Archive for 22nd September 2010

Fixing broken isomorphisms — details for non-strict memoization, part 2

22nd September 2010, 03:02 pm

The post Details for non-strict memoization, part 1 works out a systematic way of doing non-strict memoization, i.e., correct memoization of non-strict (and more broadly, non-hyper-strict) functions. As I mentioned at the end, there was an awkward aspect, which is that the purported “isomorphisms” used for regular types are not quite isomorphisms.

For instance, functions from triples are memoized by converting to and from nested pairs:

untriple ∷ (a,b,c) -> ((a,b),c)
untriple (a,b,c) = ((a,b),c)

triple ∷ ((a,b),c) -> (a,b,c)
triple ((a,b),c) = (a,b,c)

Then untriple and triple form an embedding/projection pair, i.e.,

triple ∘ untriple ≡ id
untriple ∘ triple ⊑ id

The reason for the inequality is that the nested-pair form permits (⊥,c), which does not correspond to any triple.

untriple (triple (⊥,c)) ≡ untriple ⊥ ≡ ⊥

Can we patch this problem by simply using an irrefutable (lazy) pattern in the definition of triple, i.e., triple (~(a,b),c) = (a,b,c)? Let’s try:

untriple (triple (⊥,c)) ≡ untriple (⊥,⊥,c) ≡ ((⊥,⊥),c)

So isomorphism fails and so does even the embedding/projection property.

Similarly, to deal with regular algebraic data types, I used a class that describes regular data types as repeated applications of a single, associated pattern functor (following A Lightweight Approach to Datatype-Generic Rewriting):

class Functor (PF t) ⇒ Regular t where
  type PF t ∷ * → *
  unwrap ∷ t → PF t t
  wrap   ∷ PF t t → t

Here unwrap converts a value into its pattern functor form, and wrap converts back. For example, here is the Regular instance I had used for lists:

instance Regular [a] where
  type PF [a] = Const () :+: Const a :*: Id

  unwrap []     = InL (Const ())
  unwrap (a:as) = InR (Const a :*: Id as)

  wrap (InL (Const ()))          = []
  wrap (InR (Const a :*: Id as)) = a:as

Again, we have an embedding/projection pair, rather than a genuine isomorphism:

wrap ∘ unwrap ≡ id
unwrap ∘ wrap ⊑ id

The inequality comes from ⊥ values occurring in PF [a] [a] at type Const () [a], (), (Const a :*: Id) [a], Const a [a], or Id [a].

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