffar and J.L. Lassez, "Constraint Logic Programming," Proc. ACM Principles of Programming Languages, pp. 111-119, 1987.
[10] J. Jaffar, S. Michaylov, P. Stuckey, and R. Yap, "The${\rm CLP}(\Re)$Language and System," ACM Trans. Programming Languages and Systems, 1992.
[11] M. Jarke, "Common Subexpression Isolation in Multiple Query Optimization," W. Kim., D. Reiner, and D. Batory, eds., Query Processing in Database Systems.New York: Springer-Verlag, 1984.
[12] D. Johnson and A. Klug, "Optimizing Conjunctive Queries that Contain Untyped Variables," SIAM J. Computing, vol. 12, no. 4, pp. 616-640, Nov. 1983.
[13] D. Johnson and A. Klug, "Testing Containment of Conjunctive Queries under Functional and Inclusion Dependencies, J. Computing Systems Science, vol. 28, no. 1, pp. 167-189, Feb. 1984.
[14] P. Kanellakis, G. Kuper, and P. Revesz, "Constraint Query Languages, Proc. Ninth ACM Symp. Principles of Database Systems, pp. 299-313, 1990.
[15] W. Kim, "Global Optimization of Relational Queries: A First Step," W. Kim, D. Reiner, and D. Batory, eds., Query Proccessing in Database Systems,New York: Springer-Verlag, 1984.
[16] J.J. King, Query Optimization by Semantic Reasoning.Ann Arbor, Mich.: UMI Research Press, 1984.
[17] A. Klug, "On Conjunctive Queries Containing Inequalities," J. ACM vol. 35, no. 1, pp. 146-160, Jan. 1988.
[18] D. Maier and J.D. Ullman, "Fragments of Relations," Proc. ACM SIGMOD Int'l Conf. Management of Data, 1983.
[19] V.R. Pratt, "Two Easy Theories Whose Combination is Hard," technical Report, Cambridge, Mass.: MIT, Sept. 1977.
[20] D.J. Rosenkrantz and H.B. Hunt III, "Processing Conjunctive Predicates and Queries," Proc. Sixth Int'l Conf. Very Large Databases, pp. 64-72, 1980.
[21] T. Sellis,“Global query optimization,” Proc. SIGMOD, pp. 191-205, 1986.
[22] W. Sun and M.A. Weiss, "An Efficient Algorithm for Testing Implication Involving Arithmetic Inequalities," IEEE Trans. Knowledge and Data Engineering, vol. 6, no. 5, Oct. 1994.
[23] W. Sun and C.T. Yu, “Semantic Query Optimization for Tree and Chain Queries,” IEEE Trans. Knowledge and Data Eng., vol. 6, no. 1, pp. 136–151, 1994.
[24] X. Sun, N.N. Kamell, and L.M. Ni, "Processing Implication on Queries," IEEE Trans. Software Engineering, vol. 15, no. 10, pp. 1,168-1,175, Oct. 1989.
[25] R. Tarjan, "Depth-First Search and Linear Graph Algorithms," SIAM J. Computing, vol. 1, no. 2, pp. 146-160, June 1972.
[26] R. Tarjan, "Data Structures and Network Algorithms," SIAM,Philadelphia, Penn., 1983.
[27] J.D. Ullman, Principles of Database and Knowledge-Base Systems, vol. II: The New Tech nologies. New York: Computer Science Press, 1989.
[28] J. Ullman, Principles of Database and Knowledge-Base Systems, vol. 1. Computer Science Press, 1988.
[29] J.D. Ullman and A. Van Gelder, "Efficient Tests for Top-Down Termination of Logical Rules," J. ACM, vol. 35, no. 2, pp. 345-373, Apr. 1988.
[30] M.A. Weiss, Data Structures and Algorithm Analysis. Benjamin/Cummings, 1992.
[31] E. Wong and K. Youssef, "Decomposition—A Strategy for Query Processing," ACM Trans. Database Systems, vol. 1, no. 3, Sept. 1976.
[32] C.T. Yu,K-C. Guh,D. Brill,, and A.L.P. Chen,“Partition strategy for distributed query processing in fast local area networks,” IEEE Trans. on Software Engineering, vol. 15, no. 6, pp. 780-793, June 1989.
[33] C.T. Yu and W. Sun, “Automatic Knowledge Acquisition and Maintenance for Semantic Query Optimization,” IEEE Trans. Knowledge and Data Eng., vol. 1, no. 3, pp. 362–375, 1989

Index Terms:
Database, derivability problem, implication problem, satisfiability, NP-hard, conjunctive queries.
Citation:
Sha Guo, Wei Sun, Mark A. Weiss, "On Satisfiability, Equivalence, and Implication Problems Involving Conjunctive Queries in Database Systems," IEEE Transactions on Knowledge and Data Engineering, vol. 8, no. 4, pp. 604-616, Aug. 1996, doi:10.1109/69.536253
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