This library is a straightforward combinatorization of the well-known technique of memoization. Let's start with the canonical example:
fib = (map fib' [0..] !!)
where
fib' 0 = 0
fib' 1 = 1
fib' n = fib (n-1) + fib (n-2)
I interpret what you said to mean that you know how and why this works. So I'll focus on the combinatorization.
We are essentiallly trying to capture and generalize the idea of (map f [0..] !!). The type of this function is (Int -> r) -> (Int -> r), which makes sense: it takes a function from Int -> r and returns a memoized version of the same function. Any function which is semantically the identity and has this type is called a "memoizer for Int" (even id, which doesn't memoize). We generalize to this abstraction:
type Memo a = forall r. (a -> r) -> (a -> r)
So a Memo a, a memoizer for a, takes a function from a to anything, and returns a semantically identical function that has been memoized (or not).
The idea of the different memoizers is to find a way to enumerate the domain with a data structure, map the function over them, and then index the data structure. bool is a good example:
bool :: Memo Bool
bool f = table (f True, f False)
where
table (t,f) True = t
table (t,f) False = f
Functions from Bool are equivalent to pairs, except a pair will only evaluate each component once (as is the case for every value that occurs outside a lambda). So we just map to a pair and back. The essential point is that we are lifting the evaluation of the function above the lambda for the argument (here the last argument of table) by enumerating the domain.
Memoizing Maybe a is a similar story, except now we need to know how to memoize a for the Just case. So the memoizer for Maybe takes a memoizer for a as an argument:
maybe :: Memo a -> Memo (Maybe a)
maybe ma f = table (f Nothing, ma (f . Just))
where
table (n,j) Nothing = n
table (n,j) (Just x) = j x
The rest of the library is just variations on this theme.
The way it memoizes integral types uses a more appropriate structure than [0..]. It's a bit involved, but basically just creates an infinite tree (representing the numbers in binary to elucidate the structure):
1
10
100
1000
1001
101
1010
1011
11
110
1100
1101
111
1110
1111
So that looking up a number in the tree has running time proportional to the number of bits in its representation.
As sclv points out, Conal's MemoTrie library uses the same underlying technique, but uses a typeclass presentation instead of a combinator presentation. We released our libraries independently at the same time (indeed, within a couple hours!). Conal's is easier to use in simple cases (there is only one function, memo, and it will determine the memo structure to use based on the type), whereas mine is more flexible, as you can do things like this:
boundedMemo :: Integer -> Memo Integer
boundedMemo bound f = z -> if z < bound then memof z else f z
where
memof = integral f
Which only memoizes values less than a given bound, needed for the implementation of one of the project euler problems.
There are other approaches, for example exposing an open fixpoint function over a monad:
memo :: MonadState ... m => ((Integer -> m r) -> (Integer -> m r)) -> m (Integer -> m r)
Which allows yet more flexibility, eg. purging caches, LRU, etc. But it is a pain in the ass to use, and also it puts strictness constraints on the function to be memoized (e.g. no infinite left recursion). I don't believe there are any libraries that implement this technique.
Did that answer what you were curious about? If not, perhaps make explicit the points you are confused about?
