# Changes between Version 2 and Version 3 of TypeFunctions/Ambiguity

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Timestamp:
Jan 20, 2010 9:05:57 PM (10 years ago)
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• ## TypeFunctions/Ambiguity

 v2 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. `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 == 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 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 w '''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