Conal Elliott

In Non-strict memoization, I sketched out a means of memoizing non-strict functions. I gave the essential insight but did not show the details of how a nonstrict memoization library comes together. In this new post, I give details, which are a bit delicate, in terms of the implementation described in Elegant memoization with higher-order types.

Near the end, I run into some trouble with regular data types, which I don’t know how to resolve cleanly and efficiently.

Continue reading ‘Details for non-strict memoization, part 1’ »

Memoization incrementally converts functions into data structures. It pays off when a function is repeatedly applied to the same arguments and applying the function is more expensive than accessing the corresponding data structure.

In lazy functional memoization, the conversion from function to data structure happens all at once from a denotational perspective, and incrementally from an operational perspective. See Elegant memoization with functional memo tries and Elegant memoization with higher-order types.

As Ralf Hinze presented in Generalizing Generalized Tries, trie-based memoization follows from three simple isomorphisms involving functions types:

which correspond to the familiar laws of exponents

When applied as a transformation from left to right, each law simplifies the domain part of a function type. Repeated application of the rules then eliminate all function types or reduce them to functions of atomic types. These atomic domains are eliminated as well by additional mappings, such as between a natural number and a list of bits (as in patricia trees). Algebraic data types corresponding to sums of products and so are eliminated by the sum and product rules. Recursive algebraic data types (lists, trees, etc) give rise to correspondingly recursive trie types.

So, with a few simple and familiar rules, we can memoize functions over an infinite variety of common types. Have we missed any?

Yes. What about functions over functions?

Edits:

• 2010-07-22: Made the memoization example polymorphic and switched from pairs to lists. The old example accidentally coincided with a specialized version of trie itself.

Continue reading ‘Memoizing higher-order functions’ »

A while back, I got interested in functional memoization, especially after seeing some code from Spencer Janssen using the essential idea of Ralf Hinze’s paper Generalizing Generalized Tries. The blog post Elegant memoization with functional memo tries describes a library, MemoTrie, based on both of these sources, and using associated data types. I would have rather used associated type synonyms and standard types, but I couldn’t see how to get the details to work out. Recently, while playing with functor combinators, I realized that they might work for memoization, which they do quite nicely.

This blog post shows how functor combinators lead to an even more elegant formulation of functional memoization. The code is available as part of the functor-combo package.

The techniques in this post are not so much new as they are ones that have recently been sinking in for me. See Generalizing Generalized Tries, as well as Generic programming with fixed points for mutually recursive datatypes.

Continue reading ‘Elegant memoization with higher-order types’ »

I’ve written a few posts about functional memoization. In one of them, Luke Palmer commented that the memoization methods are correct only for strict functions, which I had not noticed before. In this note, I correct this flaw, extending correct memoization to non-strict functions as well. The semantic notion of least upper bound (which can be built of unambiguous choice) plays a crucial role.

Edits:

• 2010-07-13: Fixed the non-strict memoization example to use an argument of undefined (⊥) as intended.
• 2010-07-23: Changed spelling from “nonstrict” to the much more popular “non-strict”.

Continue reading ‘Non-strict memoization’ »

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.

Continue reading ‘Memoizing polymorphic functions – part two’ »

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.

Continue reading ‘Memoizing polymorphic functions – part one’ »

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).

Edits:

• 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.

Continue reading ‘Simpler, more efficient, functional linear maps’ »