Tree-Series-to-Tree-Series Transformations

We investigate the tree-series-to-tree-series (ts-ts) transformation computed by tree series transducers. Unless the used semiring is complete, this transformation is, in general, not well-defined. In practice, many used semirings are not complete (like the probability semiring). We establish a syntactical condition that guarantees well-definedness of the ts-ts transformation in arbitrary commutative semirings. For positive (i. e., zero-sum and zero-divisor free) semirings the condition actually characterizes the well-definedness, so that well-definedness is decidable in this scenario.