Abstract

This paper describes a general framework for designing purely functional datatypes that automatically satisfy given size or structural constraints. Using the framework we develop implementations of different matrix types (eg square matrices) and implementations of several tree types (eg Braun trees, 2-3 trees). Consider, for instance, representing square n \Theta n matrices. The usual representation using lists of lists fails to meet the structural constraints: there is no way to ensure that the outer list and the inner lists have the same length. The main idea of our approach is to solve in a first step a related, but simpler problem, namely to generate the multiset of all square numbers. In order to describe this multiset we employ recursion equations involving finite multisets, multiset union, addition and multiplication lifted to multisets. In a second step we mechanically derive datatype definitions from these recursion equations which enforce the `squareness' constraint. The tra...

Citations

7321	Introduction to Algorithms - Cormen, Leiserson, et al. - 2001
644	Data Structures and Algorithms - Aho, Hopcroft, et al. - 1983
275	Functional Programming with Bananas, Lenses, Envelopes and Barbed - Meijer, Fokkinga, et al. - 1991
212	A dichromatic framework for balanced trees - Guibas, Sedgewick - 1978
196	Purely Functional Data Structures - Okasaki - 1998
154	An algorithm for the organization of information - Adelson-Velskii, Landis - 1962
140	Algorithm 232: heapsort - Williams - 1964
95	Report on Haskell 98: A non-strict, purely functional language - Jones, Hughes - 1999
89	de Moor, O.: Algebra of Programming - Bird - 1997
84	A new approach to generic functional programming - Hinze - 2000
82	Polymorphic type inference - Leivant - 1983
74	Algebra of Programming - BIRD, MOOR - 1997
67	Nested datatypes - Bird, Meertens - 1998
64	Functional programming with overloading and higher-order polymorphism - Jones - 1995
27	From fast exponentiation to square matrices: an adventure in types - Okasaki - 1999
26	An algorithm for the organization of information. Doklady Akademii Nauk SSSR - Adel'son-Vel'skii, Landis - 1962
13	2000c) Polytypic programming with ease - Hinze
12	Polytypic functions over nested datatypes - Hinze - 1999
11	A logarithmic implementation of flexible arrays. Memorandum MR83/4 - Braun, Rem - 1983
9	Functional Pearl: Perfect trees and bit-reversal permutations - Hinze - 2000
9	Nested datatypes. Pages 52–67 of - Bird, Meertens - 1998
9	Container types categorically - Hoogendijk, Moor
7	international conference on mathematics of program construction - Fourth
6	A simple, efficient, and flexible implementation of flexible arrays - Dielissen, Kaldewaij
6	Polymorphic type schemes and recursive - Mycroft - 1984
5	Numerical Representations as Higher-Order Nested Datatypes - Hinze - 1998
5	Indexed types. Theoretical Computer Science, 187:147–165 - Zenger - 1997
4	Perfect trees and bit-reversal permutations - Hinze - 2000
2	A diochromatic framework for balanced trees. Pages 8--21 of - Guibas - 1978
2	Constructing red-black trees. Okasaki, C. (ed - Hinze - 1999
2	Functional Pearl: Three algorithms on Braun trees - Okasaki - 1997
2	On uniquely represented data structures (extended abstract - Snyder - 1977
1	Dependently typed data structures. Okasaki, C. (ed - Xi - 1999