What are type quantifiers?

Question

Many statically typed languages have parametric polymorphism. For example in C# one can define:

T Foo<T>(T x){ return x; }

In a call site you can do:

int y = Foo<int>(3);

These types are also sometimes written like this:

Foo :: forall T. T -> T

I have heard people say "forall is like lambda-abstraction at the type level". So Foo is a function that takes a type (for example int), and produces a value (for example a function of type int -> int). Many languages infer the type parameter, so that you can write Foo(3) instead of Foo<int>(3).

Suppose we have an object f of type forall T. T -> T. What we can do with this object is first pass it a type Q by writing f<Q>. Then we get back a value with type Q -> Q. However, certain f's are invalid. For example this f:

f<int> = (x => x+1)
f<T> = (x => x)

So if we "call" f<int> then we get back a value with type int -> int, and in general if we "call" f<Q> then we get back a value with type Q -> Q, so that's good. However, it is generally understood that this f is not a valid thing of type forall T. T -> T, because it does something different depending on which type you pass it. The idea of forall is that this is explicitly not allowed. Also, if forall is lambda for the type level, then what is exists? (i.e. existential quantification). For these reasons it seems that forall and exists are not really "lambda at the type level". But then what are they? I realize this question is rather vague, but can somebody clear this up for me?

---

A possible explanation is the following:

If we look at logic, quantifiers and lambda are two different things. An example of a quantified expression is:

forall n in Integers: P(n)

So there are two parts to forall: a set to quantify over (e.g. Integers), and a predicate (e.g. P). Forall can be viewed as a higher order function:

forall n in Integers: P(n) == forall(Integers,P)

With type:

forall :: Set<T> -> (T -> bool) -> bool

Exists has the same type. Forall is like an infinite conjunction, where S[n] is the n-th elemen to of the set S:

forall(S,P) = P(S[0]) ∧ P(S[1]) ∧ P(S[2]) ...

Exists is like an infinite disjunction:

exists(S,P) = P(S[0]) ∨ P(S[1]) ∨ P(S[2]) ...

If we do an analogy with types, we could say that the type analogue of ∧ is computing the intersection type ∩, and the type analogue of ∨ computing the union type ∪. We could then define forall and exists on types as follows:

forall(S,P) = P(S[0]) ∩ P(S[1]) ∩ P(S[2]) ...
exists(S,P) = P(S[0]) ∪ P(S[1]) ∪ P(S[2]) ...

So forall is an infinite intersection, and exists is an infinite union. Their types would be:

forall, exists :: Set<T> -> (T -> Type) -> Type

For example the type of the polymorphic identity function. Here Types is the set of all types, and -> is the type constructor for functions and => is lambda abstraction:

forall(Types, t => (t -> t))

Now a thing of type forall T:Type. T -> T is a value, not a function from types to values. It is a value whose type is the intersection of all types T -> T where T ranges over all types. When we use such a value, we do not have to apply it to a type. Instead, we use a subtype judgement:

id :: forall T:Type. T -> T
id = (x => x)

id2 = id :: int -> int

This downcasts id to have type int -> int. This is valid because int -> int also appears in the infinite intersection.

This works out nicely I think, and it clearly explains what forall is and how it is different from lambda, but this model is incompatible with what I have seen in languages like ML, F#, C#, etc. For example in F# you do id<int> to get the identity function on ints, which does not make sense in this model: id is a function on values, not a function on types that returns a function on values.

---

Can somebody with knowledge of type theory explain what exactly are forall and exists? And to what extent is it true that "forall is lambda at the type level"?

Answer 1

Let me address your questions separately.

• Calling forall "a lambda at the type level" is inaccurate for two reasons. First, it is the type of a lambda, not the lambda itself. Second, that lambda lives on the term level, even though it abstracts over types (lambdas on the type level exist as well, they provide what is often called generic types).

• Universal quantification does not necessarily imply "same behaviour" for all instantiations. That is a particular property called "parametricity" that may or may not be present. The plain polymorphic lambda calculus is parametric, because you simply cannot express any non-parametric behaviour. But if you add constructs like typecase (a.k.a. intensional type analysis) or checked casts as a weaker form of that, then you loose parametricity. Parametricity implies nice properties, e.g. it allows a language to be implemented without any runtime representation of types. And it induces very strong reasoning principles, see e.g. Wadler's paper "Theorems for free!". But it's a trade-off, sometimes you want dispatch on types.

• Existential types essentially denote pairs of a type (the so-called witness) and a term, sometimes called packages. One common way to view these is as implementation of abstract data types. Here is a simple example:

  pack (Int, (λx. x, λx. x)) : ∃ T. (Int → T) × (T → Int)

  This is a simple ADT whose representation is Int and that only provides two operations (as a nested tuple), for converting ints in and out of the abstract type T. This is the basis of type theories for modules, for example.

• In summary, universal quantification provides client-side data abstraction, while existential types dually provides implementor-side data abstraction.

• As an additional remark, in the so-called lambda cube, forall and arrow are generalised to the unified notion of Π-type (where T1→T2 = Π(x:T1).T2 and ∀A.T = Π(A:*).T) and likewise exists and tupling can be generalised to Σ-types (where T1×T2 = Σ(x:T1).T2 and ∃A.T = Σ(A:*).T). Here, the type * is the "type of types".

Answer 2

A few remarks to complement the two already-excellent answers.

First, one cannot say that forall is lambda at the type-level because there already is a notion of lambda at the type level, and it is different from forall. It appears in system F_omega, an extension of System F with type-level computation, that is useful to explain ML modules systems for example (F-ing modules, by Andreas Rossberg, Claudio Russo and Derek Dreyer, 2010).

In (a syntax for) System F_omega you can write for example:

type prod =
  lambda (a : *). lambda (b : *).
    forall (c : *). (a -> b -> c) -> c

This is a definition of the "type constructor" prod, such as prod a b is the type of the church-encoding of the product type (a, b). If there is computation at the type level, then you need to control it if you want to ensure termination of type-checking (otherwise you could define the type (lambda t. t t) (lambda t. t t). This is done by using a "type system at the type level", or a kind system. prod would be of kind * -> * -> *. Only the types at kind * can be inhabited by values, types at higher-kind can only be applied at the type level. lambda (c : k) . .... is a type-level abstraction that cannot be the type of a value, and may live at any kind of the form k -> ..., while forall (c : k) . .... classify values that are polymorphic in some type c : k and is necessarily of ground kind *.

Second, there is an important difference between the forall of System F and the Pi-types of Martin-Löf type theory. In System F, polymorphic values do the same thing on all types. As a first approximation, you could say that a value of type forall a . a -> a will (implicitly) take a type t as input and return a value of type t -> t. But that suggest that there may be some computation happening in the process, which is not the case. Morally, when you instantiate a value of type forall a. a -> a into a value of type t -> t, the value does not change. There are three (related) ways to think about it:

• System F quantification has type erasure, you can forget about the types and you will still know what the dynamic semantic of the program is. When we use ML type inference to leave the polymorphism abstraction and instantiation implicit in our programs, we don't really let the inference engine "fill holes in our program", if you think of "program" as the dynamic object that will be run and compute.

• A forall a . foo is not a something that "produces an instance of foo for each type a, but a single type foo that is "generic in an unknown type a".

• You can explain universal quantification as an infinite conjunction, but there is an uniformity condition that all conjuncts have the same structure, and in particular that their proofs are all alike.

By contrast, Pi-types in Martin-Löf type theory are really more like function types that take something and return something. That's one of the reason why they can easily be used not only to depend on types, but also to depend on terms (dependent types).

This has very important implications once you're concerned about the soundness of those formal theories. System F is impredicative (a forall-quantified type quantifies on all types, itself included), and the reason why it's still sound is this uniformity of universal quantification. While introducing non-parametric constructs is reasonable from a programmer's point of view (and we can still reason about parametricity in an generally-non-parametric language), it very quickly destroys the logical consistency of the underlying static reasoning system. Martin-Löf predicative theory is much simpler to prove correct and to extend in correct way.

For a high-level description of this uniformity/genericity aspect of System F, see Fruchart and Longo's 97 article Carnap's remarks on Impredicative Definitions and the Genericity Theorem. For a more technical study of System F failure in presence of non-parametric constructs, see Parametricity and variants of Girard's J operator by Robert Harper and John Mitchell (1999). Finally, for a description, from a language design point of view, on how to abandon global parametricity to introduce non-parametric constructs but still be able to locally discuss parametricity, see Non-Parametric Parametricity by George Neis, Derek Dreyer and Andreas Rossberg, 2011.

This discussion of the difference between "computational abstraction" and "uniform abstract" has been revived by the large amount of work on representing variable binders. A binding construction feels like an abstraction (and can be modeled by a lambda-abstraction in HOAS style) but has an uniform structure that makes it rather like a data skeleton than a family of results. This has been much discussed, for example in the LF community, "representational arrows" in Twelf, "positive arrows" in Licata&Harper's work, etc.

Recently there have been several people working on the related notion of "irrelevance" (lambda-abstractions where the result "does not depend" on the argument), but it's still not totally clear how closely this is related to parametric polymorphism. One example is the work of Nathan Mishra-Linger with Tim Sheard (eg. Erasure and Polymorphism in Pure Type Systems).

Answer 3

if forall is lambda ..., then what is exists

Why, tuple of course!

In Martin-Löf type theory you have Π types, corresponding to functions/universal quantification and Σ-types, corresponding to tuples/existential quantification.

Their types are very similar to what you have proposed (I am using Agda notation here):

Π : (A : Set) -> (A -> Set) -> Set
Σ : (A : Set) -> (A -> Set) -> Set

Indeed, Π is an infinite product and Σ is infinite sum. Note that they are not "intersection" and "union" though, as you proposed because you can't do that without additionally defining where the types intersect. (which values of one type correspond to which values of the other type)

From these two type constructors you can have all of normal, polymorphic and dependent functions, normal and dependent tuples, as well as existentially and universally-quantified statements:

-- Normal function, corresponding to "Integer -> Integer" in Haskell
factorial : Π ℕ (λ _ → ℕ)

-- Polymorphic function corresponding to "forall a . a -> a"
id : Π Set (λ A -> Π A (λ _ → A))

-- A universally-quantified logical statement: all natural numbers n are equal to themselves
refl : Π ℕ (λ n → n ≡ n)


-- (Integer, Integer)
twoNats : Σ ℕ (λ _ → ℕ)

-- exists a. Show a => a
someShowable : Σ Set (λ A → Σ A (λ _ → Showable A))

-- There are prime numbers
aPrime : Σ ℕ IsPrime

However, this does not address parametricity at all and AFAIK parametricity and Martin-Löf type theory are independent.

For parametricity, people usually refer to the Philip Wadler's work.