On Mon, 13 Aug 2012, Alexander Solla <alex.so...@gmail.com> wrote:
On Mon, Aug 13, 2012 at 6:25 PM, Jay Sulzberger <j...@panix.com> wrote:
On Mon, 13 Aug 2012, Ryan Ingram <ryani.s...@gmail.com> wrote:
On Mon, Aug 13, 2012 at 12:30 PM, Jay Sulzberger <j...@panix.com> 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)
Yes.
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".
I am becoming sensitized to this distinction. I now, I think,
feel the impulse toward "constructivism", that is, the
assumption/delusion^Waspiration that all functions from the reals
to the reals are continuous. One argument that helped me goes:
The reals between 0 and 1 are functions from the integers to say {0, 1}.
They are attained as limits of functions f: iota(n) -> {0, 1}, as
n becomes larger and larger and ... , where iota(n) is a set with
n elements, n a finite integer.
So, our objects, the reals, are attained as limits. And the
process of proceeding toward the limit is "natural", "functorial"
in the sense of category theory.
Thus so also our morphisms, that is functions from the reals to
the reals, must be produced functorially as limits of maps
between objects f: iota(n) -> {0, 1}.
Put these ideas together, and you will recover the duality as expressed
earlier.
Thanks! I am reading some blog posts and I was impressed by the
buffalo hair here:
http://existentialtype.wordpress.com/2012/08/11/extensionality-intensionality-and-brouwers-dictum/
oo--JS.
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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