[Haskell-cafe] partially applied data constructor and corresponding type

[TP]

Hello,

I ask myself if there is a way to do the following.
Imagine that I have a dummy type:

data Tensor = TensorVar Int String

where the integer is the order, and the string is the name of the tensor.
I would like to make with the constructor TensorVar a type "Vector", such
that, in "pseudo-language":

data Vector = TensorVar 1 String

Because a vector is a tensor of order 1.

Is this possible? I have tried type synonyms and newtypes without any success.

Thanks a lot,

TP

_______________________________________________
Haskell-Cafe mailing list
Haskel...@haskell.org
http://www.haskell.org/mailman/listinfo/haskell-cafe

[Yury Sulsky]

Hi TP,

Are you looking to use a phantom types here? Here's an example:

data One

data Two

data Tensor ndims a = Tensor { dims :: [Int], items :: [a] }

type Vector = Tensor One

type Matrix = Tensor Two

etc.

[TP]

Thanks Yury,

 

The problem with this solution is that if I have written a method for the Tensor type (for example a method of a typeclass of which Tensor is an instance) that uses the order of the tensor (your "ndims") in a general way, I cannot reuse it easily for a vector defined like that.

 

In fact, my problem is to be able to define:

* from my type "Tensor", a type "Vector", by specifying that the dimension must be one to have a "Vector" type.

* from my constructor "TensorVar", a constructor "VectorVar", which corresponds to TensorVar with the integer equal to 1.

 

The idea is to avoid duplicating code, by reusing the tensor type and data constructor. At some place in my code, in some definition (say, of a vector product), I want vectors and not more general tensors.

 

TP

[Stephen Tetley]

What you probably want are type level integers (naturals)

Yury Sulsky used them in the message above - basically you can't use
literal numbers 1,2,3,... etc as they are values of type Int (or
Integer, etc...) instead you have to use type level numbers:

data One
data Two

Work is ongoing for type level numbers in GHC and there are user
libraries on Hackage so there is a lot of work to crib from.

[TP]

Thanks for pointing to "type level integers". With that I have found:

http://www.haskell.org/haskellwiki/The_Monad.Reader/Issue5/Number_Param_Types

For example:

-------------------------------
data Zero = Zero
data Succ a = Succ a

class Card c where
c2num:: c -> Integer

cpred::(Succ c) -> c
cpred = undefined

instance Card Zero where
c2num _ = 0

instance (Card c) => Card (Succ c) where
c2num x = 1 + c2num (cpred x)

main = do

putStrLn $ show $ c2num (Succ (Succ Zero))
-------------------------------

I will continue to examine the topic in the following days, according to my
needs.

Thanks a lot,

TP

[Richard Eisenberg]

There's a lot of recent work on GHC that might be helpful to you. Is it possible for your application to use GHC 7.6.x? If so, you could so something like this:

{-# LANGUAGE DataKinds, GADTs, KindSignatures #-}

data Nat = Zero | Succ Nat

type One = Succ Zero
type Two = Succ One
type Three = Succ Two

-- connects the type-level Nat with a term-level construct
data SNat :: Nat -> * where
SZero :: SNat Zero
SSucc :: SNat n -> SNat (Succ n)

zero = SZero
one = SSucc zero
two = SSucc one
three = SSucc two

data Tensor (n :: Nat) a = MkTensor { dims :: SNat n, items :: [a] }

type Vector = Tensor One
type Matrix = Tensor Two

mkVector :: [a] -> Vector a
mkVector v = MkTensor { dims = one, items = v }

vector_prod :: Num a => Vector a -> Vector a
vector_prod (MkTensor { items = v }) = ...

specializable :: Tensor n a -> Tensor n a
specializable (MkTensor { dims = SSucc SZero, items = vec }) = ...
specializable (MkTensor { dims = SSucc (SSucc SZero), items = mat }) = ...


This is similar to other possible approaches with type-level numbers, but it makes more use of the newer features of GHC that assist with type-level computation. Unfortunately, there are no "constructor synonyms" or "pattern synonyms" in GHC, so you can't pattern match on "MkVector" or something similar in specializable. But, the pattern matches in specializable are GADT pattern-matches, and so GHC knows what the value of n, the type variable, is on the right-hand sides. This will allow you to write and use instances of Tensor defined only at certain numbers of dimensions.

I hope this is helpful. Please write back if this technique is unclear!

Richard

[TP]

Thanks a lot for your message.
I can use a recent version of GHC 7.6.x (I will install the last version of
Kubuntu for that purpose).
However, it will take me some time to understand correctly this code (e.g. I
do not know "data kinds"), I will go back to you if I encounter difficulties.

Thanks,

TP

[TP]

Thanks a lot! Those days I have read about "DataKinds" extension (with help
of Haskell-Cafe guys), and now I am able to understand your code. The
technique to connect the type-level and term-level integers allows to
duplicate information (duplicate information needed because of my more or
less clear requirements in my previous posts), but in a safe way (i.e. with
no copy/paste error): if I change "one" in "two" in the definition of the
smart constructor mkVector, I obtain an error, as expected because of the
use of type variable n on both sides of the equality in Tensor type
definition (and then we understand why the type constructor SNat has been
introduced).

This is a working example (this is not exactly what I will do at the end,
but it is very instructive! One more time, thanks!):
--------------------------------------
{-# LANGUAGE DataKinds, GADTs, KindSignatures, StandaloneDeriving #-}

data Nat = Zero | Succ Nat

type One = Succ Zero
type Two = Succ One
type Three = Succ Two

-- connects the type-level Nat with a term-level construct
data SNat :: Nat -> * where
SZero :: SNat Zero
SSucc :: SNat n -> SNat (Succ n)

deriving instance Show (SNat a)

zero = SZero
one = SSucc zero
two = SSucc one
three = SSucc two

data Tensor (n :: Nat) a = MkTensor { order :: SNat n, items :: [a] }
deriving Show

type Vector = Tensor One
type Matrix = Tensor Two

mkVector :: [a] -> Vector a

mkVector v = MkTensor { order = one, items = v }

-- some dummy operation defined between two Vectors (and not other order
-- tensors), which results in a Vector.
some_operation :: Num a => Vector a -> Vector a -> Vector a
some_operation (MkTensor { items = v1 }) (MkTensor { items = v2 }) =
mkVector (v1 ++ v2)

main = do

let va = mkVector ([1,2,3] :: [Integer])
let vb = mkVector ([4,5,6] :: [Integer])

print $ some_operation va vb
print $ order va
print $ order vb
---------------------------------