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We study the definability problem for first-order logic, denoted by FO-Def. The input of FO-Def is a relational database instance I and a relation R; the question to answer is whether there exists a first-order query Q (or, equivalently, a relational algebra expression Q) such that Q evaluated on I gives R as an answer.

Although the study of FO-Def dates back to 1978, when the decidability of this problem was shown, the exact complexity of FO-Def remains as a fundamental open problem. In this article, we provide a polynomial-time algorithm for solving FO-Def that uses calls to a graph-isomorphism subroutine (or oracle). As a consequence, the first-order definability problem is found to be complete for the class GI of all problems that are polynomial-time Turing reducible to the graph isomorphism problem, thus closing the open question about the exact complexity of this problem. The technique used is also applied to a …