[Haskell-cafe] Subsets and supersets
andrewcoppin at btinternet.com
Wed Mar 16 13:13:35 CET 2011
The other day I saw the "green field Haskell" discussion on Reddit. Of
course, I could give you a very long list (ha!) of things that I would
change about Haskell. (This would probably begin with expunging
Monad.fail with extreme prejudice...) But there's one particular feature
which is non-trivial enough to be worth discussing here...
Haskell has ADTs. Most of the time, these work great. As I've written in
several other places (but possibly not here), OO languages tend to
"factor" the problem the other way. That is, if I want a binary tree, an
OO language makes me split the type and all of its operations into three
parts (an abstract base class, a branch subclass, and a leaf subclass).
Adding each new operation requires adding an abstract version of it to
the abstract base class, and putting half of the implementation into
each concrete subclass.
Haskell is factored the other way. The definition of the type goes in
one place. The definition of each operation over it goes in one place.
More can be added at any time. In particular, the OO approach doesn't
let you add new tree operations without altering the source code for the
tree classes. The ADT version *does* let you add new operations -
provided the tree structure is visible.
This is not to say that the OO approach is wrong, of course. The
important thing about a binary tree is that there are exactly two kinds
of nodes - leaves and branches - and there will *always* be exactly two
kinds. That's what makes it a binary tree! [OK, no it isn't, but bare
If, instead of a binary tree, or an abstract syntax tree, or some other
fixed, unchanging structure, we wanted to deal with, say, bank
accounts... Well, there are a bazillion kinds of bank account. And more
might be added at any time. Here an ADT is utterly the wrong thing to
do. It would have a bazillion constructors, and all the processing for a
particular bank account type would be split between dozens of seperate
functions. By contrast, an OO language would put each account type and
all its associated processing in the same place - the class definition.
In summary, problems can be factored two ways. ADTs do it one way,
classes do it the other way, and both can be appropriate. OO languages
use classes, but, happily, Haskell has ADTs *and* classes! :-D And not
classes in the same sense as OO classes; the way Haskell does it is
actually better, IMHO.
Anyway, coming back on-topic... ADTs are great, but sometimes they don't
let me easily express exactly what I want. Specifically, sometimes I
have one type which is really a subset of another type, or perhaps a
superset of several times.
You can build subsets using GADTs, after a fashion:
data Foobar x where
Foobar1 ... :: Foobar Foo
Foobar2 ... :: Foobar Bar
Foobar3 ... :: Foobar x
Foobar4 ... :: Foobar Bar
Foobar5 ... :: Foobar Foo
Now any expression of type Foobar Foo is guaranteed to contain only
Foobar1, Foobar3 or Foobar5, while any expression of type Foobar Bar is
guaranteed to contain only Foobar2, Foobar3 or Foobar4. As the tangle of
sets becomes more complicated, this approach becomes more difficult to
You can of course build supersets using an even more mundane Haskell
construct. The canonical example is Either. If I want a field that holds
a Foo *or* a Bar, I can trivially write Either Foo Bar.
The trouble is, this entails an extra level of indirection. In order to
access a Foo or a Bar, I now have to strip off the Left or Right
constructor to get to it. To construct a value, I have to stick in the
constructor. Sometimes this is just fine, even desirable. For example, I
recently wrote a function who's type is
spans :: (x -> Bool) -> [x] -> [Either [x] [x]]
Here the Left and Right constructors are actually /telling/ me
something. They convey /information/.
On the other hand, in the case of Either Foo Bar, the constructor tells
me nothing. Consider:
data Foo = Foo1 | Foo2 | Foo3
data Bar = Bar1 | Bar2 | Bar3
foobar :: Either Foo Bar -> x
foobar e =
case e of
Left Foo1 -> ...
Left Foo2 -> ...
Left Foo3 -> ...
Right Bar1 -> ...
Right Bar2 -> ...
Right Bar3 -> ...
There is, logically, no particular reason why we shouldn't simplify the
function to just
foobar Foo + Bar -> x
foobar fb =
case fb of
Foo1 -> ...
Foo2 -> ...
Foo3 -> ...
Bar1 -> ...
Bar2 -> ...
Bar3 -> ...
The set of possible constructors for Foo and for Bar are disjoint. There
is no possibility of ambiguity here. The type system itself even /uses/
this fact to infer the type of literal constructor applications! And
yet, you cannot write code such as the above. Not in Haskell, anyway.
So what is the use case here? Well, consider for a moment the problem of
parsing Haskell source code. Haskell contains both "patters" and
"expressions". Now actually, it turns out that a pattern and an
expression have a very similar structure. The set of possible patterns
is /almost/ a subset of the set of possible expressions. (But not
quite.) That gives me design choices:
* Design one ADT which encodes both patterns and expressions. Since this
removes the ability to statically distinguish between them, this is
almost certainly the wrong choice.
* Design one ADT for patterns and a seperate, unrelated ADT for
expressions. I have now violated the DRY principle. And, hypothetically,
if I wanted to treat a pattern as an expression, I would have to write a
lot of boilerplate code that does nothing more than replace one
constructor name with another (recursively).
* Do something with GADTs, and hope the type system doesn't explode.
Note that a literal value such as 5 or "foo" is both a valid expression
and a valid pattern. In fact, I probably want an ADT just for literal
values (since there's quite a few possible forms, and often I don't care
The net result of all this is usually something like
data Literal =
LInteger Integer |
LChar Char |
LString String |
data Pattern =
PLiteral Literal |
PConstructor Name [Pattern] |
PVariable Name |
data Expression =
ELiteral Literal |
EConstructor Name [Expression] |
EVariable Name |
ELambda [Pattern] Expression |
expr1 = EConstructor "Foo" [ELiteral (LString "bar")]
patt1 = PConstructor "Foo" [PLiteral (LString "bar")]
The entire Literal type just looks plain wrong. The set of possible
Integer values and the set of possible Char values are utterly disjoint.
Really I ought to be able to just say which types Literal is the union
of, and not worry about it any further.
(There is the minor detail that the set of Integer values and Double
values are *not* disjoint. There is also the minor detail that if any
valid Char is also a valid Literal, then the type of 'x' is now no
longer unambiguously Char. This makes type inference "interesting".)
The case of patterns and expressions is more interesting, since here we
have two overlapping sets, yet neither is strictly a superset of the
other. Today we could probably solve this as just
data Haskell x where
HLiteral Literal :: Haskell x
HConstructor Name [Haskell x] :: Haskell x -- Is this right?
HVariable Name :: Haskell x
HWildcard :: Haskell Pattern
HLambda [Haskell Pattern] (Haskell Expression) :: Haskell
In other words, by carefully applying GADTs, the pattern/expression
problem is probably solveable today. But it's tricky. And it requires me
to create two useless nullary types that exist only to let the type
checker distinguish patterns from expressions. (Fortunately empty data
declarations can be enabled quite easily.)
It would just be nice if I would trivially and easily take the union of
several types, and if this was a fundamental operation. Then we would
probably build type hierachies in a quite different way. Of course,
there are several "minor details" which would need to be overcome first:
1. The Haskell 2010 type system guarantees that 'x' has type Char. If we
let people define supersets of types, how is the poor type checker
supposed to know exactly which type 'x' is meant to belong to?
2. I might not want to explicitly notate every single pattern and
expression with its exact type, but /the compiled code/ needs to know!
When a Char value becomes a Literal value, how does a case expression
handle that? Currently (AFAIK) each constructor in a type is assigned a
unique ID, and case expressions match against that. But if you take the
union of two types, now what? Constructor IDs are only unique within a
single type, not within some arbitrary union of types.
3. It strikes me that the RTS (particularly the GC) is going to need to
know what the hell we're doing too.
The most obvious solution to #1 is that the type of every term is the
simplest possible type, unless you try to unify it with some other type,
at which point the type for both such terms becomes the union of types.
I rather suspect that this will instantly make every possible program
well-typed, however. I suppose an alternative is to require explicit
type signatures, but then we just lost the entire advantage of this
As to #2, maybe this whole type union insanity could be considered
merely syntax sugar, with the whole edifice being desugared into the
multiply constructor wrapping craziness we have now. But it seems a
pitty. If we could find a way to assign a globally unique ID to every
constructor (code address pointer perhaps?), then we could increase
program efficiency at the same take as making code less verbose.
As to #3... my name is not Simon.
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