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<p>Implication testing of arithmetic inequalities has been widely used in different areas in database systems and has received extensive research as well. Klug and Ullman (A. Klug, 1988; and J.D. Ullman, 1989) proposed an algorithm that determines whether S implies T, where T and S consist of inequalities of form (X op Y), X and Y are two variables, and opε{=, <, ≤, ≠, >, ≥};. The complexity of the algorithm is O(n³), where n is the number of inequalities in S. We reduce the problem to matrix multiplication, thus improving the time bound to O(n^2.376). We also demonstrate an O(n²) algorithm if the number of inequalities in T is bounded by O(n). Since matrix multiplication has been well studied, our reduction allows the possibility of directly adopting many practical results for managing matrices and their operations, such as parallel computation and efficient representation of sparse matrices.</p>
query processing; computational complexity; matrix multiplication; sparse matrices; implication testing; arithmetic inequalities; database systems; complexity; matrix multiplication; parallel computation; sparse matrices; query optimization; satisfiability; equivalence

M. Weiss and W. Sun, "An Improved Algorithm for Implication Testing Involving Arithmetic Inequalities," in IEEE Transactions on Knowledge & Data Engineering, vol. 6, no. , pp. 997-1001, 1994.
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