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Fuzzing is an increasingly popular technique for verifying software functionalities and finding security vulnerabilities. However, current mutation-based fuzzers cannot effectively test database management systems (DBMSs), which strictly check inputs for valid syntax and semantics. Generation-based testing can guarantee the syntax correctness of the inputs, but it does not utilize any feedback, like code coverage, to guide the path exploration. In this paper, we develop Squirrel, a novel fuzzing framework that considers both language validity and coverage feedback to test DBMSs. We design an intermediate representation (IR) to maintain SQL queries in a structural and informative manner. To generate syntactically correct queries, we perform type-based mutations on IR, including statement insertion, deletion and replacement. To mitigate semantic errors, we analyze each IR to identify the logical dependencies between arguments, and generate queries that satisfy these dependencies. We evaluated Squirrel on four popular DBMSs: SQLite, MySQL, PostgreSQL and MariaDB. Squirrel found 51 bugs in SQLite, 7 in MySQL and 5 in MariaDB. 52 of the bugs are fixed with 12 CVEs assigned. In our experiment, Squirrel achieves 2.4x-243.9x higher semantic correctness than state-of-the-art fuzzers, and explores 2.0x-10.9x more new edges than mutation-based tools. These results show that Squirrel is effective in finding memory errors of database management systems.