type-level integers using type families
Has anyone else tried implementing type-level integers using type families? I tried using a couple of other type level arithmetic libraries (including type-level on Hackage) and they felt a bit clumsy to use. I started looking at type families and realized I could pretty much build an entire Scheme-like language based on them. In short, I've got addition, subtraction, & multiplication working after just a days worth of hacking. I'm going to post the darcs archive sometime, sooner if anyone's interested. I really like the type-families based approach to this, it's a lot easier to understand, and you can think about things functionally instead of relationally. (Switching back and forth between Prolog-ish thinking and Haskell gets old quick.) Plus you can do type arithmetic directly in place, instead of using type classes everywhere. One thing that I'd like to be able to do is lazy unification on type instances, so that things like data True data False type family Cond x y z type instance Cond True y z = y type instance Cond False y z = z will work if the non-taken branch can't be unified with anything. Is this planned? Is it even feasible? I'm pretty sure it would be possible to implement a Lambda like this, but I'm not seeing it yet. Any ideas? Pete
Peter Gavin wrote:
Has anyone else tried implementing type-level integers using type families?
I tried using a couple of other type level arithmetic libraries (including type-level on Hackage) and they felt a bit clumsy to use. I started looking at type families and realized I could pretty much build an entire Scheme-like language based on them.
In short, I've got addition, subtraction, & multiplication working after just a days worth of hacking. I'm going to post the darcs archive sometime, sooner if anyone's interested.
I really like the type-families based approach to this, it's a lot easier to understand, and you can think about things functionally instead of relationally. (Switching back and forth between Prolog-ish thinking and Haskell gets old quick.) Plus you can do type arithmetic directly in place, instead of using type classes everywhere.
nice, it's been tried before, etc. etc.. And of course it doesn't work with a released version of GHC, so maybe it's hoping too much that it would be on Hackage. What I was going to say was, see if there is one on hackage, otherwise there should be one there to be polished. But I guess searching haskell-cafe is your man :-) (your way to try to find any. Or the Haskell blogosphere too.)
One thing that I'd like to be able to do is lazy unification on type instances, so that things like
...
will work if the non-taken branch can't be unified with anything. Is this planned? Is it even feasible?
I'm pretty sure it would be possible to implement a Lambda like this, but I'm not seeing it yet. Any ideas?
Yeah -- that would be neat but generally tends to lead to undecidability (unless you're really careful making it a lot(?) less useful). That is, potential nontermination in the type inferencer/checker, not just in runtime. Then you'll want it to be well-defined when something is type-level-lazy, so you can reliably write your type-level algorithms. And *that* is bound to be rather difficult to define and to implement and maintain.
Peter Gavin:
Has anyone else tried implementing type-level integers using type families?
I tried using a couple of other type level arithmetic libraries (including type-level on Hackage) and they felt a bit clumsy to use. I started looking at type families and realized I could pretty much build an entire Scheme-like language based on them.
In short, I've got addition, subtraction, & multiplication working after just a days worth of hacking. I'm going to post the darcs archive sometime, sooner if anyone's interested.
I really like the type-families based approach to this, it's a lot easier to understand, and you can think about things functionally instead of relationally. (Switching back and forth between Prolog- ish thinking and Haskell gets old quick.) Plus you can do type arithmetic directly in place, instead of using type classes everywhere.
I am glad to hear that type families work for you.
One thing that I'd like to be able to do is lazy unification on type instances, so that things like
data True data False
type family Cond x y z type instance Cond True y z = y type instance Cond False y z = z
will work if the non-taken branch can't be unified with anything. Is this planned? Is it even feasible?
I don't think i entirely understand the question. Do you mean that from an equality like Cond b Int Bool ~ Int you want the type checker to infer that (b ~ True)? This is generally not correct reasoning as type families are open; ie, subsequent modules might add data Bogus type instance Bogus y z = Int and now there are two solutions to (Cond b Int Bool ~ Int). Manuel
Manuel M T Chakravarty wrote:
Peter Gavin:
will work if the non-taken branch can't be unified with anything. Is this planned? Is it even feasible?
I don't think i entirely understand the question.
Maybe he wants, given cond :: Cond x y z => x -> y -> z tt :: True true_exp :: a false_exp :: <<<untypable>>> that cond tt true_exp false_exp :: a That is the type of false_exp is "lazily inferred", so that type errors do not make inference fail if they show up in an unneeded place. If we had subtyping, typing that as Top would suffice, I believe. Zun.
Roberto Zunino wrote:
Manuel M T Chakravarty wrote:
Peter Gavin:
will work if the non-taken branch can't be unified with anything. Is this planned? Is it even feasible? I don't think i entirely understand the question.
Maybe he wants, given
cond :: Cond x y z => x -> y -> z tt :: True true_exp :: a false_exp :: <<<untypable>>>
that
cond tt true_exp false_exp :: a
That is the type of false_exp is "lazily inferred", so that type errors do not make inference fail if they show up in an unneeded place.
Yes, that's exactly what I want, but for type families (not MPTC). I think it could be done if the type arguments were matched one at a time, across all visible instances. So given
data True data False
type family Cond x y z type instance Cond True y z = y type instance Cond False y z = z
the first argument (x) of Cond would be matched before y and z are inferred. Then if x is True, the type checker would see that z is never used on the RHS, so it would not bother trying to infer it at all.
If we had subtyping, typing that as Top would suffice, I believe.
Zun.
Peter Gavin wrote:
Roberto Zunino wrote:
Maybe he wants, given cond :: Cond x y z => x -> y -> z tt :: True true_exp :: a false_exp :: <<<untypable>>> that cond tt true_exp false_exp :: a That is the type of false_exp is "lazily inferred", so that type errors do not make inference fail if they show up in an unneeded place.
Yes, that's exactly what I want, but for type families (not MPTC). I think it could be done if the type arguments were matched one at a time, across all visible instances.
What do you think of the following idea? Using naive type level natural numbers,
data Zero newtype Succ a = Succ a
Booleans,
data True data False
comparison,
type family (:<:) x y type instance (Zero :<: Succ a) = True type instance (Zero :<: Zero ) = False type instance (Succ a :<: Zero ) = False type instance (Succ a :<: Succ b) = a :<: b
difference,
type family Minus x y type instance Minus a Zero = a type instance Minus (Succ a) (Succ b) = Minus a b
and a higher order type level conditional,
type family Cond2 x :: (* -> * -> *) -> (* -> * -> *) -> * -> * -> * type First2 (x :: * -> * -> *) (y :: * -> * -> *) = x type Second2 (x :: * -> * -> *) (y :: * -> * -> *) = y type instance Cond2 True = First2 type instance Cond2 False = Second2
we can define division as follows:
type family Div x y type DivZero x y = Zero type DivStep x y = Succ (Div (Minus0 x y) y) type instance Div x y = Cond2 (x :<: y) DivZero DivStep x y
It's not exactly what you asked for, but I believe it gets the effect that you wanted. Enjoy, Bertram
Peter Gavin schrieb:
Has anyone else tried implementing type-level integers using type families?
I tried using a couple of other type level arithmetic libraries (including type-level on Hackage) and they felt a bit clumsy to use. I started looking at type families and realized I could pretty much build an entire Scheme-like language based on them.
In short, I've got addition, subtraction, & multiplication working after just a days worth of hacking. I'm going to post the darcs archive sometime, sooner if anyone's interested.
I really like the type-families based approach to this, it's a lot easier to understand, and you can think about things functionally instead of relationally. (Switching back and forth between Prolog-ish thinking and Haskell gets old quick.) Plus you can do type arithmetic directly in place, instead of using type classes everywhere.
I tried to rewrite Alfonso Acosta's type-level library (the one on hackage) using type-families too, because, right, they are much nicer to use than MPTCs. It is a straigtforward translation, I've uploaded it to http://code.haskell.org/~bhuber/type-level-tf-0.2.1.tar.gz now (relevant files: src/Data/TypeLevel/{Bool.hs,Num/Ops.hs}). I've also added a type-level ackermann to torture ghc a little bit ;), but left out logarithms. Plus, I did very little testing.
One thing that I'd like to be able to do is lazy unification on type instances, so that things like
data True data False
type family Cond x y z type instance Cond True y z = y type instance Cond False y z = z
I'm not sure if this is what you had in mind, but I also found that e.g.
type instance Mod x y = Cond (y :>: x) x (Mod (Sub x y) y)
won't terminate, as
Mod D0 D1 ==> Cond True D0 (Mod (Sub D0 D1) D0) ==> <loop>
rather than
Mod D0 D1 ==> Cond True D0 ? ==> D0
For Mod, I used the following (usual) encoding:
type family Mod' x y x_gt_y type instance Mod' x y False = x type instance Mod' x y True = Mod' (Sub x y) y ((Sub x y) :>=: y) type family Mod x y type instance Mod x y = Mod' x y (x :>=: y)
There are few other things you cannot do with type families, and some for which you need type classes anyway (Alfonso pointed me to http://www.haskell.org/pipermail/haskell-cafe/2008-February/039489.html ). Here's what I found: 1) One cannot define type equality (unless total type families become available), i.e. use the overlapping instances trick:
instance Eq e e True instance Eq e e' False
Consequently, all type-level functions which depend on type equality (see HList) need to be encoded using type classes. 2) One cannot use superclass contexts to derive instances e.g. to define
instance Succ (s,s') => Pred (s',s)
In constrast, when using MPTC + FD, one can establish more than one TL function in one definition
class Succ x x' | x -> x', x' -> x
3) Not sure if this is a problem in general, but I think you cannot restrict the set of type family instances easily. E.g., if you have an instance
type instance Mod10 (x :* D0) = D0
then you also have
Mod10 (FooBar :* D0) ~ D0
What would be nice is something like
type instance (IsPos x) => Mod10 (x :* D0) = D0
though
type family AssertThen b x type instance AssertThen True x = x type instance Mod10 (x :* D0) = AssertThen (IsPos x) D0
seems to work as well. 4) Not really a limitation, but if you want to use instance methods of Nat or Bool (e.g. toBool) on the callee site, you have to provide context that the type level functions are closed w.r.t. to the type class:
test_1a :: forall a b b1 b2 b3. (b1 ~ And a b, b2 ~ Not (Or a b), b3 ~ Or b1 b2, Bool b3) => a -> b -> Prelude.Bool test_1a a b = toBool (undefined :: b3)
Actually, I think the 'a ~ b' syntax is very nice. I'm really looking forward to type families. best regards, benedikt
Benedikt Huber wrote:
data True data False
type family Cond x y z type instance Cond True y z = y type instance Cond False y z = z
I'm not sure if this is what you had in mind, but I also found that e.g.
type instance Mod x y = Cond (y :>: x) x (Mod (Sub x y) y)
won't terminate, as
Mod D0 D1 ==> Cond True D0 (Mod (Sub D0 D1) D0) ==> <loop>
Right, because it always tries to infer the (Mod (Sub x y) y) part no matter what the result of (y :>: x) is.
rather than
Mod D0 D1 ==> Cond True D0 ? ==> D0
For Mod, I used the following (usual) encoding:
type family Mod' x y x_gt_y type instance Mod' x y False = x type instance Mod' x y True = Mod' (Sub x y) y ((Sub x y) :>=: y) type family Mod x y type instance Mod x y = Mod' x y (x :>=: y)
Yes, it's possible to terminate a loop by matching the type argument directly.
1) One cannot define type equality (unless total type families become available), i.e. use the overlapping instances trick:
instance Eq e e True instance Eq e e' False
I didn't want to mix type classes and families in my implementation. All the predicates are implemented like so:
type family Eq x y type instance Eq D0 D0 = True type instance Eq D1 D1 = True ... type instance Eq (xh :* xl) (yh :* yl) = And (Eq xl yl) (Eq xh yh)
then I've added a single type-class
class Require b instance Require True
so you can do stuff like
f :: (Require (Eq (x :+: y) z)) => x -> y -> z
or whatever. I haven't yet tested it (but I think it should work) :) Pete
Consequently, all type-level functions which depend on type equality (see HList) need to be encoded using type classes.
2) One cannot use superclass contexts to derive instances e.g. to define
instance Succ (s,s') => Pred (s',s)
In constrast, when using MPTC + FD, one can establish more than one TL function in one definition
class Succ x x' | x -> x', x' -> x
3) Not sure if this is a problem in general, but I think you cannot restrict the set of type family instances easily.
E.g., if you have an instance
type instance Mod10 (x :* D0) = D0
then you also have
Mod10 (FooBar :* D0) ~ D0
What would be nice is something like
type instance (IsPos x) => Mod10 (x :* D0) = D0
though
type family AssertThen b x type instance AssertThen True x = x type instance Mod10 (x :* D0) = AssertThen (IsPos x) D0
seems to work as well.
4) Not really a limitation, but if you want to use instance methods of Nat or Bool (e.g. toBool) on the callee site, you have to provide context that the type level functions are closed w.r.t. to the type class:
test_1a :: forall a b b1 b2 b3. (b1 ~ And a b, b2 ~ Not (Or a b), b3 ~ Or b1 b2, Bool b3) => a -> b -> Prelude.Bool test_1a a b = toBool (undefined :: b3)
Actually, I think the 'a ~ b' syntax is very nice.
I'm really looking forward to type families.
best regards,
benedikt
Replying to myself... I put a copy of the darcs repo at http://code.haskell.org/~pgavin/tfp, if anyone is interested. Pete Peter Gavin wrote:
Has anyone else tried implementing type-level integers using type families?
I tried using a couple of other type level arithmetic libraries (including type-level on Hackage) and they felt a bit clumsy to use. I started looking at type families and realized I could pretty much build an entire Scheme-like language based on them.
In short, I've got addition, subtraction, & multiplication working after just a days worth of hacking. I'm going to post the darcs archive sometime, sooner if anyone's interested.
I really like the type-families based approach to this, it's a lot easier to understand, and you can think about things functionally instead of relationally. (Switching back and forth between Prolog-ish thinking and Haskell gets old quick.) Plus you can do type arithmetic directly in place, instead of using type classes everywhere.
One thing that I'd like to be able to do is lazy unification on type instances, so that things like
data True data False
type family Cond x y z type instance Cond True y z = y type instance Cond False y z = z
will work if the non-taken branch can't be unified with anything. Is this planned? Is it even feasible?
I'm pretty sure it would be possible to implement a Lambda like this, but I'm not seeing it yet. Any ideas?
Pete
On Thu, May 29, 2008 at 5:15 AM, Peter Gavin
Has anyone else tried implementing type-level integers using type families?
When I started to work on thetype-level and parameterized data packages, I considered using type-families and GADTs, but I found quite a few problems which have been nicely summarized by Benedikt in this thread.
participants (7)
-
Alfonso Acosta -
Benedikt Huber -
Bertram Felgenhauer -
Isaac Dupree -
Manuel M T Chakravarty -
Peter Gavin -
Roberto Zunino