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# 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 deducible (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.

## 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. Usually a

**Problem 1**: how can w