Syntactic composition of top-down tree transducers is short cut fusion

We compare two deforestation techniques: short cut fusion formalized in category theory and the syntactic composition of tree transducers. The former strongly depends on types and uses the parametricity property or free theorem whereas the latter makes no use of types at all and allows more general compositions. We introduce the notion of a categorical transducer which is a generalization of a catamorphism and show a respective fusion result which is a generalization of the ‘acid rain theorem’. We prove the following main theorems: (i) The class of all categorical transducers builds a category where composition is fusion. (ii) The semantics of categorical transducers is a functor. (iii) The subclass of top-down categorical transducers is a subcategory. (iv) Syntactic composition of top-down tree transducers is equivalent to the fusion of top-down categorical transducers.