Suppose you have a large number of types and a large number of functions that each return "subsets" of these types.
Let's use a small example to make the situation more explicit. Here's a simple algebraic data type:
data T = A | B | C
and there are two functions f, g that return a T
f :: T
g :: T
For the situation at hand, assume it is important that f can only return a A or B and g can only return a B or C.
I would like to encode this in the type system. Here are a few reasons/circumstances why this might be desirable:
Let the functions f and g have a more informative signature than just ::T
Enforce that implementations of f and g do not accidentally return a forbidden type that users of the implementation then accidentally use
Allow code reuse, e.g. when helper functions are involved that only operate on subsets of type T
Avoid boilerplate code (see below)
Make refactoring (much!) easier
One way to do this is to split up the algebraic datatype and wrap the individual types as needed:
data A = A
data B = B
data C = C
data Retf = RetfA A | RetfB B
data Retg = RetgB B | RetgC C
f :: Retf
g :: Retg
This works, and is easy to understand, but carries a lot of boilerplate for frequent unwrapping of the return types Retf and Retg.
I don't see polymorphism being of any help, here.
So, probably, this is a case for dependent types. It's not really a type-level list, rather a type-level set, but I've never seen a type-level set.
The goal, in the end, is to encode the domain knowledge via the types, so that compile-time checks are available, without having excessive boilerplate. (The boilerplate gets really annoying when there are lots of types and lots of functions.)
Define an auxiliary sum type (to be used as a data kind) where each branch corresponds to a version of your main type:
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE StandaloneKindSignatures #-}
{-# LANGUAGE StandaloneDeriving #-}
{-# LANGUAGE DataKinds #-}
import Data.Kind
import Data.Void
import GHC.TypeLits
data Version = AllEnabled | SomeDisabled
Then define a type family that maps the version and the constructor name (given as a type-level Symbol) to the type () if that branch is allowed, and to the empty type Void if it's disallowed.
type Enabled :: Version -> Symbol -> Type
type family Enabled v ctor where
Enabled SomeDisabled "C" = Void
Enabled _ _ = ()
Then define your type as follows:
type T :: Version -> Type
data T v = A !(Enabled v "A")
| B !(Enabled v "B")
| C !(Enabled v "C")
(The strictness annotations are there to help the exhaustivity checker.)
Typeclass instances can be derived, but separately for each version:
deriving instance Show (T AllEnabled)
deriving instance Eq (T AllEnabled)
deriving instance Show (T SomeDisabled)
deriving instance Eq (T SomeDisabled)
Here's an example of use:
noC :: T SomeDisabled
noC = A ()
main :: IO ()
main = print $ case noC of
A _ -> "A"
B _ -> "B"
-- this doesn't give a warning with -Wincomplete-patterns
This solution makes pattern-matching and construction more cumbersome, because those () are always there.
A variation is to have one type family per branch (as in Trees that Grow) instead of a two-parameter type family.
I tried to achieve something like this in the past, but without much success -- I was not too satisfied with my solution.
Still, one can use GADTs to encode this constraint:
data TagA = IsA | NotA
data TagC = IsC | NotC
data T (ta :: TagA) (tc :: TagC) where
A :: T 'IsA 'NotC
B :: T 'NotA 'NotC
C :: T 'NotA 'IsC
-- existential wrappers
data TnotC where TnotC :: T ta 'NotC -> TnotC
data TnotA where TnotA :: T 'NotA tc -> TnotA
f :: TnotC
g :: TnotA
This however gets boring fast, because of the wrapping/unwrapping of the exponentials. Consumer functions are more convenient since we can write
giveMeNotAnA :: T 'NotA tc -> Int
to require anything but an A. Producer functions instead need to use existentials.
In a type with many constructors, it also gets inconvenient since we have to use a GADT with many tags/parameters. Maybe this can be streamlined with some clever typeclass machinery.
Giving each individual value its own type scales extremely badly, and is quite unnecessarily fine-grained.
What you probably want is just restrict the types by some property on their values. In e.g. Coq, that would be a subset type:
Inductive T: Type :=
| A
| B
| C.
Definition Retf: Type := { x: T | x<>C }.
Definition Retg: Type := { x: T | x<>A }.
Well, Haskell has no way of expressing such value constraints, but that doesn't stop you from creating types that conceptually fulfill them. Just use newtypes:
newtype Retf = Retf { getRetf :: T }
mkRetf :: T -> Maybe Retf
mkRetf C = Nothing
mkRetf x = Retf x
newtype Retg = Retg { getRetg :: T }
mkRetg :: ...
Then in the implementation of f, you match for the final result of mkRetf and raise an error if it's Nothing. That way, an implementation mistake that makes it give a C will unfortunately not give a compilation error, but at least a runtime error from within the function that's actually at fault, rather than somewhere further down the line.
An alternative that might be ideal for you is Liquid Haskell, which does support subset types. I can't say too much about it, but it's supposedly pretty good (and will in new GHC versions have direct support).
I'm covering polymorphism and I'm trying to see the practical uses of such a feature.
My basic understanding of Rank 2 is:
type MyType = ∀ a. a -> a
subFunction :: a -> a
subFunction el = el
mainFunction :: MyType -> Int
mainFunction func = func 3
I understand that this is allowing the user to use a polymorphic function (subFunction) inside mainFunction and strictly specify it's output (Int). This seems very similar to GADT's:
data Example a where
ExampleInt :: Int -> Example Int
ExampleBool :: Bool -> Example Bool
1) Given the above, is my understanding of Rank 2 polymorphism correct?
2) What are the general situations where Rank 2 polymorphism can be used, as opposed to GADT's, for example?
If you pass a polymorphic function as and argument to a Rank2-polymorphic function, you're essentially passing not just one function but a whole family of functions – for all possible types that fulfill the constraints.
Typically, those forall quantifiers come with a class constraint. For example, I might wish to do number arithmetic with two different types simultaneously (for comparing precision or whatever).
data FloatCompare = FloatCompare {
singlePrecision :: Float
, doublePrecision :: Double
}
Now I might want to modify those numbers through some maths operation. Something like
modifyFloat :: (Num -> Num) -> FloatCompare -> FloatCompare
But Num is not a type, only a type class. I could of course pass a function that would modify any particular number type, but I couldn't use that to modify both a Float and a Double value, at least not without some ugly (and possibly lossy) converting back and forth.
Solution: Rank-2 polymorphism!
modifyFloat :: (∀ n . Num n => n -> n) -> FloatCompare -> FloatCompare
mofidyFloat f (FloatCompare single double)
= FloatCompare (f single) (f double)
The best single example of how this is useful in practice are probably lenses. A lens is a “smart accessor function” to a field in some larger data structure. It allows you to access fields, update them, gather results... while at the same time composing in a very simple way. How it works: Rank2-polymorphism; every lens is polymorphic, with the different instantiations corresponding to the “getter” / “setter” aspects, respectively.
The go-to example of an application of rank-2 types is runST as Benjamin Hodgson mentioned in the comments. This is a rather good example and there are a variety of examples using the same trick. For example, branding to maintain abstract data type invariants across multiple types, avoiding confusion of differentials in ad, a region-based version of ST.
But I'd actually like to talk about how Haskell programmers are implicitly using rank-2 types all the time. Every type class whose methods have universally quantified types desugars to a dictionary with a field with a rank-2 type. In practice, this is virtually always a higher-kinded type class* like Functor or Monad. I'll use a simplified version of Alternative as an example. The class declaration is:
class Alternative f where
empty :: f a
(<|>) :: f a -> f a -> f a
The dictionary representing this class would be:
data AlternativeDict f = AlternativeDict {
empty :: forall a. f a,
(<|>) :: forall a. f a -> f a -> f a }
Sometimes such an encoding is nice as it allows one to use different "instances" for the same type, perhaps only locally. For example, Maybe has two obvious instances of Alternative depending on whether Just a <|> Just b is Just a or Just b. Languages without type classes, such as Scala, do indeed use this encoding.
To connect to leftaroundabout's reference to lenses, you can view the hierarchy there as a hierarchy of type classes and the lens combinators as simply tools for explicitly building the relevant type class dictionaries. Of course, the reason it isn't actually a hierarchy of type classes is that we usually will have multiple "instances" for the same type. E.g. _head and _head . _tail are both "instances" of Traversal' s a.
* A higher-kinded type class doesn't necessarily lead to this, and it can happen for a type class of kind *. For example:
-- Higher-kinded but doesn't require universal quantification.
class Sum c where
sum :: c Int -> Int
-- Not higher-kinded but does require universal quantification.
class Length l where
length :: [a] -> l
If you are using modules in Haskell, you are already using Rank-2 types. Theoretically speaking, modules are records with rank-2 type properties.
For example, the Foo module below in Haskell ...
module Foo(id) where
id :: forall a. a -> a
id x = x
import qualified Foo
main = do
putStrLn (Foo.id "hello")
return ()
... can actually be thought as a record as follows:
type FooType = FooType {
id :: forall a. a -> a
}
Foo :: FooType
Foo = Foo {
id = \x -> x
}
P/S (unrelated this question): from a language design perspective, if you are going to support module system, then you might as well support higher-rank types (i.e. allow arbitrary quantification of type variables on any level) to reduce duplication of efforts (i.e. type checking a module should be almost the same as type checking a record with higher rank types).
here's my question:
this works perfectly:
type Asdf = [Integer]
type ListOfAsdf = [Asdf]
Now I want to do the same but with the Integral class restriction:
type Asdf2 a = (Integral a) => [a]
type ListOfAsdf2 = (Integral a) => [Asdf2 a]
I got this error:
Illegal polymorphic or qualified type: Asdf2 a
Perhaps you intended to use -XImpredicativeTypes
In the type synonym declaration for `ListOfAsdf2'
I have tried a lot of things but I am still not able to create a type with a class restriction as described above.
Thanks in advance!!! =)
Dak
Ranting Against the Anti-Existentionallists
I always dislike the anti-existential type talk in Haskell as I often find existentials useful. For example, in some quick check tests I have code similar to (ironically untested code follows):
data TestOp = forall a. Testable a => T String a
tests :: [TestOp]
tests = [T "propOne:" someProp1
,T "propTwo:" someProp2
]
runTests = mapM runTest tests
runTest (T s a) = putStr s >> quickCheck a
And even in a corner of some production code I found it handy to make a list of types I'd need random values of:
type R a = Gen -> (a,Gen)
data RGen = forall a. (Serialize a, Random a) => RGen (R a)
list = [(b1, str1, random :: RGen (random :: R Type1))
,(b2, str2, random :: RGen (random :: R Type2))
]
Answering Your Question
{-# LANGUAGE ExistentialQuantification #-}
data SomeWrapper = forall a. Integral a => SW a
If you need a context, the easiest way would be to use a data declaration:
data (Integral a) => IntegralData a = ID [a]
type ListOfIntegralData a = [IntegralData a]
*Main> :t [ ID [1234,1234]]
[ID [1234,1234]] :: Integral a => [IntegralData a]
This has the (sole) effect of making sure an Integral context is added to every function that uses the IntegralData data type.
sumID :: Integral a => IntegralData a -> a
sumID (ID xs) = sum xs
The main reason a type synonym isn't working for you is that type synonyms are designed as
just that - something that replaces a type, not a type signature.
But if you want to go existential the best way is with a GADT, because it handles all the quantification issues for you:
{-# LANGUAGE GADTs #-}
data IntegralGADT where
IG :: Integral a => [a] -> IntegralGADT
type ListOfIG = [ IntegralGADT ]
Because this is essentially an existential type, you can mix them up:
*Main> :t [IG [1,1,1::Int], IG [234,234::Integer]]
[IG [1,1,1::Int],IG [234,234::Integer]] :: [ IntegralGADT ]
Which you might find quite handy, depending on your application.
The main advantage of a GADT over a data declaration is that when you pattern match, you implicitly get the Integral context:
showPointZero :: IntegralGADT -> String
showPointZero (IG xs) = show $ (map fromIntegral xs :: [Double])
*Main> showPointZero (IG [1,2,3])
"[1.0,2.0,3.0]"
But existential quantification is sometimes used for the wrong reasons,
(eg wanting to mix all your data up in one list because that's what you're
used to from dynamically typed languages, and you haven't got used to
static typing and its advantages yet).
Here I think it's more trouble than it's worth, unless you need to mix different
Integral types together without converting them. I can't see a reason
why this would help, because you'll have to convert them when you use them.
For example, you can't define
unIG (IG xs) = xs
because it doesn't even type check. Rule of thumb: you can't do stuff that mentions the type a on the right hand side.
However, this is OK because we convert the type a:
unIG :: Num b => IntegralGADT -> [b]
unIG (IG xs) = map fromIntegral xs
Here existential quantification has forced you convert your data when I think your original plan was to not have to!
You may as well convert everything to Integer instead of this.
If you want things simple, keep them simple. The data declaration is the simplest way of ensuring you don't put data in your data type unless it's already a member of some type class.
When metaprogramming, it may be useful (or necessary) to pass along to Haskell's type system information about types that's known to your program but not inferable in Hindley-Milner. Is there a library (or language extension, etc) that provides facilities for doing this—that is, programmatic type annotations—in Haskell?
Consider a situation where you're working with a heterogenous list (implemented using the Data.Dynamic library or existential quantification, say) and you want to filter the list down to a bog-standard, homogeneously typed Haskell list. You can write a function like
import Data.Dynamic
import Data.Typeable
dynListToList :: (Typeable a) => [Dynamic] -> [a]
dynListToList = (map fromJust) . (filter isJust) . (map fromDynamic)
and call it with a manual type annotation. For example,
foo :: [Int]
foo = dynListToList [ toDyn (1 :: Int)
, toDyn (2 :: Int)
, toDyn ("foo" :: String) ]
Here foo is the list [1, 2] :: [Int]; that works fine and you're back on solid ground where Haskell's type system can do its thing.
Now imagine you want to do much the same thing but (a) at the time you write the code you don't know what the type of the list produced by a call to dynListToList needs to be, yet (b) your program does contain the information necessary to figure this out, only (c) it's not in a form accessible to the type system.
For example, say you've randomly selected an item from your heterogenous list and you want to filter the list down by that type. Using the type-checking facilities supplied by Data.Typeable, your program has all the information it needs to do this, but as far as I can tell—this is the essence of the question—there's no way to pass it along to the type system. Here's some pseudo-Haskell that shows what I mean:
import Data.Dynamic
import Data.Typeable
randList :: (Typeable a) => [Dynamic] -> IO [a]
randList dl = do
tr <- randItem $ map dynTypeRep dl
return (dynListToList dl :: [<tr>]) -- This thing should have the type
-- represented by `tr`
(Assume randItem selects a random item from a list.)
Without a type annotation on the argument of return, the compiler will tell you that it has an "ambiguous type" and ask you to provide one. But you can't provide a manual type annotation because the type is not known at write-time (and can vary); the type is known at run-time, however—albeit in a form the type system can't use (here, the type needed is represented by the value tr, a TypeRep—see Data.Typeable for details).
The pseudo-code :: [<tr>] is the magic I want to happen. Is there any way to provide the type system with type information programatically; that is, with type information contained in a value in your program?
Basically I'm looking for a function with (pseudo-) type ??? -> TypeRep -> a that takes a value of a type unknown to Haskell's type system and a TypeRep and says, "Trust me, compiler, I know what I'm doing. This thing has the value represented by this TypeRep." (Note that this is not what unsafeCoerce does.)
Or is there something completely different that gets me the same place? For example, I can imagine a language extension that permits assignment to type variables, like a souped-up version of the extension enabling scoped type variables.
(If this is impossible or highly impractical,—e.g., it requires packing a complete GHCi-like interpreter into the executable—please try to explain why.)
No, you can't do this. The long and short of it is that you're trying to write a dependently-typed function, and Haskell isn't a dependently typed language; you can't lift your TypeRep value to a true type, and so there's no way to write down the type of your desired function. To explain this in a little more detail, I'm first going to show why the way you've phrased the type of randList doesn't really make sense. Then, I'm going to explain why you can't do what you want. Finally, I'll briefly mention a couple thoughts on what to actually do.
Existentials
Your type signature for randList can't mean what you want it to mean. Remembering that all type variables in Haskell are universally quantified, it reads
randList :: forall a. Typeable a => [Dynamic] -> IO [a]
Thus, I'm entitled to call it as, say, randList dyns :: IO [Int] anywhere I want; I must be able to provide a return value for all a, not simply for some a. Thinking of this as a game, it's one where the caller can pick a, not the function itself. What you want to say (this isn't valid Haskell syntax, although you can translate it into valid Haskell by using an existential data type1) is something more like
randList :: [Dynamic] -> (exists a. Typeable a => IO [a])
This promises that the elements of the list are of some type a, which is an instance of Typeable, but not necessarily any such type. But even with this, you'll have two problems. First, even if you could construct such a list, what could you do with it? And second, it turns out that you can't even construct it in the first place.
Since all that you know about the elements of the existential list is that they're instances of Typeable, what can you do with them? Looking at the documentation, we see that there are only two functions2 which take instances of Typeable:
typeOf :: Typeable a => a -> TypeRep, from the type class itself (indeed, the only method therein); and
cast :: (Typeable a, Typeable b) => a -> Maybe b (which is implemented with unsafeCoerce, and couldn't be written otherwise).
Thus, all that you know about the type of the elements in the list is that you can call typeOf and cast on them. Since we'll never be able to usefully do anything else with them, our existential might just as well be (again, not valid Haskell)
randList :: [Dynamic] -> IO [(TypeRep, forall b. Typeable b => Maybe b)]
This is what we get if we apply typeOf and cast to every element of our list, store the results, and throw away the now-useless existentially typed original value. Clearly, the TypeRep part of this list isn't useful. And the second half of the list isn't either. Since we're back to a universally-quantified type, the caller of randList is once again entitled to request that they get a Maybe Int, a Maybe Bool, or a Maybe b for any (typeable) b of their choosing. (In fact, they have slightly more power than before, since they can instantiate different elements of the list to different types.) But they can't figure out what type they're converting from unless they already know it—you've still lost the type information you were trying to keep.
And even setting aside the fact that they're not useful, you simply can't construct the desired existential type here. The error arises when you try to return the existentially-typed list (return $ dynListToList dl). At what specific type are you calling dynListToList? Recall that dynListToList :: forall a. Typeable a => [Dynamic] -> [a]; thus, randList is responsible for picking which a dynListToList is going to use. But it doesn't know which a to pick; again, that's the source of the question! So the type that you're trying to return is underspecified, and thus ambiguous.3
Dependent types
OK, so what would make this existential useful (and possible)? Well, we actually have slightly more information: not only do we know there's some a, we have its TypeRep. So maybe we can package that up:
randList :: [Dynamic] -> (exists a. Typeable a => IO (TypeRep,[a]))
This isn't quite good enough, though; the TypeRep and the [a] aren't linked at all. And that's exactly what you're trying to express: some way to link the TypeRep and the a.
Basically, your goal is to write something like
toType :: TypeRep -> *
Here, * is the kind of all types; if you haven't seen kinds before, they are to types what types are to values. * classifies types, * -> * classifies one-argument type constructors, etc. (For instance, Int :: *, Maybe :: * -> *, Either :: * -> * -> *, and Maybe Int :: *.)
With this, you could write (once again, this code isn't valid Haskell; in fact, it really bears only a passing resemblance to Haskell, as there's no way you could write it or anything like it within Haskell's type system):
randList :: [Dynamic] -> (exists (tr :: TypeRep).
Typeable (toType tr) => IO (tr, [toType tr]))
randList dl = do
tr <- randItem $ map dynTypeRep dl
return (tr, dynListToList dl :: [toType tr])
-- In fact, in an ideal world, the `:: [toType tr]` signature would be
-- inferable.
Now, you're promising the right thing: not that there exists some type which classifies the elements of the list, but that there exists some TypeRep such that its corresponding type classifies the elements of the list. If only you could do this, you would be set. But writing toType :: TypeRep -> * is completely impossible in Haskell: doing this requires a dependently-typed language, since toType tr is a type which depends on a value.
What does this mean? In Haskell, it's perfectly acceptable for values to depend on other values; this is what a function is. The value head "abc", for instance, depends on the value "abc". Similarly, we have type constructors, so it's acceptable for types to depend on other types; consider Maybe Int, and how it depends on Int. We can even have values which depend on types! Consider id :: a -> a. This is really a family of functions: id_Int :: Int -> Int, id_Bool :: Bool -> Bool, etc. Which one we have depends on the type of a. (So really, id = \(a :: *) (x :: a) -> x; although we can't write this in Haskell, there are languages where we can.)
Crucially, however, we can never have a type that depends on a value. We might want such a thing: imagine Vec 7 Int, the type of length-7 lists of integers. Here, Vec :: Nat -> * -> *: a type whose first argument must be a value of type Nat. But we can't write this sort of thing in Haskell.4 Languages which support this are called dependently-typed (and will let us write id as we did above); examples include Coq and Agda. (Such languages often double as proof assistants, and are generally used for research work as opposed to writing actual code. Dependent types are hard, and making them useful for everyday programming is an active area of research.)
Thus, in Haskell, we can check everything about our types first, throw away all that information, and then compile something that refers only to values. In fact, this is exactly what GHC does; since we can never check types at run-time in Haskell, GHC erases all the types at compile-time without changing the program's run-time behavior. This is why unsafeCoerce is easy to implement (operationally) and completely unsafe: at run-time, it's a no-op, but it lies to the type system. Consequently, something like toType is completely impossible to implement in the Haskell type system.
In fact, as you noticed, you can't even write down the desired type and use unsafeCoerce. For some problems, you can get away with this; we can write down the type for the function, but only implement it with by cheating. That's exactly how fromDynamic works. But as we saw above, there's not even a good type to give to this problem from within Haskell. The imaginary toType function allows you to give the program a type, but you can't even write down toType's type!
What now?
So, you can't do this. What should you do? My guess is that your overall architecture isn't ideal for Haskell, although I haven't seen it; Typeable and Dynamic don't actually show up that much in Haskell programs. (Perhaps you're "speaking Haskell with a Python accent", as they say.) If you only have a finite set of data types to deal with, you might be able to bundle things into a plain old algebraic data type instead:
data MyType = MTInt Int | MTBool Bool | MTString String
Then you can write isMTInt, and just use filter isMTInt, or filter (isSameMTAs randomMT).
Although I don't know what it is, there's probably a way you could unsafeCoerce your way through this problem. But frankly, that's not a good idea unless you really, really, really, really, really, really know what you're doing. And even then, it's probably not. If you need unsafeCoerce, you'll know, it won't just be a convenience thing.
I really agree with Daniel Wagner's comment: you're probably going to want to rethink your approach from scratch. Again, though, since I haven't seen your architecture, I can't say what that will mean. Maybe there's another Stack Overflow question in there, if you can distill out a concrete difficulty.
1 That looks like the following:
{-# LANGUAGE ExistentialQuantification #-}
data TypeableList = forall a. Typeable a => TypeableList [a]
randList :: [Dynamic] -> IO TypeableList
However, since none of this code compiles anyway, I think writing it out with exists is clearer.
2 Technically, there are some other functions which look relevant, such as toDyn :: Typeable a => a -> Dynamic and fromDyn :: Typeable a => Dynamic -> a -> a. However, Dynamic is more or less an existential wrapper around Typeables, relying on typeOf and TypeReps to know when to unsafeCoerce (GHC uses some implementation-specific types and unsafeCoerce, but you could do it this way, with the possible exception of dynApply/dynApp), so toDyn doesn't do anything new. And fromDyn doesn't really expect its argument of type a; it's just a wrapper around cast. These functions, and the other similar ones, don't provide any extra power that isn't available with just typeOf and cast. (For instance, going back to a Dynamic isn't very useful for your problem!)
3 To see the error in action, you can try to compile the following complete Haskell program:
{-# LANGUAGE ExistentialQuantification #-}
import Data.Dynamic
import Data.Typeable
import Data.Maybe
randItem :: [a] -> IO a
randItem = return . head -- Good enough for a short and non-compiling example
dynListToList :: Typeable a => [Dynamic] -> [a]
dynListToList = mapMaybe fromDynamic
data TypeableList = forall a. Typeable a => TypeableList [a]
randList :: [Dynamic] -> IO TypeableList
randList dl = do
tr <- randItem $ map dynTypeRep dl
return . TypeableList $ dynListToList dl -- Error! Ambiguous type variable.
Sure enough, if you try to compile this, you get the error:
SO12273982.hs:17:27:
Ambiguous type variable `a0' in the constraint:
(Typeable a0) arising from a use of `dynListToList'
Probable fix: add a type signature that fixes these type variable(s)
In the second argument of `($)', namely `dynListToList dl'
In a stmt of a 'do' block: return . TypeableList $ dynListToList dl
In the expression:
do { tr <- randItem $ map dynTypeRep dl;
return . TypeableList $ dynListToList dl }
But as is the entire point of the question, you can't "add a type signature that fixes these type variable(s)", because you don't know what type you want.
4 Mostly. GHC 7.4 has support for lifting types to kinds and for kind polymorphism; see section 7.8, "Kind polymorphism and promotion", in the GHC 7.4 user manual. This doesn't make Haskell dependently typed—something like TypeRep -> * example is still out5—but you will be able to write Vec by using very expressive types that look like values.
5 Technically, you could now write down something which looks like it has the desired type: type family ToType :: TypeRep -> *. However, this takes a type of the promoted kind TypeRep, and not a value of the type TypeRep; and besides, you still wouldn't be able to implement it. (At least I don't think so, and I can't see how you would—but I am not an expert in this.) But at this point, we're pretty far afield.
What you're observing is that the type TypeRep doesn't actually carry any type-level information along with it; only term-level information. This is a shame, but we can do better when we know all the type constructors we care about. For example, suppose we only care about Ints, lists, and function types.
{-# LANGUAGE GADTs, TypeOperators #-}
import Control.Monad
data a :=: b where Refl :: a :=: a
data Dynamic where Dynamic :: TypeRep a -> a -> Dynamic
data TypeRep a where
Int :: TypeRep Int
List :: TypeRep a -> TypeRep [a]
Arrow :: TypeRep a -> TypeRep b -> TypeRep (a -> b)
class Typeable a where typeOf :: TypeRep a
instance Typeable Int where typeOf = Int
instance Typeable a => Typeable [a] where typeOf = List typeOf
instance (Typeable a, Typeable b) => Typeable (a -> b) where
typeOf = Arrow typeOf typeOf
congArrow :: from :=: from' -> to :=: to' -> (from -> to) :=: (from' -> to')
congArrow Refl Refl = Refl
congList :: a :=: b -> [a] :=: [b]
congList Refl = Refl
eq :: TypeRep a -> TypeRep b -> Maybe (a :=: b)
eq Int Int = Just Refl
eq (Arrow from to) (Arrow from' to') = liftM2 congArrow (eq from from') (eq to to')
eq (List t) (List t') = liftM congList (eq t t')
eq _ _ = Nothing
eqTypeable :: (Typeable a, Typeable b) => Maybe (a :=: b)
eqTypeable = eq typeOf typeOf
toDynamic :: Typeable a => a -> Dynamic
toDynamic a = Dynamic typeOf a
-- look ma, no unsafeCoerce!
fromDynamic_ :: TypeRep a -> Dynamic -> Maybe a
fromDynamic_ rep (Dynamic rep' a) = case eq rep rep' of
Just Refl -> Just a
Nothing -> Nothing
fromDynamic :: Typeable a => Dynamic -> Maybe a
fromDynamic = fromDynamic_ typeOf
All of the above is pretty standard. For more on the design strategy, you'll want to read about GADTs and singleton types. Now, the function you want to write follows; the type is going to look a bit daft, but bear with me.
-- extract only the elements of the list whose type match the head
firstOnly :: [Dynamic] -> Dynamic
firstOnly [] = Dynamic (List Int) []
firstOnly (Dynamic rep v:xs) = Dynamic (List rep) (v:go xs) where
go [] = []
go (Dynamic rep' v:xs) = case eq rep rep' of
Just Refl -> v : go xs
Nothing -> go xs
Here we've picked a random element (I rolled a die, and it came up 1) and extracted only the elements that have a matching type from the list of dynamic values. Now, we could have done the same thing with regular boring old Dynamic from the standard libraries; however, what we couldn't have done is used the TypeRep in a meaningful way. I now demonstrate that we can do so: we'll pattern match on the TypeRep, and then use the enclosed value at the specific type the TypeRep tells us it is.
use :: Dynamic -> [Int]
use (Dynamic (List (Arrow Int Int)) fs) = zipWith ($) fs [1..]
use (Dynamic (List Int) vs) = vs
use (Dynamic Int v) = [v]
use (Dynamic (Arrow (List Int) (List (List Int))) f) = concat (f [0..5])
use _ = []
Note that on the right-hand sides of these equations, we are using the wrapped value at different, concrete types; the pattern match on the TypeRep is actually introducing type-level information.
You want a function that chooses a different type of values to return based on runtime data. Okay, great. But the whole purpose of a type is to tell you what operations can be performed on a value. When you don't know what type will be returned from a function, what do you do with the values it returns? What operations can you perform on them? There are two options:
You want to read the type, and perform some behaviour based on which type it is. In this case you can only cater for a finite list of types known in advance, essentially by testing "is it this type? then we do this operation...". This is easily possible in the current Dynamic framework: just return the Dynamic objects, using dynTypeRep to filter them, and leave the application of fromDynamic to whoever wants to consume your result. Moreover, it could well be possible without Dynamic, if you don't mind setting the finite list of types in your producer code, rather than your consumer code: just use an ADT with a constructor for each type, data Thing = Thing1 Int | Thing2 String | Thing3 (Thing,Thing). This latter option is by far the best if it is possible.
You want to perform some operation that works across a family of types, potentially some of which you don't know about yet, e.g. by using type class operations. This is trickier, and it's tricky conceptually too, because your program is not allowed to change behaviour based on whether or not some type class instance exists – it's an important property of the type class system that the introduction of a new instance can either make a program type check or stop it from type checking, but it can't change the behaviour of a program. Hence you can't throw an error if your input list contains inappropriate types, so I'm really not sure that there's anything you can do that doesn't essentially involve falling back to the first solution at some point.