“未定义”的类型签名在Haskell中意味着什么？

Question

I am a beginner in Haskell and I am taken aback by the undefined function's type signature.

I expected something more simple, but I found this on Hackage:

undefined :: forall (r :: RuntimeRep). forall (a :: TYPE r). HasCallStack => a

A special case of error. It is expected that compilers will recognize this and insert error messages which are more appropriate to the context in which undefined appears.

Could you please explain what does this signature mean?

Thanks!

Answer 1

For all regards a beginner needs to know, the signature is simply

undefined :: a

which means, as always with type variables (ie any lowercase letters) that a is universally quantified , which can also be made explicit:

{-# LANGUAGE ExplicitForall #-}
undefined :: forall a. a

...or, as I prefer to write it

{-# LANGUAGE ExplicitForall, UnicodeSyntax #-}
undefined :: ∀ a. a

The quantification is infered to be over all types , ie all things with kind * (read: “type”, more accurate the kind of lifted types – lifted meaning it can be lazy thunk). Therefore, you can use undefined in any expression, no matter what type is required.

Now, undefined is of course a “partial function” like thing, basically a function which zero arguments which is defined nowhere. (FTR, it's not a function, as a function by definition has argument[s].)
You'd like to get a useful error message when it is actually evaluated, but GHC doesn't by default produce a call stack for everything (for performance reasons), so it used to be the case that the error message was almost completely useless. That's where the HasCallStack comes in: that's a constraint which essentially tells the context in which some code might incur undefined that it should note the place where it happens, in order for the error message to actually show it up. So,

undefined :: ∀ a. HasCallStack => a

It's a bit confusing that the HasCallStack appears after the ∀ a – this doesn't really have anything to do with a but with the context in which undefined will be used. Just, the form of signatures is always

Identifier :: Quantifiers. Constraints => Type

and HasCallStack is a constraint, that's why it appears in the middle. (More often, constraints actually apply to one of the type variables you've quantified over.)

Finally, this RunTimeRep stuff is about levity polymorphism . I don't really understand that myself, but it's discussed in Why is the undefined function levity-polymorphic when using it with unboxed types?

Answer 2

This is a continuation of @leftaroundabout's answer.

In Haskell, values have types. But types themselves also have types. When a type is acting as the type of another type, we call it a "kind".

The most important and common kind in Haskell is Type , often denoted in signatures as * . It is the kind of types that are "lifted", that is, whose values can be thunks that, when evaluated, can diverge, throw an error, etc... For example the type Int has kind Type .

There are also other types like Int# that are not lifted. A value of type Int# is never a thunk, it's always an actual value in memory.

In short, the in-memory representation of values is controlled by their type's kind.

RuntimeRep is another kind. It is the kind of types like LiftedRep and IntRep . These types don't have any values, they exist merely to express things at the type level , as we shall see.

There is a super-magical type-level entity called TYPE that, when parameterized with a type of kind RuntimeRep (that is with a type that describes an in-memory representation) returns the kind of types whose values have that representation. For example, Type is TYPE LiftedRep , while the kind of Int# is TYPE IntRep .

ghci> :set -XMagicHash
ghci> import GHC.Prim 
ghci> import GHC.Types
ghci> import Data.Kind
ghci> :kind (Int :: TYPE 'LiftedRep)
(Int :: TYPE 'LiftedRep) :: *
ghci> :kind Int#
Int# :: TYPE 'IntRep

Now we can get back to why undefined has such a weird signature. The thing is, we want to be able to use undefined in all functions, be it functions that return types of kind Type , but also functions that return types of kind TYPE IntRep (in other words: the Int# type) or functions that return another unlifted type. Otherwise, we would need multiple different versions of undefined , which would be annoying.

The solution is to make undefined levity-polymorphic . The signature says: for any possible in-memory representation ( RuntimeRep ) and for any possible type whose values have that representation, undefined counts a member of that type.