# Ambiguity

The question of *ambiguity* in Haskell is a tricky one. This wiki page is a summary of thoughts and definitions, in the hope of gaining clarity. I'm using a wiki because it's easy to edit, and many people can contribute, even though you can't typeset nice rules.

[Started Jan 2010.] **Please edit to improve.**

## Terminology

A type system is usually specified by

- A
**specification**, in the form of some**declarative typing rules**. These rules often involve "guessing types". Here is a typical example, for variables:(f : forall a1,..,an. C => tau) \in G theta = [t1/a1, ..., tn/an] -- Substitution, guessing ti Q |= theta( C ) ------------------------- (VAR) Q, G |- f :: theta(tau)

The preconditions say that f is in the environment G with a suitable polymorphic type. We "guess" types t1..tn, and use them to instantiate f's polymorphic type variables a1..an, via a substitution`theta`. Under this substitution f's instantiated constraints`theta(C)`must be*satisfiable*(using`|=`) from the ambient constraints Q.

The point is that we "guess" the ai.

- An
**inference algorithm**, often also presented using similar-looking rules, but in a form that can be read as an algorithm with no "guessing". Typically- The "guessing" is replaced by generating fresh unification variables.
- The algorithm carries an ever-growing substitution that instantiates these unification variables.

We want the inference algorithm to be

**sound**(if it succeeds, then the program is well typed according to the specification) and**complete**(if the program is well typed according to the specification, the algorithm succeeds).

## Coherence

Suppose we have (I conflate classes `Read` and `Show` into one class `Text` for brevity):

class Text a where read :: String -> a show :: a -> String x :: String x = show (read "3.7")

The trouble is that there is a constraint `(Text t)`, where `t` is a type variable that is otherwise unconstrained. Moreover, the type that we choose for `t` affects the semantics of the program. For example, if we chose `t = Int` then we might get `x = "3"`, but if we choose `t = Float` we might get `x = "3.7"`. This is bad: we want our type system to be **coherent** in the sense that every well-typed program has but a single value.

In practice, the Haskell Report, and every Haskell implementation, rejects such a program saying something like

Cannot deduce (Text t) from ()

In *algorithmic* terms this is very natural: we indeed have a constraint `(Text t)` for some unification variable `t`, and no way to solve it, except by searching for possible instantiations of `t`. So we simply refrain from trying such a search.

But in terms of the type system *specification* it is harder. We can simply guess `a=Int` when we instantiate `read` and `show` and lo, the program is well typed. But we do not *want* this program to be well-typed.

**Problem 1**: How can we write the specification so as to reject programs such as that above.

## Digression: open and closed world

Suppose there was precisely one instance for `Text`:

instance Text Char where ...

Then you might argue that there is only one way for the algorithm to succeed, namely by instantiating `read` and `show` at `Char`.

It's pretty clear that this is a Bad Idea:

- In general it is hard to say whether there is a unique substitution that would make a collection of constraints satisfiable.
- If you add just one more instance, the program would become untypeable, which seems fragile.

To avoid this nasty corner we use the **open-world assumption**; that is, we assume that the programmer may add new instances at any time, and that doing so should not make a well-typed program become ill-typed. (We ignore overlapping instances for now.

## Early detection of errors

Suppose, with the above class `Text` I write

f x = show (read x)

What type should we infer for `f`? Well, a simple-minded inference algorithm works as follows for a let-definition `f=e`: typecheck `e`, collecting whatever constraints it generates. Now simply abstract over them.

In this example we'd get

f :: (Text a) => String -> String

And indeed this is a perfectly fine type for