Helmut Seidl. **Finite Tree Automata with Cost Functions**. In Jean-Claude Raoult, editor, *Trees in Algebra and Programming*, volume 581 of *Lecture Notes in Computer Science*, pages 279-299, Rennes, France, February 1992. Springer.

Cost functions for tree automata are mappings from transitions to (tuples of) polynomials over some semiring. We consider four semirings, namely the semiring of nonnegative integers, the ldquoarctical semiringrdquo, the tropical semiring and the semiring of finite subsets of nonnegative integers. We show: for semirings and it is decidable in polynomial time whether or not the costs of accepting computations is bounded; for 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 we are able to derive similar results at least in case of polynomials of degree at most 1. For and we extend our results to multi-dimensional cost functions.

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