Finite sets

[winterkoninkje]

So, I just encountered a most delicious type the other day:

class Finite a where
    assemble :: Applicative f => (a -> f b) -> f (a -> b)

What's so nice about it is that the only way you can implement it is if the type a is in fact finite. (But see the notes.) So the questions are:

• Can you see why?
• Can you figure out how to implement it for some chosen finite type?
• Can you figure out how to implement it in general, given a list of all the values? (you may assume Eq a for this one)
• Can you figure out how to get a list of all the values, given some arbitrary implementation of assemble?

A trivial note: Well, of course you can implement anything in Haskell by using undefined or equivalent. For the sake of these problems, assume you're constrained to use only fully-defined functions.

A big note, also a hint perhaps: And really, there's nothing here that truly forces the type to be finite. In order to get some real logical guarantees, you should assume you're in a total language (preferably a provably-total language), and you should use the following dependent type instead:

class Finite a where
    assemble :: Applicative f => ((x::a) -> f (b x)) -> f ((x::a) -> b x)

Comments

Missing some laws

[https://www.google.com/accounts/o8/id?id=AItOawnNP5_3eRT_BuDtOzsEuFk1mSKmdeQ6mt8]

Without some laws this type doesn't tell us anything about the finiteness of a type (as far as I can tell).

import Control.Applicative

assemble1 :: Applicative f => a -> (a -> f b) -> f (a -> b)
assemble1 a f = fmap const (f a)

class Assemble a where
  assemble :: Applicative f => (a -> f b) -> f (a -> b)

instance Assemble Integer where
  assemble = assemble1 0

As long as we have one element of the type we can implement the rest, finite or not. Perhaps there were other constraints that an implementation should satisfy.

Re: Missing some laws

[winterkoninkje]

That's exactly what I mentioned in the second note. Of course, even though the collections of all values of a given type may not be finite; this encoding does give a specification of a finite subset of values from that type. Thus, it can be considered an alternative to strict lists for ensuring finiteness of a collection.

Of course, as an attempt to enumerate all values in the type, your instance isn't morally correct. In this case, the morality in question can be enforced with dependent types. Can you find any way to break the dependent version? (nontermination doesn't count.)