haskell - singletons - idris dependent type
Why not be dependently typed? (3)
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
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
Natty n is the type of run-time copies of the type-level
:: 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
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!
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
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)
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
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
computations in the typechecker: remember that the typechecker doesn't have to give
launchMissiles the same meaning that the run-time system does!
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
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
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
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
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
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)
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.
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.
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.