Families Of Datatypes Research Articles

An induction principle for nested datatypes in intensional type theory

AbstractNested datatypes are families of datatypes that are indexed over all types such that the constructors may relate different family members (unlike the homogeneous lists). Moreover, the argument types of the constructors refer to indices given by expressions in which the family name may occur. Especially in this case of true nesting, termination of functions that traverse these data structures is far from being obvious. A joint paper with A. Abel and T. Uustalu (Theor. Comput. Sci., 333 (1–2), 2005, pp. 3–66) proposed iteration schemes that guarantee termination not by structural requirements but just by polymorphic typing. They are generic in the sense that no specific syntactic form of the underlying datatype “functor” is required. However, there was no induction principle for the verification of the programs thus obtained, although they are well known in the usual model of initial algebras on endofunctor categories. The new contribution is a representation of nested datatypes in intensional type theory (more specifically, in the calculus of inductive constructions) that is still generic and covers true nesting, guarantees termination of all expressible programs, and has an induction principle that allows to prove functoriality of monotonicity witnesses (maps for nested datatypes) and naturality properties of iteratively defined polymorphic functions.

A pattern for almost compositional functions

AbstractThis paper introduces a pattern for almost compositional functions over recursive data types, and over families of mutually recursive data types. Here “almost compositional” means that for all of the constructors in the type(s), except a limited number of them, the result of the function depends only on the constructor and the results of calling the function on the constructor's arguments. The pattern consists of a generic part constructed once for each data type or family of data types, and a task-specific part. The generic part contains the code for the predictable compositional cases, leaving the interesting work to the task-specific part. Examples of the pattern are given, implemented in dependent type theory with inductive families, in Haskell with generalized algebraic data types and rank-2 polymorphism, and in Java using a variant of the Visitor design pattern. The relationships to the “Scrap Your Boilerplate” approach to generic programming, and to general tree types in dependent type theory, are investigated by reimplementing our operations using those frameworks.

A pattern for almost compositional functions

This paper introduces a pattern for almost compositional functions over recursive data types, and over families of mutually recursive data types. Here "almost compositional" means that for a number of the constructors in the type(s), the result of the function depends only on the constructor and the results of calling the function on the constructor's arguments. The pattern consists of a generic part constructed once for each data type or family of data types, and a task-specific part. The generic part contains the code for the predictable compositional cases, leaving the interesting work to the task-specific part. Examples of the pattern implemented in dependent type theory with inductive families, in Haskell with generalized algebraic data types and rank-2 polymorphism, and in Java using a variant of the Visitor design pattern are given. The relationship to the "Scrap Your Boilerplate" approach to generic programming, and to general tree types in dependent type theory are also investigated.