## Memoizing polymorphic functions – part two

Part one of this series introduced the problem of memoizing functions involving polymorphic recursion. The caching data structures used in memoization typically handle only one type of argument at a time. For instance, one can have finite maps of differing types, but each concrete finite map holds just one type of key and one type of value.

I extended memoization to handle polymorphic recursion by using an existential type together with a reified type of types. This extension works (afaik), but it is restricted to a particular form for the type of the polymorphic function being memoized, namely

``````-- Polymorphic function
type k :--> v = forall a. HasType a => k a -> v a
``````

My motivating example is a GADT-based representation of typed lambda calculus, and some of the functions I want to memoize do not fit the pattern. After writing part one, I fooled around and found that I could transform these awkwardly typed polymorphic functions into isomorphic form that does indeed fit the restricted pattern of polymorphic types I can handle.

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## 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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