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Generalized Iteration and CoIteration for Higher-Order Nested Datatypes (2003)

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by Andreas Abel , Ralph Matthes , Tarmo Uustalu , Department Of Computer Science
Citations:8 - 5 self
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BibTeX

@MISC{Abel03generalizediteration,
    author = {Andreas Abel and Ralph Matthes and Tarmo Uustalu and Department Of Computer Science},
    title = {Generalized Iteration and CoIteration for Higher-Order Nested Datatypes},
    year = {2003}
}

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Abstract

The problem of de ning iteration for higher-order nested datatypes of arbitrary ( nite) rank is solved within the framework of System F of higher-order parametric polymorphism. The proposed solution heavily relies on a general notion of monotonicity as opposed to a syntactic criterion on the shape of the type constructors such as positivity or even being polynomial. Its use is demonstrated for some rank-2 heterogeneous/nested datatypes such as powerlists and de Bruijn terms with explicit substitutions. An important feature is the availability of an iterative de nition of the mapping operation (the functoriality) for those rank-1 type transformers (i. e., functions from types to types) arising as least xed-points of monotone rank-2 type transformers. Strong normalization is shown by an embedding into F . The results dualize to greatest xed-points, hence to coinductive constructors with coiteration.

Citations

474 Lambda Calculi with Types - Barendregt - 1992
395 The formulae-as-types notion of construction - Howard - 1980
221 Interprétation fonctionnelle et élimination des coupures de l’arithmétique d’ordre supérieur. Thèse de Doctorat d’ État, Université de Paris VII - Girard - 1972
182 Categories for the working mathematician, volume 5 of Graduate Texts - Lane
102 Polytypic Values Possess Polykinded Types - Hinze - 2000
68 Monadic presentations of lambda terms using generalized inductive types - Altenkirch, Reus - 1999
67 Nested datatypes - Bird, Meertens - 1998
62 De Bruijn notation as a nested datatype - Bird, Paterson - 1999
49 Inductive and Coinductive Types with Iteration and Recursion - Geuvers - 1992
45 Generic programming within dependently typed programming - Altenkirch, McBride - 2003
45 A typed lambda calculus with categorical type constructors - Hagino - 1987
31 Contracting proofs to programs - Leivant - 1990
30 Type-based termination of recursive definitions - Barthe, Frade, et al.
28 Generalised folds for nested datatypes - Bird, Paterson - 1999
28 Structural recursive definitions in type theory - Giménez - 1998
25 2004, Termination Checking with Types - Abel
21 Extensions of System F by Iteration and Primitive Recursion on Monotone Inductive Types - Matthes - 1998
9 Structural recursive de in type theory - Gimenez - 1998
9 Efficient generalized folds - Hinze - 2000
9 Monotone inductive and coinductive constructors of rank 2 - Matthes - 2001
8 Co-)iteration for higher-order nested datatypes - Abel, Matthes - 2003
7 Program optimisation, naturally - Bird, Gibbons, et al. - 2000
3 Type-based termination of recursive de - Barthe, Frade, et al. - 2004
3 Representations of order function types as terminal coalgebras - Altenkirch - 2001
1 Efficient generalized folds - Coll - 1998
1 examples for iteration and coiteration on higher-order datatypes. Haskell code accompanying this article, available on the author's homepage - Haskell - 2003
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