Version 16 (modified by goldfire, 4 years ago) (diff) 

Patternmatching axioms
This page describes an extension to type families that supports overlap.
 We'll use GHC branch overlappingtyfams for development work.
 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.
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.
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
So the deficiency is in System FC, and it seems fundamental. We've been working on an extension to System FC, with a corresponding sourcelanguage extension, that does allow overlapping type families, with care. Here's the idea of the surfacelevel extension, but see the attached PDF for the details on the FC extension. (The attached PDF uses the FC formalism fully presented here.)
All of the following is currently implemented in the overlappingtyfams branch.
 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 "group") 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 groups for F may not overlap. That is, there must be no type t such that (F t) matches both groups. 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 (Aug 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 group, special care must be taken not to accidentally introduce incoherence. Consider the following example:
type instance where F Int = Bool F a = Char
and we try to simplify the type F b. The naive implementation would just simplify F b to Char, but this would be wrong. The problem is that b may later be unified with Int, meaning F b should simplify to Bool, not Char. So, the correct behavior is not to simplify F b at all; it is stuck for now. Note that the second equation above is not useless: we will still simplify, say, F Double to Char.
More formally, we only match a type against an equation in an instance group when no previous equation can unify against the type.
 Taking the above point into account, we will still simplify a type family use when those previous unifying equations produce coincident righthand sides. For example,
type instance where And True x = x And y True = y
and we want to simplify And z True. The first equation does not match. The second one does, with the substitution y > z! But then, the first one unifies with the substitution x > True, z > True. Applying both substitutions to the second righthand side (y) and just the second substitution to the first (x}), we see that both righthand sides reduce to True in the problematic case (which is when z will unify with True at some later point). So, we can indeed reduce And z True to z as desired. This is indeed a little fiddly, but it seems useful enough to include.
 Optional extra (not yet (August 2012) implemented): It would make sense to allow the type family and type instance declaration to be combined into one, in cases where all the equations can be given at the definition site. For example:
type family Equal a b :: Bool where Equal a a = True Equal a b = False type family Member (a :: k) (b :: '[k]) :: Bool where Member a '[] = False  (not overlapping) Member a ( a ': bs ) = True Member a ( b ': bs ) = Member a bs
Questions of syntax
What should the "header" look like?
(A) type instance where  Use "where" F (a,b) = [Char] F (Int,b) = Char F Bool = Char (B) type instance of  Use "of" (yuk) F (a,b) = [Char] F (Int,b) = Char F Bool = Char (C) type instance F where  Redundantly mention F in the header F (a,b) = [Char] F (Int,b) = Char F Bool = Char
We need one of the existing "layout herald" keywords (of, let, where) to smoothly support the nested block of equations. It's not clear whether or not it is useful to mention the name of the function in the header.
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 will be 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.
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
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added by goldfire 4 years ago.
Description of FC extension to support overlapping type family instances
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