I was thinking about how I could save myself from undefinition, and one idea I had was to enumerate all possible sources of partiality. At least I would know what of to beware. I found three yet:
Incomplete pattern matches or guards.
Recursion. (Optionally excluding structural recursion on algebraic types.)
If a function is unsafe, any use of that function infects the user code. (Should I be saying "partiality is transitive"?)
I have heard of other ways to obtain a logical contradiction, for instance by using negative types, but I am not sure if anything of that sort applies to Haskell. There are many logical paradoxes out there, and some of them can be encoded in Haskell, but may it be true that any logical paradox requires the use of recursion, and is therefore covered by the point 2 above?
For instance, if it were proven that a Haskell expression free of recursion can always be evaluated to normal form, then the three points I give would be a complete list. I fuzzily remember seeing something like a proof of this in one of Simon Peyton Jones's books, but that was written like 30 years ago, so even if I remember correctly and it used to apply to a prototype Haskell back then, it may be false today, seeing how many a language extension we have. Possibly some of them enable other ways to undefine a program?
And then, if it were so easy to detect expressions that cannot be partial, why do we not do that? How easier would life be!
This is a partial answer (pun intended), where I'll only list a few arguably non obvious ways one can achieve non termination.
First, I'll confirm that negative-recursive types can indeed cause non termination. Indeed, it is known that allowing a recursive type such as
data R a = R (R a -> a)
allows one to define fix, and obtain non termination from there.
{-# LANGUAGE ScopedTypeVariables #-}
{-# OPTIONS -Wall #-}
data R a = R (R a -> a)
selfApply :: R a -> a
selfApply t#(R x) = x t
-- Church's fixed point combinator Y
-- fix f = (\x. f (x x))(\x. f (x x))
fix :: forall a. (a -> a) -> a
fix f = selfApply (R (\x -> f (selfApply x)))
Total languages like Coq or Agda prohibit this by requiring recursive types to use only strictly-positive recursion.
Another potential source of non-termination is that Haskell allows Type :: Type. As far as I can see, that makes it possible to encode System U in Haskell, where Girard's paradox can be used to cause a logical inconsistency, constructing a term of type Void. That term (as far as I understand) would be non terminating.
Girard's paradox is unfortunately rather complex to fully describe, and I have not completely studied it yet. I only know it is related to the so-called hypergame, a game where the first move is to choose a finite game to play. A finite game is one which causes every match to terminate after finitely many moves. The next moves after that would correspond to a match according to the chosen finite game at step one. Here's the paradox: since the chosen game must be finite, no matter what it is, the whole hypergame match will always terminate after a finite amount of moves. This makes hypergame itself a finite game, making the infinite sequence of moves "I choose hypergame, I choose hypergame, ..." a valid play, in turn proving that hypergame is not finite.
Apparently, this argument can be encoded in a rich enough pure type system like System U, and Type :: Type allows to embed the same argument.
I would like to define a type for infinite number sequence in haskell. My idea is:
type MySeq = Natural -> Ratio Integer
However, I would also like to be able to define some properties of the sequence on the type level. A simple example would be a non-decreasing sequence like this. Is this possible to do this with current dependent-type capabilities of GHC?
EDIT: I came up with the following idea:
type PositiveSeq = Natural -> Ratio Natural
data IncreasingSeq = IncreasingSeq {
start :: Ratio Natural,
diff :: PositiveSeq}
type IKnowItsIncreasing = [Ratio Natural]
getSeq :: IncreasingSeq -> IKnowItsIncreasing
getSeq s = scanl (+) (start s) [diff s i | i <- [1..]]
Of course, it's basically a hack and not actually type safe at all.
This isn't doing anything very fancy with types, but you could change how you interpret a sequence of naturals to get essentially the same guarantee.
I think you are thinking along the right lines in your edit to the question. Consider
data IncreasingSeq = IncreasingSeq (Integer -> Ratio Natural)
where each ratio represents how much it has increased from the previous number (starting with 0).
Then you can provide a single function
applyToIncreasing :: ([Ratio Natural] -> r) -> IncreasingSeq -> r
applyToIncreasing f (IncreasingSeq s) = f . drop 1 $ scanl (+) 0 (map (s $) [0..])
This should let you deconstruct it in any way, without allowing the function to inspect the real structure.
You just need a way to construct it: probably a fromList that just sorts it and an insert that performs a standard ordered insertion.
It pains part of me to say this, but I don't think you'd gain anything over this using fancy type tricks: there are only three functions that could ever possibly go wrong, and they are fairly simple to correctly implement. The implementation is hidden so anything that uses those is correct as a result of those functions being correct. Just don't export the data constructor for IncreasingSeq.
I would also suggest considering making [Ratio Natural] be the underlying representation. It simplifies things and guarantees that there are no "gaps" in the sequence (so it is guaranteed to be a sequence).
If you want more safety and can take the performance hit, you can use data Nat = Z | S Nat instead of Natural.
I will say that if this was Coq, or a similar language, instead of Haskell I would be more likely to suggest doing some fancier type-level stuff (depending on what you are trying to accomplish) for a couple reasons:
In systems like Coq, you are usually proving theorems about the code. Because of this, it can be useful to have a type-level proof that a certain property holds. Since Haskell doesn't really have a builtin way to prove those sorts of theorems, the utility diminishes.
On the other hand, we can (sometimes) construct data types that essentially must have the properties we want using a small number of trusted functions and a hidden implementation. In the context of a system with more theorem proving capability, like Coq, this might be harder to convince theorem prover of the property than if we used a dependent type (possibly, at least). In Haskell, however, we don't have that issue in the first place.
I'm trying to get my head around F-algebras, and this article does a pretty good job. I understand the notion of a dual in category theory, but I'm having a hard time understanding how F-coalgebras (the dual of F-algebras) relate to lazy data structures in Haskell.
F-algebras are described with an endofunctor with the function: F a -> a, which makes sense if you think of F a as an expression, and a as the result of evaluating that expression, as the linked article explains it.
Being the dual of F-algebras, the corresponding function for a F-coalgebra would be a -> F a. Wikipedia says that F-coalgebras can be used to create infinite, lazy data structures. How does the a -> F a functon allow one to create infinite, lazy data structures? Also, with that in mind, since Haskell is at it's core lazy, are most data-types in Haskell F-coalgebras instead of F-algebras? Are F-algebras not lazily evaluated?
If data types (or at least the ones that are capable of infinite data) are based on F-coalgebras in Haskell, what is the a -> F a function for lists, for example? What is the terminal F-coalgebra for lists?
Making an infinite list [1,2,3,4...] might look like this in Haskell:
list = 1 : map (+ 1) list
Does this use F-coalgebras somehow? Do infinite data structures require a notion of lazy evaluation and recursion alongside the use of F-coalgebras? Am I missing something here?
A coalgebra A -> F A can be used to peel away the outer layer of a (possibly infinite) data structure. For lists of X, the functor is F a = Maybe (X, a), the same as in the algebraic view. In haskell the function for the coalgebra is
headView :: [a] -> Maybe (a, [a])
headView [] = Nothing
headView (x:xs) = Just (x,xs)
unfoldr is the unfold corresponding to this coalgebra, just like foldr is the fold corresponding to this algebra.
If you consider [a] not as the type of lists, but as a the type of descriptions of lists or programs, then this allows you to construct (seemingly) infinite values, just with a necessarily finite description.
As you can see, a Haskell list looks like both an F-algebra and an F-coalgebra. This is possible because Haskell is not actually consistent. You can fold an unfold, and get an infinite loop. Languages like coq and agda make the distinction between data types (F-algebras) and codata types (F-coalgebras) explicit. In those languages you have two list types, an algebraic List and a coalgebraic Colist.
A list in Haskell might look like this:
data List a = Nil | Cons a (List a)
A type theoretic interpretation is:
λα.μβ.1+αβ
which encodes the list type as the fixed point of a functor. In Haskell this could be represented:
data Fix f = In (f (Fix f))
data ListF a b = Nil | Cons a b
type List a = Fix (ListF a)
I'm curious about the scope of the earlier μ binder. Can a name bound in an outer scope remain available in an inner scope? Is, say, the following a valid expression:
μγ.1+(μβ.1+γβ)γ
...perhaps it's the same as:
μβ.μγ.1+(1+γβ)γ
...but then how would things change when the name is reused:
μβ.μγ.1+(μβ.1+γβ)γ
Are the above all regular types?
Scoping of μ works no different from other binders, so yes, all your examples are valid. They are also regular, because they do not even contain a λ. (*)
As for equality, that depends on what sort of μ-types you have. There are basically two different notions:
equi-recursive: in that case, the typing rules assume an equivalence
μα.T = T[μα.T / α]
i.e., a recursive type is considered equal to its one-level 'unrolling', where the μ is removed and the μ-bound variable is replaced by the type itself (and because this rule can be applied repeatedly, one can unroll arbitrary many times).
iso-recursive: here, no such equivalence exists. Instead, a μ-type is a separate form of type with its own expression forms to introduce and eliminate it -- they are usually called roll and unroll (or fold and unfold), and are typed as follows:
roll : T[μα.T / α] → μα.T
unroll : μα.T → T[μα.T / α]
These must be applied explicitly on the term level to mirror the equation above (once for each level of unrolling).
Functional languages like ML or Haskell usually use the latter for their interpretation of datatypes. However, the roll/unroll is built into the use of the data constructors. So each constructor is an injection into an iso-recursive type composed with an injection into a sum type (and inversely when matched).
Your examples are all different under the iso-recursive interpretation. The first and the third are the same under an equi-recursive interpretation, because the outer μ just disappears when you apply the above equivalence.
(*) Edit: An irregular recursive type is one whose infinite expansion does not correspond to a regular tree (or equivalently, can not be represented by a finite cyclic graph). Such a case can only be expressed with recursive type constructors, i.e., a λ that occurs under a μ. For example, μα.λβ.1+α(β×β) -- corresponding to the recursive equation t(β) = 1+t(β×β) -- would be irregular, because the recursive type constructor α is recursively applied to a type "larger" than its argument, and hence every application is a recursive type that "grows" indefinitely (and consequently, you cannot draw it as a graph).
It's worth noting, however, that in most type theories with μ, its bound variable is restricted to ground kind, so cannot express irregular types at all. In particular, a theory with unrestricted equi-recursive type constructors would have non-terminating type normalisation, so type equivalence (and thus type checking) would be undecidable. For iso-recursive types you'd need higher-order roll/unroll, which is possible, but I'm not aware of much literature investigating it.
Your μ type expressions are valid. I believe your types are regular as well since you only use recursion, sum, and products.
The type
T1 = μγ.1+(μβ.1+γβ)γ
does not look equal to
T2 = μβ.μγ.1+(1+γβ)γ
since inr (inr (inl *, inr (inl *)), inl *) has the second type but not the first.
The last type
T3 = μβ.μγ.1+(μβ.1+γβ)γ
is equal to (α-converting the first β)
μ_.μγ.1+(μβ.1+γβ)γ
which is, unfolding the top-level μ,
μγ.1+(μβ.1+γβ)γ
which is T1.
Basically, the scope of μ-bound variables follows the same rules of λ-bound variables. That is, the value of each occurrence of a variable β is provided by the closest μβ on top of it.
I have seen several sources echo the opinion that "Haskell is gradually becoming a dependently-typed language". The implication seems to be that with more and more language extensions, Haskell is drifting in that general direction, but isn't there yet.
There are basically two things I would like to know. The first is, quite simply, what does "being a dependently-typed language" actually mean? (Hopefully without being too technical about it.)
The second question is... what's the drawback? I mean, people know we're heading that way, so there must be some advantage to it. And yet, we're not there yet, so there must be some downside stopping people going all the way. I get the impression that the problem is a steep increase in complexity. But, not really understanding what dependent typing is, I don't know for sure.
What I do know is that every time I start reading about a dependently-typed programming language, the text is utterly incomprehensible... Presumably that's the problem. (?)
Dependently Typed Haskell, Now?
Haskell is, to a small extent, a dependently typed language. There is a notion of type-level data, now more sensibly typed thanks to DataKinds, and there is some means (GADTs) to give a run-time
representation to type-level data. Hence, values of run-time stuff effectively show up in types, which is what it means for a language to be dependently typed.
Simple datatypes are promoted to the kind level, so that the values
they contain can be used in types. Hence the archetypal example
data Nat = Z | S Nat
data Vec :: Nat -> * -> * where
VNil :: Vec Z x
VCons :: x -> Vec n x -> Vec (S n) x
becomes possible, and with it, definitions such as
vApply :: Vec n (s -> t) -> Vec n s -> Vec n t
vApply VNil VNil = VNil
vApply (VCons f fs) (VCons s ss) = VCons (f s) (vApply fs ss)
which is nice. Note that the length n is a purely static thing in
that function, ensuring that the input and output vectors have the
same length, even though that length plays no role in the execution of
vApply. By contrast, it's much trickier (i.e., impossible) to
implement the function which makes n copies of a given x (which
would be pure to vApply's <*>)
vReplicate :: x -> Vec n x
because it's vital to know how many copies to make at run-time. Enter
singletons.
data Natty :: Nat -> * where
Zy :: Natty Z
Sy :: Natty n -> Natty (S n)
For any promotable type, we can build the singleton family, indexed
over the promoted type, inhabited by run-time duplicates of its
values. Natty n is the type of run-time copies of the type-level n
:: Nat. We can now write
vReplicate :: Natty n -> x -> Vec n x
vReplicate Zy x = VNil
vReplicate (Sy n) x = VCons x (vReplicate n x)
So there you have a type-level value yoked to a run-time value:
inspecting the run-time copy refines static knowledge of the
type-level value. Even though terms and types are separated, we can
work in a dependently typed way by using the singleton construction as
a kind of epoxy resin, creating bonds between the phases. That's a
long way from allowing arbitrary run-time expressions in types, but it ain't nothing.
What's Nasty? What's Missing?
Let's put a bit of pressure on this technology and see what starts
wobbling. We might get the idea that singletons should be manageable a
bit more implicitly
class Nattily (n :: Nat) where
natty :: Natty n
instance Nattily Z where
natty = Zy
instance Nattily n => Nattily (S n) where
natty = Sy natty
allowing us to write, say,
instance Nattily n => Applicative (Vec n) where
pure = vReplicate natty
(<*>) = vApply
That works, but it now means that our original Nat type has spawned
three copies: a kind, a singleton family and a singleton class. We
have a rather clunky process for exchanging explicit Natty n values
and Nattily n dictionaries. Moreover, Natty is not Nat: we have
some sort of dependency on run-time values, but not at the type we
first thought of. No fully dependently typed language makes dependent
types this complicated!
Meanwhile, although Nat can be promoted, Vec cannot. You can't
index by an indexed type. Full on dependently typed languages impose
no such restriction, and in my career as a dependently typed show-off,
I've learned to include examples of two-layer indexing in my talks,
just to teach folks who've made one-layer indexing
difficult-but-possible not to expect me to fold up like a house of
cards. What's the problem? Equality. GADTs work by translating the
constraints you achieve implicitly when you give a constructor a
specific return type into explicit equational demands. Like this.
data Vec (n :: Nat) (x :: *)
= n ~ Z => VNil
| forall m. n ~ S m => VCons x (Vec m x)
In each of our two equations, both sides have kind Nat.
Now try the same translation for something indexed over vectors.
data InVec :: x -> Vec n x -> * where
Here :: InVec z (VCons z zs)
After :: InVec z ys -> InVec z (VCons y ys)
becomes
data InVec (a :: x) (as :: Vec n x)
= forall m z (zs :: Vec x m). (n ~ S m, as ~ VCons z zs) => Here
| forall m y z (ys :: Vec x m). (n ~ S m, as ~ VCons y ys) => After (InVec z ys)
and now we form equational constraints between as :: Vec n x and
VCons z zs :: Vec (S m) x where the two sides have syntactically
distinct (but provably equal) kinds. GHC core is not currently
equipped for such a concept!
What else is missing? Well, most of Haskell is missing from the type
level. The language of terms which you can promote has just variables
and non-GADT constructors, really. Once you have those, the type family machinery allows you to write type-level programs: some of
those might be quite like functions you would consider writing at the
term level (e.g., equipping Nat with addition, so you can give a
good type to append for Vec), but that's just a coincidence!
Another thing missing, in practice, is a library which makes
use of our new abilities to index types by values. What do Functor
and Monad become in this brave new world? I'm thinking about it, but
there's a lot still to do.
Running Type-Level Programs
Haskell, like most dependently typed programming languages, has two
operational semanticses. There's the way the run-time system runs
programs (closed expressions only, after type erasure, highly
optimised) and then there's the way the typechecker runs programs
(your type families, your "type class Prolog", with open expressions). For Haskell, you don't normally mix
the two up, because the programs being executed are in different
languages. Dependently typed languages have separate run-time and
static execution models for the same language of programs, but don't
worry, the run-time model still lets you do type erasure and, indeed,
proof erasure: that's what Coq's extraction mechanism gives you;
that's at least what Edwin Brady's compiler does (although Edwin
erases unnecessarily duplicated values, as well as types and
proofs). The phase distinction may not be a distinction of syntactic category
any longer, but it's alive and well.
Dependently typed languages, being total, allow the typechecker to run
programs free from the fear of anything worse than a long wait. As
Haskell becomes more dependently typed, we face the question of what
its static execution model should be? One approach might be to
restrict static execution to total functions, which would allow us the
same freedom to run, but might force us to make distinctions (at least
for type-level code) between data and codata, so that we can tell
whether to enforce termination or productivity. But that's not the only
approach. We are free to choose a much weaker execution model which is
reluctant to run programs, at the cost of making fewer equations come
out just by computation. And in effect, that's what GHC actually
does. The typing rules for GHC core make no mention of running
programs, but only for checking evidence for equations. When
translating to the core, GHC's constraint solver tries to run your type-level programs,
generating a little silvery trail of evidence that a given expression
equals its normal form. This evidence-generation method is a little
unpredictable and inevitably incomplete: it fights shy of
scary-looking recursion, for example, and that's probably wise. One
thing we don't need to worry about is the execution of IO
computations in the typechecker: remember that the typechecker doesn't have to give
launchMissiles the same meaning that the run-time system does!
Hindley-Milner Culture
The Hindley-Milner type system achieves the truly awesome coincidence
of four distinct distinctions, with the unfortunate cultural
side-effect that many people cannot see the distinction between the
distinctions and assume the coincidence is inevitable! What am I
talking about?
terms vs types
explicitly written things vs implicitly written things
presence at run-time vs erasure before run-time
non-dependent abstraction vs dependent quantification
We're used to writing terms and leaving types to be inferred...and
then erased. We're used to quantifying over type variables with the
corresponding type abstraction and application happening silently and
statically.
You don't have to veer too far from vanilla Hindley-Milner
before these distinctions come out of alignment, and that's no bad thing. For a start, we can have more interesting types if we're willing to write them in a few
places. Meanwhile, we don't have to write type class dictionaries when
we use overloaded functions, but those dictionaries are certainly
present (or inlined) at run-time. In dependently typed languages, we
expect to erase more than just types at run-time, but (as with type
classes) that some implicitly inferred values will not be
erased. E.g., vReplicate's numeric argument is often inferable from the type of the desired vector, but we still need to know it at run-time.
Which language design choices should we review because these
coincidences no longer hold? E.g., is it right that Haskell provides
no way to instantiate a forall x. t quantifier explicitly? If the
typechecker can't guess x by unifiying t, we have no other way to
say what x must be.
More broadly, we cannot treat "type inference" as a monolithic concept
that we have either all or nothing of. For a start, we need to split
off the "generalisation" aspect (Milner's "let" rule), which relies heavily on
restricting which types exist to ensure that a stupid machine can
guess one, from the "specialisation" aspect (Milner's "var" rule)
which is as effective as your constraint solver. We can expect that
top-level types will become harder to infer, but that internal type
information will remain fairly easy to propagate.
Next Steps For Haskell
We're seeing the type and kind levels grow very similar (and they
already share an internal representation in GHC). We might as well
merge them. It would be fun to take * :: * if we can: we lost
logical soundness long ago, when we allowed bottom, but type
soundness is usually a weaker requirement. We must check. If we must have
distinct type, kind, etc levels, we can at least make sure everything
at the type level and above can always be promoted. It would be great
just to re-use the polymorphism we already have for types, rather than
re-inventing polymorphism at the kind level.
We should simplify and generalise the current system of constraints by
allowing heterogeneous equations a ~ b where the kinds of a and
b are not syntactically identical (but can be proven equal). It's an
old technique (in my thesis, last century) which makes dependency much
easier to cope with. We'd be able to express constraints on
expressions in GADTs, and thus relax restrictions on what can be
promoted.
We should eliminate the need for the singleton construction by
introducing a dependent function type, pi x :: s -> t. A function
with such a type could be applied explicitly to any expression of type s which
lives in the intersection of the type and term languages (so,
variables, constructors, with more to come later). The corresponding
lambda and application would not be erased at run-time, so we'd be
able to write
vReplicate :: pi n :: Nat -> x -> Vec n x
vReplicate Z x = VNil
vReplicate (S n) x = VCons x (vReplicate n x)
without replacing Nat by Natty. The domain of pi can be any
promotable type, so if GADTs can be promoted, we can write dependent
quantifier sequences (or "telescopes" as de Briuijn called them)
pi n :: Nat -> pi xs :: Vec n x -> ...
to whatever length we need.
The point of these steps is to eliminate complexity by working directly with more general tools, instead of making do with weak tools and clunky encodings. The current partial buy-in makes the benefits of Haskell's sort-of dependent types more expensive than they need to be.
Too Hard?
Dependent types make a lot of people nervous. They make me nervous,
but I like being nervous, or at least I find it hard not to be nervous
anyway. But it doesn't help that there's quite such a fog of ignorance
around the topic. Some of that's due to the fact that we all still
have a lot to learn. But proponents of less radical approaches have
been known to stoke fear of dependent types without always making sure
the facts are wholly with them. I won't name names. These "undecidable typechecking", "Turing incomplete", "no phase distinction", "no type erasure", "proofs everywhere", etc, myths persist, even though they're rubbish.
It's certainly not the case that dependently typed programs must
always be proven correct. One can improve the basic hygiene of one's
programs, enforcing additional invariants in types without going all
the way to a full specification. Small steps in this direction quite
often result in much stronger guarantees with few or no additional
proof obligations. It is not true that dependently typed programs are
inevitably full of proofs, indeed I usually take the presence of any
proofs in my code as the cue to question my definitions.
For, as with any increase in articulacy, we become free to say foul
new things as well as fair. E.g., there are plenty of crummy ways to
define binary search trees, but that doesn't mean there isn't a good way. It's important not to presume that bad experiences cannot be
bettered, even if it dents the ego to admit it. Design of dependent
definitions is a new skill which takes learning, and being a Haskell
programmer does not automatically make you an expert! And even if some
programs are foul, why would you deny others the freedom to be fair?
Why Still Bother With Haskell?
I really enjoy dependent types, but most of my hacking projects are
still in Haskell. Why? Haskell has type classes. Haskell has useful
libraries. Haskell has a workable (although far from ideal) treatment
of programming with effects. Haskell has an industrial strength
compiler. The dependently typed languages are at a much earlier stage
in growing community and infrastructure, but we'll get there, with a
real generational shift in what's possible, e.g., by way of
metaprogramming and datatype generics. But you just have to look
around at what people are doing as a result of Haskell's steps towards
dependent types to see that there's a lot of benefit to be gained by
pushing the present generation of languages forwards, too.
Dependent typing is really just the unification of the value and type levels, so you can parametrize values on types (already possible with type classes and parametric polymorphism in Haskell) and you can parametrize types on values (not, strictly speaking, possible yet in Haskell, although DataKinds gets very close).
Edit: Apparently, from this point forward, I was wrong (see #pigworker's comment). I'll preserve the rest of this as a record of the myths I've been fed. :P
The issue with moving to full dependent typing, from what I've heard, is that it would break the phase restriction between the type and value levels that allows Haskell to be compiled to efficient machine code with erased types. With our current level of technology, a dependently typed language must go through an interpreter at some point (either immediately, or after being compiled to dependently-typed bytecode or similar).
This is not necessarily a fundamental restriction, but I'm not personally aware of any current research that looks promising in this regard but that has not already made it into GHC. If anyone else knows more, I would be happy to be corrected.
John that's another common misconception about dependent types: that they don't work when data is only available at run-time. Here's how you can do the getLine example:
data Some :: (k -> *) -> * where
Like :: p x -> Some p
fromInt :: Int -> Some Natty
fromInt 0 = Like Zy
fromInt n = case fromInt (n - 1) of
Like n -> Like (Sy n)
withZeroes :: (forall n. Vec n Int -> IO a) -> IO a
withZeroes k = do
Like n <- fmap (fromInt . read) getLine
k (vReplicate n 0)
*Main> withZeroes print
5
VCons 0 (VCons 0 (VCons 0 (VCons 0 (VCons 0 VNil))))
Edit: Hm, that was supposed to be a comment to pigworker's answer. I clearly fail at SO.
pigworker gives an excellent discussion of why we should be headed towards dependent types: (a) they're awesome; (b) they would actually simplify a lot of what Haskell already does.
As for the "why not?" question, there are a couple points I think. The first point is that while the basic notion behind dependent types is easy (allow types to depend on values), the ramifications of that basic notion are both subtle and profound. For example, the distinction between values and types is still alive and well; but discussing the difference between them becomes far more nuanced than in yer Hindley--Milner or System F. To some extent this is due to the fact that dependent types are fundamentally hard (e.g., first-order logic is undecidable). But I think the bigger problem is really that we lack a good vocabulary for capturing and explaining what's going on. As more and more people learn about dependent types, we'll develop a better vocabulary and so things will become easier to understand, even if the underlying problems are still hard.
The second point has to do with the fact that Haskell is growing towards dependent types. Because we're making incremental progress towards that goal, but without actually making it there, we're stuck with a language that has incremental patches on top of incremental patches. The same sort of thing has happened in other languages as new ideas became popular. Java didn't use to have (parametric) polymorphism; and when they finally added it, it was obviously an incremental improvement with some abstraction leaks and crippled power. Turns out, mixing subtyping and polymorphism is inherently hard; but that's not the reason why Java Generics work the way they do. They work the way they do because of the constraint to be an incremental improvement to older versions of Java. Ditto, for further back in the day when OOP was invented and people started writing "objective" C (not to be confused with Objective-C), etc. Remember, C++ started out under the guise of being a strict superset of C. Adding new paradigms always requires defining the language anew, or else ending up with some complicated mess. My point in all of this is that, adding true dependent types to Haskell is going to require a certain amount of gutting and restructuring the language--- if we're going to do it right. But it's really hard to commit to that kind of an overhaul, whereas the incremental progress we've been making seems cheaper in the short term. Really, there aren't that many people who hack on GHC, but there's a goodly amount of legacy code to keep alive. This is part of the reason why there are so many spinoff languages like DDC, Cayenne, Idris, etc.