A type for partial values

In simplifying my Eros implementation, I came across a use for a type that represents partial information about values. I came up with a very simple implementation, though perhaps not quite ideal. In this post, I’ll present the interface and invite ideas for implementation. In the next post, I’ll give the implementation I use.

First the type:

type Partial a

I require that Partial a be a monoid, for which mempty is the completely undefined value, and in u `mappend` v, v selectively replaces parts of u. (Lattice-wise, mempty is bottom, and mappend is not quite lub, as it lacks commutativity).

Now the programming interface:

-- Treat a full value as a partial one.  Fully overrides any "previous" (earlier
-- argument to mappend) partial value.
valp :: c -> Partial c

-- Force a partial value into a full one, filling in bottom for any missing parts.
pval :: Partial c -> c

-- Inverse to fst, on partial values.  Inject info into the the first half of a pair,
-- and leave the second half alone.
unFst :: Partial a -> Partial (a,b)

-- Inverse to snd.
unSnd :: Partial b -> Partial (a,b)

-- Inverse to "element" access, on all elements.  A way to inject some info about every
-- element.  For f, consider [], (->) a, Event, etc.
unElt :: Functor f => Partial a -> Partial (f a)

That’s it. I’d love to hear ideas for representing Partial a and implementing the functions above. In the next post, I’ll give mine.