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<p><b>Abstract</b>—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 (<it>X</it><b>op</b><it>C</it>), (<it>X</it><b>op</b><it>Y</it>), and/or (<it>X</it><b>op</b><it>Y</it> + <it>C</it>), where <it>X</it> and <it>Y</it> are two attributes, <it>C</it> is a constant, and <b>op</b>∈ {< , ≤ , =, ≠, >, ≥}. These types of inequalities are widely used in database systems, since the first type is a <it>selection</it>, the second type is a θ-<it>join</it>, 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 [<ref rid="bibk060420" type="bib">20</ref>], [<ref rid="bibk060424" type="bib">24</ref>]. 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 "<it>modulo closure</it>" and a set of sound and complete axioms with respect to the "<it>modulo closure</it>" are also proposed to infer all correct and necessary inequalities from a given query. The proposed axioms generalize Ullman's axioms [<ref rid="bibk060427" type="bib">27</ref>] where queries only consist of θ-joins.</p>
Database, derivability problem, implication problem, satisfiability, NP-hard, conjunctive queries.

W. Sun, M. A. Weiss and S. Guo, "On Satisfiability, Equivalence, and Implication Problems Involving Conjunctive Queries in Database Systems," in IEEE Transactions on Knowledge & Data Engineering, vol. 8, no. , pp. 604-616, 1996.
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