The power of symbolic automata and transducers

Symbolic automata and transducers extend finite automata and transducers by allowing transitions to carry predicates and functions over rich alphabet theories,  such as linear arithmetic.  Therefore, these models extend their classic counterparts to operate over infinite alphabets, such as the set of rational numbers. Due to their expressiveness, symbolic automata and transducers  have been used to verify functional programs operating over lists and trees, to prove the correctness of complex implementations of BASE64 and UTF encoders, and to expose data parallelism in computations that may otherwise seem inherently sequential. In this paper, we give an overview of what is currently known about symbolic automata and transducers as well as their variants. We discuss what makes these models different from their finite-alphabet counterparts, what kind of applications symbolic models can enable, and what challenges arise when reasoning about these formalisms. Finally, we present a list of open problems and research directions that relate to both the theory and practice of symbolic automata and transducers.