Version 22 (modified by 4 years ago) (diff)  ,

Patternmatching axioms
This page describes an extension to type families that supports overlap.
 See also the Discussion Page added May 2012, for comment/suggestions/requests for clarification/alternative solutions, to explore the design space.
 See also the Coincident Overlap page (added August 2012) for a discussion around the usefulness of allowing certain overlaps when the righthand sides coincide.
 See also the Template Haskell page (added December 2012) for a proposal for the Template Haskell changes necessary to support this change.
 See also the Nonlinearity and Type Spaces pages (added May 2013) for discussion and a proposal around type unsoundness that can be caused by repeated variables on the lefthand side of an instance. The proposal on that page will likely be implemented and will then be copied here.
Status (Jan 12): the groundwork is done, in HEAD; mainly making CoAxiom
a more fundamental data type. Not yet started on the details.
Status (Aug 12): A working prototype implementation is in overlappingtyfams
.
Status (Dec 12): A working implementation has been pushed to HEAD.
Background
One might imagine that it would be a simple matter to have a typelevel function
type family Equal a b :: Bool
so that (Equal t1 t2)
was True
if t1
=t2
and False
otherwise. But it isn't.
You can't write
type instance Equal a a = True type instance Equal a b = False
because System FC (rightly) prohibits overlapping family instances.
Expanding this out, you can do it for a fixed collection of types thus:
type instance Equal Int Int = True type instance Equal Bool Bool = True type instance Equal Int Bool = False type instance Equal Bool Int = False
but this obviously gets stupid as you add more types.
Furthermore, this is not what you want. Even if we restrict the equality function to booleans
type family Equal (a :: Bool) (b :: Bool) :: Bool
we can't define instances of Equal so that a constraint like this one
Equal a a ~ True
is satisfiablethe type instances only reduce if a is known to True or False. GHC doesn't reason by cases. (Nor should it, Any also inhabits Bool. No kinds really are closed.)
The only way to work with this sort of reasoning is to use Overlapping Instances, as suggested in the HList paper.
What to do about it
A new version of axioms is now implemented. The formal treatment can be found in docs/corespec/corespec.pdf.
Here are the changes to source Haskell.
 A
type instance
declaration can define multiple equations, not just one:type instance where Equal a a = True Equal a b = False
 Patterns within a single
type instance
declaration (henceforth "branches") may overlap, and are matched top to bottom.
 A single type family may, as now, have multiple
type instance
declarations:type family F a :: * type instance where F [Int] = Int F [a] = Bool type instance where F (Int,b) = Char F (a,b) = [Char]
 The instances for
F
may not overlap. That is, there must be no typet
such that(F t)
matches more than one instance. This rule explicitly excludes overlaps among group members, even if the righthand sides coincide (but see the Coincident Overlap page for discussion).
 The groups do not need to be exhaustive. If there is no equation that matches, the call is stuck. (This is exactly as at present.)
 An error is issued when a later equation is matched by a former, making the later one inaccessible.
type instance where F (a,b) = [Char] F (Int,b) = Char
Here the second equation can never match.
For closed kinds (and maybe for open ones, but I can't unravel it), it seems possible to write a set of equations that will catch all possible cases but doesn't match the general case. This situation is currently (Dec 2012) undetected, because I (Richard, eir
at cis.upenn.edu
) am unconvinced I have a strong enough handle on the details. For example, what about Any
?
 The equations do not need to share a common pattern:
type instance where F Int = Char F (a,b) = Int
 When matching a use of a type family against a branched instance, special care must be taken (by GHC) not to accidentally introduce incoherence. Consider the following example:
type instance where F Int = Bool F a = Char
and we try to simplify the typeF b
. The naive implementation would just simplifyF b
toChar
, but this would be wrong. The problem is thatb
may later be unified withInt
, meaningF b
should simplify toBool
, notChar
. So, the correct behavior is not to simplifyF b
at all; it is stuck for now. Note that the second equation above is not useless: we will still simplify, say,F Double
toChar
.
More formally, we only match a type against an equation in an instance group when no previous equation can unify against the type.
Limitations
The implementation described above does not address all desired use cases. In particular, it does not work with associated types at all. (Using something like type where
in a class definition is a parse error.) There's no set reason the approach couldn't be expanded to work with associated types, but it is not done yet. In particular, the FC extension will handle intramodule overlapping associated types without a change. The biggest reason not to add associated types into the mix is that it will be a confusing feature. Overlap among class instances is directed by specificity; overlap among family instances is ordered by the programmer. Users would likely expect the two to coincide, but they don't and can't, as it would not be type safe:
It seems that intermodule overlapping noncoincident associated types are a Bad Idea, but please add comments if you think otherwise and/or need such a feature. Why is it a Bad Idea? Because it would violate type safety: different modules with different visible instances could simplify type family applications to different ground types, perhaps concluding True ~ False
, and the world would immediately cease to exist.
This last point doesn't apply to overlapping type class instances because type class instance selection compiles to a termlevel thing (a dictionary). Using two different dictionaries for the same constraint in different places may be silly, but it won't end the world.
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axioms.pdf (212.0 KB)  added by 5 years ago.
Description of FC extension to support overlapping type family instances
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