The structure of polynomial growth for tree automata/transducers and MSO set queries

Given an $\mathbb{N}$-weighted tree automaton, we give a decision procedure for exponential vs polynomial growth (with respect to the input size) in quadratic time, and an algorithm that computes the exact polynomial degree of growth in cubic time. As a special case, they apply to the growth of the ambiguity of a nondeterministic tree automaton, i.e. the number of distinct accepting runs over a given input. Our time complexities match the recent fine-grained lower bounds for these problems restricted to ambiguity of word automata. We deduce analogous decidability results (ignoring complexity) for the growth of the number of results of set queries in Monadic Second-Order logic (MSO) over ranked trees. In the case of polynomial growth of degree $k$, we also prove a reparameterization theorem for such queries: their results can be mapped to $k$-tuples of input nodes in a finite-to-one and MSO-definable fashion. This property of MSO set queries leads directly to a generalization of the dimension minimization theorem for string-to-string polyregular functions. Our generalization applies to MSO set interpretations from trees, which subsume (as we show) tree-walking tree transducers and invisible pebble tree-to-string transducers. Finally, with a bit more work we obtain the following: * a new, short and conceptual proof that macro tree transducers (MTTs) of linear growth compute only tree-to-tree MSO transductions; * a procedure to decide polynomial size-to-height increase for MTTs and compute the degree. The paper concludes with a survey of a wide range of related work, with over a hundred references.