There are a couple kinds of functions that we can turn into totally inspectable data.

Linear functions can be reconstituted into a matrix if you give a basis of vectors.

Functions from enumerable types can be turned into a lookup table

Sufficiently polymorphic functions are another example though. forall a. a-> a is commonly known to only be id. The same goes for fst = forall a b. (a,b)->a and snd and swap and all the nesting of . These functions have exactly one inhabiting value (excluding internal churning and the possibility of going into an infinite loop).

So the type directly tells us the implementation

forall a. (a,a)->a is similar. It can only be fst or snd. Types that reuse a type parameter in the input can only be permutations.

I’ve been trying to find a way to take a written lambda and convert it to data automatically and have been having trouble.

An opaque type that we have hidden the contructors to is the same (T,T)->T can only be fst or snd specialized to T since we can’t possibly destruct on T.

We can figure out which one by giving a labeled example to that function and then inspecting a single output. This gives the permutation and duplication that was done.

Similarly for T -> Either T T

Once we have this, we can (Hopefully) reinterpret this lambda in terms of a monoidal category.

{-# LANGUAGE RankNTypes, GADTs, FlexibleInstances, DataKinds, TypeFamilies,MultiParamTypeClasses, FlexibleContexts, ScopedTypeVariables, FunctionalDependencies, GADTs, TypeOperators #-} --AllowAmbiguousTypes, -- OverlappingInstances, -- UndecidableInstances, import Data.Proxy import Unsafe.Coerce data Tag = Tag Int deriving Show type family (MonoMorphTag a) :: * where MonoMorphTag (a,b) = (MonoMorphTag a, MonoMorphTag b) MonoMorphTag (a->b) = (MonoMorphTag a) -> (MonoMorphTag b) MonoMorphTag Int = Int MonoMorphTag [a] = [a] MonoMorphTag (a,b,c) = (a,b,c) MonoMorphTag (a,b,c,d) = (a,b,c,d) MonoMorphTag Double = Double MonoMorphTag () = () MonoMorphTag Char = Char MonoMorphTag _ = Tag unsafeMonoTag :: a -> MonoMorphTag a unsafeMonoTag = unsafeCoerce -- unsafeTagLeaves :: forall a. MonoMorphTag a -> Tag -- unsafeTagLeaves = unsafeCoerce type T = Tag class GetVal a where val :: Int -> Proxy a -> (a, Int) instance GetVal Tag where val n _ = (Tag n, n+1) instance (GetVal a, GetVal b) => GetVal (a,b) where val n _ = ((v1, v2), n'') where (v1 , n') = val n (Proxy :: Proxy a) (v2 , n'') = val n' (Proxy :: Proxy b) data TagTree a = Node (TagTree a) (TagTree a) | Leaf a deriving Show -- | Apply (k a b) TagTree class Treeify a b where treeify :: a -> TagTree b instance Treeify Tag Tag where treeify x = Leaf x instance (Treeify a Tag, Treeify b Tag) => Treeify (a,b) Tag where treeify (a,b) = Node (treeify a) (treeify b) class MonoMorph a where type Mono a :: * instance MonoMorph (a,b) where type Mono (a,b) = (Mono a, Mono b) {- instance MonoMorph (MonoMorphTag a) where type Mono a = Tag -} {- -- Hmm I'm not sure how to monomorhpize this. fst' :: (TagTup a) => (a, b) -> a fst' = fst -} {- class AutoCurry a b | a -> b where autocurry :: a -> b instance AutoCurry (a->b->Tag) ((a,b)->Tag) where autocurry f = uncurry f instance AutoCurry c (a->c') => AutoCurry (b->c) ((b,a) -> c') where autocurry f = uncurry (\b -> autocurry (f b)) -} data Monoidal = Dup | Mon Monoidal Monoidal | Par Monoidal Monoidal | Fst | Snd | Id | Comp Monoidal Monoidal deriving Show data Monoidal' a b where Id' :: Monoidal' a a Dup' :: Monoidal' a (a,a) Fst' :: Monoidal' (a,b) a Snd' :: Monoidal' (a,b) b Comp' :: Monoidal' b c -> Monoidal' a b -> Monoidal' a c Mon' :: Monoidal' a a' -> Monoidal' b b' -> Monoidal' (a,b) (a',b') data FunData = FunData {inval :: TagTree Tag, outval :: TagTree Tag} deriving Show class TestIdea a b where works :: (a -> b) -> (a, b) instance (GetVal a) => TestIdea a b where works f = (inval, f inval) where inval = fst $ val 0 (Proxy :: Proxy a) -- fst $ val 0 (Proxy :: Proxy b) fuckmyshitup :: (GetVal a, Treeify a Tag, Treeify b Tag) => (a -> b) -> FunData fuckmyshitup f = let (a, b) = works f in FunData ((treeify a) :: TagTree Tag) ((treeify b):: TagTree Tag) ccc :: FunData -> Monoidal ccc (FunData x (Node y z)) = Mon (ccc $ FunData x y) (ccc $ FunData x z) ccc (FunData (Leaf _) (Leaf _)) = Id ccc (FunData (Node x y) z@(Leaf (Tag n))) = if inleft n x then Comp Fst (ccc (FunData x z)) else Comp Snd (ccc (FunData y z)) ineither :: Int -> TagTree Tag -> Bool ineither n (Node x y) = (ineither n x) || (ineither n y) ineither n (Leaf (Tag n')) = n == n' inleft :: Int -> TagTree Tag -> Bool inleft n (Node l r) = ineither n l inleft n (Leaf (Tag n')) = n == n' inright :: Int -> TagTree Tag -> Bool inright n (Node l r) = ineither n r inright n (Leaf (Tag n')) = n == n' -- Then we can compile to categories. Replacing the entire structure with dup and par and -- fst, snd, etc. -- Make an infix operator $' --data Apply k a b c = Apply (FreeCat k a b) c --type ($$) = Apply -- No, don't need getval. -- We'll just need it for treeify? {-instance GetVal c => GetVal (Apply k a b c) where val n _ = where x, n' = val n Proxy c -} -- Another Option data A data B data C -- This is basically a lambda calculus -- I could probably finitely enumerate through all the typeclasses for all the variables example = Proxy :: Proxy ((A,B) -> B) -- Hmm this would allow you to force duplicate input types though. {- class (Tagify a ~ a, Tagify b ~ b) => TestIdea a b where works :: (a -> b) -> (a, b) instance (GetVal a) => TestIdea a b where works f = (inval, f inval) where inval = fst $ val 0 (Proxy :: Proxy a) -- fst $ val 0 (Proxy :: Proxy b) -} --thisworks :: String --thisworks = works id -- fst . (val 0) {- instance (F a ~ flag, GetVal' flag a) => GetVal a where val = val' (Proxy :: Proxy flag) class GetVal' (flag :: Bool) a where val' :: Proxy flag -> a -> Tagify a instance (GetVal a, GetVal b) => GetVal' 'True (a,b) where val' _ (x,y) = (val x, val y) instance GetVal' 'False a where val' _ x = Tag 0 -}

What about TH? Also the new quantified constraints extensions might be helpful?

Ok. A Different approach. This works much better to what I had in mind. you can write aribatrary (\(x,y,) -> (y,x)) tuple like lambdas and it will convert them to a category. I really had to hack around to get the thing to compile. Like that Pick typeclass, what the heck? Why can I get defaults values in type families but not in typeclasses?

It is all decidedly not typesafe. You can get totally nonsensical things to compile to something. However if you stick to lambdas, you’ll be ok. Maybe.

No on further review this does not work. I got tricked that the type seemed ok at a certain point. A couple problems arise upon actual application. Since the idea is to drive the form based on the type variables upon actual application to something that has types of the same form it gets all screwed up. Also tons of instances are overlapping, although I think this is fixable.

Maybe what I need is existential types that can’t ever unify together accidentally.

A couple thought on typelevel programming principles:

- Typeclasses are hard to get default cases. You want to use type families if that is what you want
- Typeclasses need unique stuff to appear on the right hand side. Only 1 pattern should match. You might need to add extra parameters to match to which you can force on the left hand side of the instance
- ~ type equality is real useful

An alternative to using lambda is to use an explicit Proxy. The type variables are basically just as good for syntactic purposes (a touch more noisy).

{-# LANGUAGE RankNTypes, GADTs, FlexibleInstances, DataKinds, TypeFamilies,MultiParamTypeClasses, ImpredicativeTypes, FlexibleContexts, ScopedTypeVariables, FunctionalDependencies, UndecidableInstances, GADTs, TypeOperators #-} -- OverlappingInstances, NoImplicitPrelude -- --UndecidableInstances, --OverlappingInstances, import Data.Type.Bool import Data.Proxy --import Control.Category --import GHC.Base hiding (id,(.)) class IsId a where val :: a -> a -- toCat instance forall a. IsId (a -> a) where val _ = id {- class Catable f a b | f -> a,b where toCat :: forall k. CartesianCategory k => k a b instance forall a b. Catable ((a,b)->a) (a,b) a where toCat = fst -} class Fst ab a | ab -> a where -- toCat :: forall k. k ab a instance forall a b. Fst (a,b) a class Anything b where fun :: b -> b class Stringly a where stringly :: a -> String instance (Stringly a, Stringly b) => Stringly (a,b) where stringly (x,y) = "(" ++ (stringly x) ++ "," ++ (stringly y) ++ ")" {- instance (Stringly a, Stringly b) => Stringly (a -> b) where stringly f = "(" ++ (stringly x) ++ "->" ++ (stringly y) ++ ")" -} class Category k where dot' :: k b c -> k a b -> k a c id' :: k a a instance Category (->) where dot' = (.) id' = id class Category k => CartesianCat k where fst' :: k (a,b) a snd' :: k (a,b) b join' :: k a b -> k a c -> k a (b,c) instance CartesianCat (->) where fst' = fst snd' = snd join' = join'' class Catable a b where toCat :: CartesianCat k => (a -> b) -> (k a b) -- toCat (\x -> ((x,x),x)) . id -- it's not INSANE to just list out a finite list of possibilities ((a,b),c) etc. {- data HeldApply k a b = HeldApply (k a b) a ($$) :: Category k => k a b -> b -> HeldApply k a b f $$ x = HeldApply f instance Catable a (HeldApply a b) where toCat Doesn't seem to work. We don't have an a get get the heldapply out of the function Maybe we could pass in the approriate function as a a lambda \f x -> Apply f x instance ExponentialCategory k where apply :: k (k a b, a) b -} instance Catable a a where toCat _ = id' -- why is this okay? should these be covered by the other cases? instance Catable (a,b) a where toCat _ = fst' instance Catable (a,b) b where toCat _ = snd' dup x = (x,x) {- instance Catable a (a,a) where toCat _ = dup -} join'' f g x = (f x, g x) -- iterates down through the output instance (Catable a b, Catable a c) => Catable a (b,c) where toCat f = join' (toCat (fst . f)) (toCat (snd . f)) {- instance (InL c (a,b), Catable a c) => Catable (a,b) c where toCat f = (toCat (f . fst)) instance (InR c (a,b), Catable a c) => Catable (a,b) c where toCat f = (toCat (f . snd)) -} instance (Catable a c, Catable b c, Pick' c (a,b) (In a c)) => Catable (a,b) c where toCat f = pick' (Proxy :: Proxy (In a c)) {- instance (Catable a c, Catable b c, Pick c (a,b) (In a c)) => Catable (a,b) c where toCat f = (toCat (pick (Proxy :: Proxy (In a c)))) -} {- class In a c where find :: c -> a instance In a a find = id instance In a b => In a (b,c) find = find . fst instance In a c => In a (b,c) find = find . snd -} {- type family (LorR a c) :: Nat where LorR a (a,_) = 1 LorR a (_,a) = 2 LorR a ((b,c),d) = (LorR a (b,c)) + (LorR a d) LorR a (d,(b,c)) = (LorR a (b,c)) + (LorR a d) LorR a _ = 0 -} type family (In a c) :: Bool where In a a = 'True In a (a,_) = 'True In a (_,a) = 'True In a ((b,c),d) = In a (b,c) || In a d In a (d,(b,c)) = In a (b,c) || In a d In a _ = 'False {- type Snd = forall a b. (a,b) -> b type family (FstSnd a) :: * where FstSnd 'True = Snd FstSnd 'False = Snd -} class Pick a c (d :: Bool) where pick :: Proxy d -> c -> a instance (Pick a (e,f) (In a e), (e,f) ~ b) => Pick a (b,c) 'True where pick _ = (pick (Proxy :: Proxy (In a e))) . fst instance (Pick a (e,f) (In a e), (e,f) ~ c) => Pick a (b,c) 'False where pick _ = (pick (Proxy :: Proxy (In a e))) . snd instance Pick a (a,b) 'True where pick _ = fst instance Pick a (b,a) 'False where pick _ = snd instance Pick a a d where pick _ = id -- The bool is true if in the left branch class Pick' a c (d :: Bool) where pick' :: CartesianCat k => Proxy d -> k c a instance (Pick' a (e,f) (In a e), (e,f) ~ b) => Pick' a (b,c) 'True where pick' _ = dot' (pick' (Proxy :: Proxy (In a e))) fst' instance (Pick' a (e,f) (In a e), (e,f) ~ c) => Pick' a (b,c) 'False where pick' _ = dot' (pick' (Proxy :: Proxy (In a e))) snd' instance Pick' a (a,b) 'True where pick' _ = fst' instance Pick' a (b,a) 'False where pick' _ = snd' instance Pick' a a d where pick' _ = id' {- class InL a c where instance InL a a instance In a b => InL a (b,c) class InR a c instance InR a a instance In a b => InR a (c,b) -} {- instance (Catable a c, Catable b c) => Catable (a,b) c where toCat f = instance (Catable a c, Catable b c) => Catable a (b,c) where toCat f = -} {- instance (Stringly a, Stringly b, (a,b) ~ c, IsTup c ~ 'True) => Stringly c where stringly (x,y) = "(" ++ (stringly x) ++ "," ++ (stringly y) ++ ")" -} --instance (IsTup a ~ 'False, IsArr a ~ 'False) => Stringly a where -- stringly _ = "_" instance forall a. Anything a where fun = id example :: a -> a example = val id type family (IsTup a) :: Bool where IsTup (a,b) = 'True IsTup _ = 'False type family (IsArr a) :: Bool where IsArr (a->b) = 'True IsArr _ = 'False