
On Mon, Aug 13, 2012 at 6:25 PM, Jay Sulzberger
On Mon, 13 Aug 2012, Ryan Ingram
wrote: On Mon, Aug 13, 2012 at 12:30 PM, Jay Sulzberger
wrote: Does Haskell have a word for "existential type" declaration? I
have the impression, and this must be wrong, that "forall" does double duty, that is, it declares a "for all" in some sense like the usual "for all" of the Lower Predicate Calculus, perhaps the "for all" of system F, or something near it.
It's using the forall/exists duality: exsts a. P(a) <=> forall r. (forall a. P(a) -> r) -> r
;)
This is, I think, a good joke. It has taken me a while, but I now understand that one of the most attractive things about Haskell is its sense of humor, which is unfailing.
I will try to think about this, after trying your examples.
I did suspect that, in some sense, "constraints" in combination with "forall" could give the quantifier "exists".
In a classical logic, the duality is expressed by !E! = A, and !A! = E, where E and A are backwards/upsidedown and ! represents negation. In particular, for a proposition P, Ex Px <=> !Ax! Px (not all x's are not P) and Ax Px <=> !Ex! Px (there does not exist an x which is not P) Negation is relatively tricky to represent in a constructive logic -- hence the use of functions/implications and bottoms/contradictions. The type ConcreteType -> b represents the negation of ConcreteType, because it shows that ConcreteType "implies the contradiction". Put these ideas together, and you will recover the duality as expressed earlier.
For example: (exists a. Show a => a) <=> forall r. (forall a. Show a => a -> r) -> r
This also works when you look at the type of a constructor:
data Showable = forall a. Show a => MkShowable a -- MkShowable :: forall a. Show a => a -> Showable
caseShowable :: forall r. (forall a. Show a => a -> r) -> Showable -> r caseShowable k (MkShowable x) = k x
testData :: Showable testData = MkShowable (1 :: Int)
useData :: Int useData = case testData of (MkShowable x) -> length (show x)
useData2 :: Int useData2 = caseShowable (length . show) testData
Haskell doesn't currently have first class existentials; you need to wrap them in an existential type like this. Using 'case' to pattern match on the constructor unwraps the existential and makes any packed-up constraints available to the right-hand-side of the case.
An example of existentials without using constraints directly:
data Stream a = forall s. MkStream s (s -> Maybe (a,s))
viewStream :: Stream a -> Maybe (a, Stream a) viewStream (MkStream s k) = case k s of Nothing -> Nothing Just (a, s') -> Just (a, MkStream s' k)
takeStream :: Int -> Stream a -> [a] takeStream 0 _ = [] takeStream n s = case viewStream s of Nothing -> [] Just (a, s') -> a : takeStream (n-1) s'
fibStream :: Stream Integer fibStream = Stream (0,1) (\(x,y) -> Just (x, (y, x+y)))
Here the 'constraint' made visible by pattern matching on MkStream (in viewStream) is that the "s" type stored in the stream matches the "s" type taken and returned by the 'get next element' function. This allows the construction of another stream with the same function but a new state -- MkStream s' k.
It also allows the stream function in fibStream to be non-recursive which greatly aids the GHC optimizer in certain situations.
-- ryan
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