Courcelle (1992) proposed the idea of using logic, in particular Monadic second-order logic (MSO), to define graph to graph transformations. Transducers, on the other hand, are executable machine models to define transformations, and are typically studied in the context of string-to-string transformations. Engelfriet and Hoogeboom (2001) studied two-way finite state string-to-string transducers and showed that their expressiveness matches MSO-definable transformations (MSOT). Alur and Cerny (2011) presented streaming transducers-one-way transducers equipped with multiple registers that can store output strings, as an equi-expressive model. Natural generalizations of streaming transducers to string-to-tree (Alur and D'Antoni, 2012) and infinite-string-to-string (Alur, Filiot, and Trivedi, 2012) cases preserve MSO-expressiveness. While earlier reductions from MSOT to streaming transducers used two-way transducers as the intermediate model, we revisit the earlier reductions in a more general, and previously unexplored, setting of infinite-string-to-tree transformations, and provide a direct reduction. Proof techniques used for this new reduction exploit the conceptual tools (composition theorem and finite additive coloring theorem) presented by Shelah (1975) in his alternative proof of Bϋchi's theorem. Using such streaming string-to-tree transducers we show the decidability of functional equivalence for MSO-definable infinite-string-to-tree transducers.

Many relational structures are automatically presentable, i.e. elements of the domain can be seen as words over a finite alphabet and equality and other atomic relations are represented with finite automata. The first-order theories over such structures are known to be primitive recursive, which is shown by the inductive construction of an automaton representing any relation definable in the first-order logic. We propose a general method based on Ehrenfeucht-Fraïssé games to give upper bounds on the size of these automata and on the time required to build them. We apply this method for two different automatic structures which have elementary decision procedures, Presburger Arithmetic and automatic structures of bounded degree. For the latter no upper bound on the size of the automata was known. We conclude that the very general and simple automata-based algorithm works well to decide the first-order theories over these structures.