Memoizing polymorphic functions – part one

Memoization takes a function and gives back a semantically equivalent function that reuses rather than recomputes when applied to the same argument more than once. Variations include not-quite-equivalence due to added strictness, and replacing value equality with pointer equality.

Memoization is often packaged up polymorphically:

memo :: (???) => (k -> v) -> (k -> v)

For pointer-based (“lazy”) memoization, the type constraint (“???”) is empty. For equality-based memoization, we’d need at least Eq k, and probably Ord k or HasTrie k for efficient lookup (in a finite map or a possibly infinite memo trie).

Although memo is polymorphic, its argument is a monomorphic function. Implementations that use maps or tries exploit that monomorphism in that they use a type like Map k v or Trie k v. Each map or trie is built around a particular (monomorphic) type of keys. That is, a single map or trie does not mix keys of different types.

Now I find myself wanting to memoize polymorphic functions, and I don’t know how to do it.

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Simpler, more efficient, functional linear maps

A previous post described a data type of functional linear maps. As Andy Gill pointed out, we had a heck of a time trying to get good performance. This note describes a new representation that is very simple and much more efficient. It’s terribly obvious in retrospect but took me a good while to stumble onto.

The Haskell module described here is part of the vector-space library (version 0.5 or later) and requires ghc version 6.10 or better (for associated types).


  • 2008-11-09: Changed remarks about versions. The vector-space version 0.5 depends on ghc 6.10.
  • 2008-10-21: Fixed the vector-space library link in the teaser.

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Composing memo tries

Elegant memoization with functional memo tries

Function memoization goes back at least as far as Donald Michie’s 1968 paper. The idea is to stash away results of a function call for reuse when a function is called more than once with the same argument. Given an argument, the memoized function looks up the argument in the internal memo table (often a hash table). If found, the previously computed result is reused. Otherwise, the function is applied, and the result is stored in the table, keyed by the argument. The table and its mutation are kept private from clients of the memo function.

Perhaps surprisingly, memoization can be implemented simply and purely functionally in a lazy functional language. Laziness allows the implementation to build the memo table once and for all, filling in all the results for the function at all domain values. Thanks to laziness, the values don’t actually get computed until they’re used. As usual with lazy data structures, once a component has been evaluated, future accesses come for free.

The implementation described in this post is based one I got from Spencer Janssen. It uses the essential idea of Ralf Hinze’s paper Generalizing Generalized Tries. The library is available on Hackage and a darcs repository. See the wiki page and hackage page for documentation and source code. I’ve compiled it successfully with ghc versions 6.8.2 through 6.10.3.


  • 2009-02-12: Fixed typo.
  • 2009-11-14: Hackage page pointer.
  • 2009-11-20: Fixed source code pointer.

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