[Haskell-cafe] Proposal to solve Haskell's MPTC dilemma

[Carlos Camarao]

This message presents, informally, a proposal to solve Haskell's MPTC
(multi-parameter type class) dilemma. If this informal proposal turns
out to be acceptable, we (I am a volunteer) can proceed and make a
concrete proposal.

The proposal has been published in the SBLP'2009 proceedings and is
available at
    www.dcc.ufmg.br/~camarao/CT/solution-to-MPTC-dilemma.pdf

The well-known dilemma
(hackage.haskell.org/trac/haskell-prime/wiki/MultiParamTypeClassesDilemma)
is that it is generally accepted that MPTCs are very useful, but their
introduction is thought to require the introduction also of FDs
(Functional Dependencies) or another mechanism like ATs (Associated
Types) and FDs are tricky and ATs, somewhat in a similar situation,
have been defined more recently and there is less experience with its
use.

In
www.haskell.org/ghc/dist/current/docs/html/users_guide/type-class-extensions.html
there exists a solution to the termination problem related to the
introduction of MPTCs in Haskell. In our proposal, neither FDs nor any
other mechanism like ATs are needed in order to introduce MPTCs in
Haskell; the only change we have to make is in the ambiguity
rule. This is explained below. The termination problem is essentially
ortogonal and can be dealt with, with minor changes, as described in
the solution presented in the above mentioned (type-class-extensions)
web page.

Let us review the ambiguity rule used in Haskell-98 and after that the
ambiguity rule used in GHC. Haskell-98 ambiguity rule (which is
adequate for Haskell-98's single parameter type classes) is: a type
C => T is ambiguous iff there is a type variable v that occurs in the
"context" (constraint set) C but not in the simple (unconstrained)
type T.

For example: forall a.(Show a, Read a)=>String is ambiguous, because
"a" occurs in the constraints (Show a,Read a) but not in the simple
type (String).

In the context of MPTCs, this rule alone is not enough. Consider, for
example (Example 1):

   class F a b where f:: a->b
   class O a where o:: a
and
    k = f o:: (C a b,O a) => b

Type forall a b. (C a b,O a) => b can be considered to be not
ambiguos, since overloading resolution can be defined so that
instantiation of "b" can "determine" that "a" should also be
instantiated (as FD b|->a does), thus resolving the overloading.

GHC, since at least version 6.8, makes significant progress towards a
definition of ambiguity in the context of MPTCs; GHC 6.10.3 User’s
Guide says (section 7.8.1.1):

  "GHC imposes the following restrictions on the constraints in a type
  signature. Consider the type: forall tv1..tvn (c1, ...,cn) => type. ...
  Each universally quantified type variable tvi must be reachable from type.

  A type variable a is reachable if it appears in the same constraint as
  either a type variable free in the type, or another reachable type
variable.”

For example, type variable "a" in constraint (O a) in the example
above is reachable, because it appears in (C a b) (the same constraint
as type variable "b", which occurs in the simple type).

Our proposal is: consider unreachability not as indication of
ambiguity but as a condition to trigger overloading resolution (in a
similar way that FDs trigger overloading resolution): when there is at
least one unreachable variable and overloading is found not to be
resolved, then we have ambiguity. Overloading is resolved iff there is
a unique substitution that can be used to specialize the constraint
set to one, available in the current context, such that the
specialized constraint does not contain unreachable type variables.
(A formal definition, with full details, is in the cited SBLP'09 paper.)

Consider, in Example 1, that we have a single instance of F and
O, say:

instance F Bool Bool where f = not
instance O Bool where o = True

and consider also that "k" is used as in e.g.:

   kb = not k

According to our proposal, kb is well-typed. Its type is Bool. This
occurs because (F a b, O a)=>Bool can be simplified to Bool, since
there exists a single substitution that unifies the constraints with
instances (F Bool Bool) and (O Bool) available in the current context,
namely S = (a|->Bool,b|->Bool).

As another example (Example 2), consider:

{-# NoMonomorphismRestriction, MultiParameterTypeClasses #-}
data Matrix = ...
data Vector = ...
class Mult a b c where (*):: a ! b ! c
instance Mult Matrix Vector Matrix where (*) = ...
instance Mult Matrix Vector Vector where (*) = ...
m1:: Matrix = ...
m2:: Vector = ...
m3:: Vector = ...
m = (m1 * m2) * m3

GHC gives the following type to m:
  m :: forall a, b. (Mult Matrix Vector a, Mult a Vector b) => b

However, "m" cannot be used effectively in a program compiled with
GHC: if we annotate m::Matrix, a type error is reported. This occurs
because the type inferred for "m" includes constraints (Mult Matrix
Vector a, Mult a Vector Matrix) and, with b|->Matrix, type variable "a"
appears only in the constraint set, and this type is considered
ambiguous. Similarly, if "m" is annotated with type Vector, the type
is also considered ambiguous. There is no means of specializing type
variable "a" occurring in the type of "m" to Matrix. And no FD may be
used in order to achieve such specialization, because we do not have
here a FD: "a" and "b" do not determine "c" in class Mult: for (a,b =
Matrix,Vector), we can have "c" equal to either Matrix or Vector.

According to our proposal, type m:: forall a. (Mult Matrix Vector a,
Mult a Vector Matrix)) => Matrix is not ambiguous, in the context of
the class and instance declarations of Example 2 (since "a" can be
instantiated to Matrix), whereas type m:: forall a.(Mult Matrix Vector
a, Mult a Vector Matrix)) => Vector is ambiguous in this context.

We would like to encourage compiler front-end developers to change
Haskell's ambiguity rule, and test with as many examples as desired,
including recurring examples that appear in Haskell mailing lists
involving MPTCs, FDs etc. We would like to hear and discuss your
doubts, opinions, comments, criticisms etc.

Cheers,

Carlos
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