Finite tree automata with cost functions

Cost functions for tree automata are mappings from transitions to (tuples of) polynomials over some semiring. We consider four semirings, namely N the semiring of nonnegative integers, A the “arctical semiring”, T the tropical semiring and F the semiring of finite subsets of nonnegative integers. We show: for semirings N and A it is decidable in polynomial time whether or not the costs of accepting computations is bounded; for F it is decidable in polynomial time whether or not the cardinality of occurring cost sets is bounded. In all three cases we derive explicit upper bounds. For semiring T we are able to derive similar results at least in case of polynomials of degree at most 1. For N and A we extend our results to multi-dimensional cost functions.