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User-defined functions (UDFs) are widely used to enhance the ca- pabilities of DBMSs. However, using UDFs comes with a significant performance penalty because DBMSs treat UDFs as black boxes, which hinders their ability to optimize queries that invoke such UDFs. To mitigate this problem, in this paper we present LAMBDA, a technique and framework for improving DBMSs’ performance in the presence of UDFs. The core idea of LAMBDA is to statically infer properties of UDFs that facilitate UDF processing. Taking one such property as an example, if DBMSs know that a UDF is pure, that is it returns the same result given the same arguments, they can leverage a cache to avoid repetitive UDF invocations that have the same call arguments. We reframe the problem of analyzing UDF properties as a data flow problem. We tackle the data flow problem by building LAMBDA on top of an extensible abstract interpretation framework and de- veloping an analysis model that is tailored for UDFs. Currently, LAMBDA supports inferring four properties from UDFs that are widely used across DBMSs. We evaluate LAMBDA on a benchmark that is derived from production query workloads and UDFs. Our evaluation results show that (1) LAMBDA conservatively and ef- ficiently infers the considered UDF properties, and (2) inferring such properties improves UDF performance, with a time reduction ranging from 10% to 99%. In addition, when applied to 20 produc- tion UDFs, LAMBDA caught five instances in which developers provided incorrect UDF property annotations. We qualitatively compare LAMBDA against Froid, a state-of-the-art framework for improving UDF performance, and explain how LAMBDA can opti- mize UDFs that are not supported by Froid.