Extended symbolic finite automata and transducers

Symbolic finite automata and transducers augment classic automata and transducers with symbolic alphabets represented as parametric theories. This extension enables to succinctly represent large and potentially infinite alphabets while preserving closure and decidability properties. Extended symbolic finite automata and transducers further extend these objects by allowing transitions to read consecutive input elements in a single step. In this paper we study the properties of these models. In contrast to the case of finite alphabets, we show how reading multiple symbols increases the expressiveness of the models, which causes some closure properties to stop holding and most decision problems to become undecidable. In particular we show how extended symbolic finite transducers are not closed under composition, and the equivalence problem is undecidable for both extended symbolic finite automata and transducers. We then introduce the subclass of Cartesian extended symbolic finite transducers in which guards are limited to conjunctions of unary predicates and we propose an equivalence algorithm for this subclass in the single-valued case. We also present a heuristic algorithm for composing extended symbolic finite transducers that works for many practical cases. Finally, we model real world programs with Cartesian extended symbolic finite transducers and use the proposed algorithms to prove their correctness.