|Version 16 (modified by goldfire, 2 years ago) (diff)|
Adding dependent types to Haskell
This page is to track design and implementation ideas around adding a form of dependent types to Haskell. This work will also fix bug #7961. Richard Eisenberg (a.k.a. goldfire) is expecting to take on most (all?) of this work.
Disclaimer: Everything below represents a research proposal. While it is my (RAE's) hope that something resembling this all will actually make it into GHC, no one should read anything too strongly into words like "will happen".
Surface Language Design
It is possible to fix #7961 without any surface language changes, as that bug addresses only lifting restrictions on promotion. There is a chance that this bugfix will enter HEAD without all of the other features below, but this writeup generally will not consider fixing #7961 separate from adding dependent types.
Merging Types and Kinds
Following the work in the kind equality paper, the new Haskell will merge types and kinds into one syntactic and semantic category. Haskell would have the * :: * property. As a consequence, it will be easily possible to explicit quantify over kinds. In other words, the following type signature is allowed: forall (k :: *) (a :: k). Proxy a -> Proxy a. Furthermore, kind variables will be able to be listed explicitly when declaring datatypes and classes. Of course, if a kind variable is listed explicitly in the declaration of a type or class, then it also must be listed explicitly at the use sites. Note that this change will completely eliminate BOX.
There are actually two separate aspects to this change:
- Merge the grammar of types and kinds. This is a simplification (with a sizable caveat) of the current scenario and will fix the original motivation for #8706.
- Add * :: *. Why do this? One alternative is to go the route of Coq and Agda and have an infinite tower of type universes. But, this adds a lot of complexity. These languages take this route because * :: * makes a language inconsistent as a logic. However, Haskell is already inconsistent as a logic (because of undefined and GHC.Exts.Any) and so we don't have to worry about a new source of inconsistency. Furthermore, the type safety of Haskell does not depend on its own consistency -- unlike Coq and Agda, Haskell relies on the consistency of a coercion language, which is not threatened by * :: *. See the paper for more details.
As pointed out in the Hasochism paper, Haskell currently enjoys a confluence of design decisions. One says that compile-time arguments are elided in runtime code. For example, when calling map :: (a -> b) -> [a] -> [b], the type instantiations for a and b are properly arguments to map (and are passed quite explicitly in Core), but these arguments are always elided in surface Haskell. As the levels are mixing, we may want to revisit this. Along similar lines, type arguments in Haskell are always erasable -- that is, instantiations for types are never kept at runtime. While this is generally a Good Thing and powers much of Haskell's efficiency, dependent typing relies on keeping some types around at runtime. Here, it is even more apparent that sometimes, we want to be able to pass in values for type arguments, especially if those values can be inspected at runtime.
Haskell currently has three quantifiers: forall, ->, and =>, as classified in the following table:
|forall||yes||No (unification)||No (FVs)||No|
|->||no||Yes (as term)||Yes||Yes|
- Dependent means that the quantified thing (henceforth, quantifiee) can appear later in the type. This is clearly true for forall-quantified things and clearly not true for ->-quantified things. (That is, if we have Int -> Bool, we can't mention the Int value after the ->!)
- Visibility refers to whether or not the argument must appear at call sites in the program text. If something is not visible, the table lists how GHC is to fill in the missing bit at call sites. If something is visible, we must specify how it is parsed, noting that the term- and type-level parsers are different. For example, if f :: forall a. Ord a => a -> Int, then a call must look like f "foo", omitting the type argument and the Ord String argument.
- A required quantification is one that must textually appear in the type. Note that Haskell freely infers the type a -> a really to mean forall a. a -> a, by looking for free variables (abbreviated to FVs, above). Haskell currently does slightly more than analyze just free variables, though: it also quantifies over free kind variables that do not textually appear in a type. For example, the type Proxy a -> Proxy a really means (in today's Haskell) forall (k :: BOX) (a :: k). Proxy a -> Proxy a, even though k does not appear in the body of the type. Note that a visible quantifications impose a requirement on how a thing is used/written; required quantifications impose a requirement on how a thing's type is written.
- Relevance refers to how the quantifiee can be used in the term that follows. (This is distinct from dependence, which says how the quantifiee can be used in the type that follows!) forall-quantifiees are not relevant. While they can textually appear in the term that follows, they appear only in irrelevant positions -- that is, in type annotations and type signatures. ->- and =>-quantifiees, on the other hand, can be used freely. Relevance is something of a squirrely issue. It is (RAE believes) closely related to parametricity, in that if forall-quantifiees were relevant, Haskell would lose the parametricity property. Another way to think about this is that parametric arguments are irrelevant and non-parametric arguments are relevant.
Having explained our terms with the current Haskell, the proposed set of quantifiers for dependent Haskell is below:
|forall (...) .||yes||unification||FVs + Rel.I.||no|
|forall (...) ->||yes||as type||yes||no|
|pi (...) .||yes||unification||FVs + Rel.I.||yes|
|pi (...) ->||yes||as term||yes||yes|
Note that the current quantifiers remain and with their original meanings. This table adds three new quantifiers: forall ->, and the two pi quantifiers. The idea is that, currently, we always say forall, then some binders and then a .. If we replace the . with an ->, then we make the quantifications visible but otherwise unchanged. (Quantification not mentioned in a type defaults to invisible, thus making the visible quantification required.)
The new pi quantifiers allow for quantifiees that are both dependent and relevant. This means that the quantifiee is named in the type and can be used within its scope in the type, and also that the quantifiee can be inspected in the term. A pi-bound argument is a proper dependent type. Since a pi-quantifiee can appear in both terms and types, its instantiations must come from a restricted subset of Haskell that makes sense at both the type and term level. This issue is addressed in Adam Gundry's thesis. For now, we propose that this subset include only data constructors (perhaps applied) and other pi-quantifiees. The subset can be expanded later. For some tantalizing ideas of how far it can go, see this paper, discussing promotion of Haskell terms to types.
The table above has a new abbreviation: Rel.I. is short for relevance inference. When a type variable is used in a type without an explicit quantification, should it be forall-quantified or pi-quantified? Choosing forall quantification means that the type can be erased during compilation, while pi quantification is more powerful. Thus, we can't just have a default. We must infer the relevance of the type variable, by looking at its use sites. This seems actually quite straightforward, and is remarkably similar to role inference. (Indeed, the phantom role seems to correspond quite closely to an irrelevant argument. There is Something Interesting here -- yet another way that roles and parametricity relate.)
It is tempting to treat -> as a degenerate form of a pi -- something like pi (_ :: ...) ->. However, this is slightly misleading, in that -> quantifies over any Haskell term, and pi quantifies over only the shared term/type subset.
Quantifiers in kinds
The preceding discussion focuses mostly on classifying terms. How does any of this change when we think of classifying types?
- Relevance in types
- Relevance in a term corresponds quite closely to phase. A relevant term-level quantifiee must be kept at runtime, while an irrelevant quantifiee can be erased. But, what does relevance in a type mean? Everything in a type is (absent pi-quantifications) irrelevant in a term, and it all can be erased. Furthermore, it is all used in the same phase, at compile time. Yet, it seems useful to still have a notion of relevance in types. This allows programmers to reason about parametricity in their type-level functions, and it keeps the function space in types similar to the function space in terms.
For example, today's Haskell permits things like this:type family F (x :: k) :: k type instance F True = False type instance F False = True type instance F (x :: *) = x
Note that the behavior of F depends on the kind of its argument, k. This is an example of a non-parametric type function. Looking at the kind, k -> k, one would expect F to be the identity; yet, it is not.
Thus, we would want to distinguish pi k. k -> k (the kind of F) and forall k. k -> k (the kind of a type-level polymorphic identity). This distinction does not affect erasure or phase, but it does affect how a quantifiee can be used. Furthermore, this keeps term classifiers more in line with type classifiers.
- How is the kind of a datatype classified? (stub)
Open design questions
Parsing is a bit a nightmare for this new language and will require some compromises.
- Merging types and kinds is almost straightforward, but for one major stumbling block: *. In a kind, * is parsed as an alphanumeric identifier would be. In a type, * is parsed as an infix operator. How can we merge the type- and kind-parser given this discrepancy? As an example, what is the meaning of Foo * Int? Is it the type Foo applied to * and Int? Or is it the operator * applied to Foo and Int? The solution to this annoyance seems to be to introduce a new identifier for * (say, TYPE) and then remove * from the language, allowing it to be used for multiplication, for example.
- What name to choose for *? TYPE would appear plenty in code, and it seems a little rude. Choosing a new symbol just kicks the can down the road. Choosing Type would conflict with lots of code (including GHC's) that uses a type Type. Choosing T would conflict with lots of (example) code that uses T. The best option I'm aware of is U, short for universe. Mitigating this problem somewhat is that Dependent Haskell would come with kind synonyms, and whatever name we choose would be a "normal" name exported from the Prelude and could be hidden if desired.
- What is our migration strategy? One proposal: introduce the new name now and keep * around. Then, when Dependent Haskell is ready for release, it will come with a new extension flag which will change the parsing of *. Only when that flag is enabled would * fail to work. It is unclear whether it is worth it to fully squash * out of the language.
- The type language and the term languages are more different. Even if we could write some kind of combined parser, the renamer would have major trouble distinguishing between data constructors and type constructors. One way or the other, programmers will likely have to specify how to parse their arguments explicitly. Take id :: forall a -> a -> a. This is just like the normal id, but with the type parameter explicit. Because the parser/renamer won't know the type of id when parsing its arguments, the first argument will have to manifestly be a type. For example, id @Bool True. The @ indicates to the parser that the following thing is a type, not a term.
- We will similarly need a syntax for type patterns embedded within term patterns. It would be ideal if the pattern syntax were identical to the expression syntax.
- The choice of @ above is stolen from the ExplicitTypeApplication and TypeApplication proposals, neither of which have been implemented and will be subsumed by Dependent Haskell (if we get to that first). Is this the right choice, though? For example, the ExplicitTypeApplication page includes an example of ambiguity:
f :: Int -> forall a. a f n @a = ....
This is ambiguous because @-patterns allow a space around the @-sign. However, common usage does not use any spaces around @, and we could use the presence/absence of a space to disambiguate between an @-pattern and a type pattern.
Overriding visibility defaults
The ./-> distinction in quantifiers allows programmers to specify the visibility of arguments at use sites. But, sometimes callers will want to override the defaults.
- If a visible, dependent argument is to be elided, we could allow _ to indicate that GHC should use unification to fill in the argument. (This is similar to the approach in Coq, for example, among other languages.) Does this conflict in any way with typed holes? Perhaps a programmer wants to get an informative error message, not for GHC to plug in a value.
- Visible, non-dependent arguments cannot be inferred via unification, so _ would not be applicable here, and would retain its current meaning of a typed hole.
- How to override an invisible, dependent type argument? This might be critical if a function call would be otherwise ambiguous. (Today's Haskell would benefit from this override occasionally, too. See TypeApplication.) One proposal is that @ would serve double-duty: it would override invisibility and also indicate a type argument. If this is the case, the forall (...) -> form would be essentially useless, as it would still require users to use @ to indicate parsing. Thus, forall (...) -> seems strictly worse than forall (...) ..
- How to override an invisible, dependent term argument (that is, pi (...) .)? Using @ would not work, because the parser wouldn't be able to deal with it. Alternatively, pi-bound arguments could use type-level syntax, and then @ would work. However, this seems suboptimal, as we would more often want to use, say, a data constructor in a pi-bound argument, not a type constructor. Braces could possibly work, at least in expressions. These would not conflict with record-update syntax because record-updates require an =, but pi-bound arguments would not. Patterns might be problematic with braces, because record puns do not require =. There are no other proposals here, for the moment.
Parametric vs. Non-parametric type families
- Concrete syntax? (stub)
Readers: Please add to these lists!
There are several published works very relevant to the design:
- System FC with Explicit Kind Equality. Stephanie Weirich, Justin Hsu, and Richard A. Eisenberg. ICFP 2013.
- Type Inference, Haskell, and Dependent Types. Adam Gundry. PhD Thesis, 2013.
There are also many works addressing the use of dependent types in Haskell. Here is a selection:
- Dependently typed programming with singletons. Richard A. Eisenberg and Stephanie Weirich. Haskell Symposium 2012.
- Hasochism: The Pleasure and Pain of Dependently Typed Haskell. Sam Lindley and Conor McBride. Haskell Symposium 2013.
- Promoting Functions to Type Families in Haskell. Richard A. Eisenberg and Jan Stolarek. Draft, 2014.