Iteration and coiteration schemes for higher-order and nested datatypes

• Published in: Theoretical computer science Vol. 333; no. 1; pp. 3 - 66
• Main Authors: Abel, Andreas, Matthes, Ralph, Uustalu, Tarmo
• Format: Journal Article Conference Proceeding
• Language: English
• Published: Amsterdam Elsevier B.V 01-03-2005; Elsevier
• Subjects: Algorithmics. Computability. Computer arithmetics; Applied sciences; Coiteration; Computer science; control theory; systems; Efficient folds; Exact sciences and technology; Generalized folds; Higher-order datatypes; Higher-order polymorphism; Iteration; Strong normalization; System [formula omitted]; Theoretical computing; Higher-order polymorphism; Strong normalization; Coiteration; System F ω; Higher-order datatypes; Efficient folds; Generalized folds; Iteration; Computer theory; Standardization; Rewriting; System Fω; Higher order datatype; Generalized fold; Monotonicity; Polymorphism
• ISSN: 0304-3975; 1879-2294
• DOI: 10.1016/j.tcs.2004.10.017

This article studies the implementation of inductive and coinductive constructors of higher kinds (higher-order nested datatypes) in typed term rewriting, with emphasis on the choice of the iteration and coiteration constructions to support as primitive. We propose and compare several well-behaved extensions of System 
F
ω
with some form of iteration and coiteration uniform in all kinds. In what we call Mendler-style systems, the iterator and coiterator have a computational behavior similar to the general recursor, but their types guarantee termination. In conventional-style systems, monotonicity witnesses are used for a notion of monotonicity defined uniformly for all kinds. Our most expressive systems
GMIt
ω
and
GIt
ω
of generalized Mendler, resp. conventional (co)iteration encompass Martin, Gibbons and Bailey's efficient folds for rank-2 inductive types. Strong normalization of all systems considered is proved by providing an embedding of the basic Mendler-style system
MIt
ω
into System 
F
ω
.