Weighted finite-state transducers (FSTs) are frequently used in language processing to handle tasks such as part-of-speech tagging and speech recognition. There has been previous work using multiple CPU cores to accelerate finite state algorithms, but limited attention has been given to parallel graphics processing unit (GPU) implementations. In this paper, we introduce the first (to our knowledge) GPU implementation of the FST composition operation, and we also discuss the optimizations used to achieve the best performance on this architecture. We show that our approach obtains speedups of up to 6x over our serial implementation and 4.5x over OpenFST.