In Girard (2001), J.-Y. Girard presents a new theory, The Ludics, which is a model of realisibility of logic that associates proofs with designs, and formulas with behaviours. In this article we study the interpretation in this semantics of formulas with first-order quantifications and their proofs. We extend to the first-order quantifiers the full completeness theorem obtained in Girard (2001) for $MALL_2$. A significant part of this article is devoted to the study of a uniformity property for the families of designs that represent proofs of formulas depending on a first-order free variable.

We compare two deforestation techniques: short cut fusion formalised 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 generalisation of a catamorphism, and show a corresponding fusion result, which is a generalisation 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.