Labelled structures and combinatorial species

We describe a theory of labelled structures, which intuitively consist of a labelled shape together with a mapping from labels to data. Labelled structures thus subsume algebraic data types as well as “labelled” types such as arrays and finite maps. This idea of decomposing container structures into shapes and data is an old one; our novel contribution is to explicitly mediate the decomposition with arbitrary labels. We demonstrate the benefits of this approach, showing how it can be used, for example, to explicitly model composition of container shapes, to model value-level sharing via superimposing multiple structures on the same data, and to model issues of memory allocation and layout. The theory of labelled structures is built directly on the foundation of combinatorial species, which serve to describe labelled shapes. The theory of species bears striking similarities to the theory of algebraic data types, and it has long been suspected that a more precise and fruitful connection could be made between the two. In a larger sense, the aim of this paper is to serve as a first step in connecting combinatorial species to the theory and practice of programming. In particular, we describe a “port” of the theory of species into constructive type theory, justifying its use as a basis for computation.