Add linear map semantics and safe transformations

[snowleopard]

@bolt12 This replaces my previous PR #7. You don't have to merge it -- it's just an illustration of how the ideas from my experiment can be translated to your data type.

[bolt12]

Thanks for this Andrey :smile:

As I said in my email, it would be very nice if a version of matrixBuilder or fromF can be implemented in a type safe manner. Do you have any idea on how to do it?

[snowleopard]

@bolt12 I think we can implement matrixBuilder type-safely. I've just pushed a quick experiment, although I haven't had enough time to see how well it scales for Generic data types.

As for fromF, it's semantics is a bit obscure: it produces only 0/1 matrices, which seem more like relations to me. Using the Matrix e a b data type feels wrong here. Perhaps, you better have a separate type for relations, something like type Relation a b = Matrix Bool a b.

[bolt12]

@snowleopard

As for fromF, it's semantics is a bit obscure: it produces only 0/1 matrices, which seem more like relations to me. Using the Matrix e a b data type feels wrong here. Perhaps, you better have a separate type for relations, something like type Relation a b = Matrix Bool a b.

They are indeed relations I decided to use 1 and 0's because I could work with other quantitative matrices. I also have a Relation module that uses a similar type to type Relation a b = Matrix Bool a b!

This seems promising! There must be a cool way to work with arbitrary generic types!

[snowleopard]

I also have a Relation module that uses a similar type to type Relation a b = Matrix Bool a b!

Ah, indeed, got it!

There must be a cool way to work with arbitrary generic types!

Good, hope you'll manage to make it work :)

[bolt12]

The way I see it work with your implementation is to use Generics to create a deconstructor similar to either for each type. I have something that can work but needs TH https://github.com/bolt12/f-algebra-gen... I think I saw somewhere this solved using Generics

[bolt12]

I think I was able to do it!

toNorm :: (Enum a, Enum (Normalize a)) => a -> Normalize a
toNorm = toEnum . fromEnum
fromNorm :: (Enum a, Enum (Normalize a)) =>Normalize a -> a
fromNorm = toEnum . fromEnum

There's no need for the type-class anymore I think! Since Count a ~ Count (Normalize a), under this restrictions this isomorphism is correct!

[snowleopard]

@bolt12 Awesome! Could you show the full code?

[bolt12]

@snowleopard sure!

type ConstructNorm a = (Enum a, Enum (Normalize a))

toNorm :: ConstructNorm a => a -> Normalize a
toNorm = toEnum . fromEnum

fromNorm :: ConstructNorm a => Normalize a -> a
fromNorm = toEnum . fromEnum

rowN :: (Construct (Normalize a), ConstructNorm a, Num e) => Vector e a -> Matrix e (Normalize a) ()
rowN = row' . contramap fromNorm

linearMapN :: (Construct (Normalize a), Construct (Normalize b), ConstructNorm a, ConstructNorm b, Num e)
           => LinearMap e a b -> Matrix e (Normalize a) (Normalize b)
linearMapN = linearMap . dimap (contramap toNorm) (contramap fromNorm)

columnN :: (Construct (Normalize a), ConstructNorm a, Num e) => Vector e a -> Matrix e () (Normalize a)
columnN = tr . rowN

functionN :: (Construct (Normalize a), Construct (Normalize b), ConstructNorm a, ConstructNorm b, Enumerable a, Num e)
          => (a -> b -> e) -> Matrix e (Normalize a) (Normalize b)
functionN f = linearMapN $ \v -> Vector $ \b -> dot v $ Vector $ \a -> f a b

Everything is the same as your code but without the type class. What do you think?

Thank you once again for all your help! :smile:

[snowleopard]

@bolt12 Thanks but I meant the full code, including the implementation of various other key functions like kr, ><, etc :)

[bolt12]

I'm still working on the refactoring! I hope to open a PR soon with all the changes!

[bolt12]

@snowleopard please see #11 for the full code!

I think we are getting there :smile: the Internal.hs and Type.hs modules are good to go! The whole project still does not compile because I haven't finished refactoring yet.