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Versions [faq] 0.0.1 CHANGELOG.md array, async, base (<=5.1.0.0), bytestring, containers, deepseq, ghc-prim, hashable, mtl, mtl-compat, noether, pretty-show, safe, stm, text, transformers, vector [details] MIT Soham Chowdhury Math https://github.com/mrkgnao/noether#readme https://github.com/mrkgnao/noether/issues head: git clone https://github.com/mrkgnao/noether by mrkgnao at Sun Jun 25 05:43:00 UTC 2017 NixOS:0.0.1 noether 290 total (12 in the last 30 days) (no votes yet) [estimated by rule of succession] λ λ λ Docs available Last success reported on 2017-06-25

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# Noether

A very WIP number theory / abstract algebra playground in Haskell.

The part I'm working on at present develops a highly polymorphic numeric hierarchy. Unlike almost every other project (including the great subhask, which is by far the biggest inspiration for this project), all typeclasses representing algebraic structures are "tagged" with the operations that the base type supports. The intention is to have, without newtyping, things like automatically specified L-vector space instances for any K-vector space with K / L a (nice) field extension.

While it may not be inevitable, my inexperienced preliminary encoding of these ideas has delightful consequences like

instance {-# INCOHERENT #-}
( DotProductSpace' k v
, DotProductSpace' k w
, m ~ Mul
) => InnerProductSpace DotProduct p m k Add (v, w) where


that I don't know how to "kill with fire".

Obviously, I'm still exploring the design space to try and find a good balance between avoiding arbitrary choices (e.g. no privileged Monoid instances for Double and the like) and a useful level of type inference. In large part, this means that I'm trying not to run up against trouble with instance resolution and failing hard (see above), or discovering that associated types are sometimes less permissive than one would like.

The numeric hierarchy, at present, extends to functions like this:

(%<) :: LeftModule' r v => r -> v -> v

-- | Linear interpolation.
-- lerp λ v w = λv + (1 - λ)w
lerp
:: VectorSpace' r v
=> r -> v -> v -> v
lerp lambda v w = lambda %< v + w >% (one - lambda)

lol :: (Complex Double, Complex Double)
lol =
(1, 3) * lerp lambda (3, 3) (4, 5) + (1, 0) >% lambda + v + lambda %< w +
(lambda, -lambda)

where
lambda :: Complex Double
lambda = 0.3 :+ 1

v = (3, 3)
w = (2, 7)


A preliminary implementation of linear maps between (what should be) free modules is being developed after the design in Conal Elliott's "Reimagining matrices". The added polymorphism and lack of fixed Scalar a-esque base fields is an interesting challenge, and Conal's basic GADT decomposition of linear maps changes in my case to

data (\>) :: (* -> * -> * -> *) where


where the first "slot" is for the base field. With a nice ~> type operator (which is basically \$), a linear map between two k-vector space types a and b has the type

func :: k \> a ~> b


paving the way for the representation of the category k-Vect as (\>) k :: (* -> * -> *).

Some sample function signatures:


-- | Converts a linear map into a function.
apply :: k \> a ~> b -> a -> b

compose
:: k \> a ~> b
-> k \>      b ~> c
-> k \> a ~>      c


Usage looks like this for now:

> apply (rotate (pi / 4 :: Double)) (1,1)
(1.1102230246251565e-16,1.414213562373095)


# Other stuff

Some other stuff I'm thinking about includes polymorphic numeric literals, possibly along the lines of this:

type family NumericLit (n :: Nat) = (c :: * -> Constraint) where
NumericLit 1 = Neutral Mul
-- NumericLit 2 = Field Add Mul
-- NumericLit n = NumericLit (n - 1)
NumericLit n = Ring Add Mul

fromIntegerP :: forall n a. (KnownNat n, NumericLit n a) => Proxy n -> a
fromIntegerP p =
case sameNat p (Proxy :: Proxy 0) of
Just prf -> gcastWith prf zero'
Nothing -> case sameNat p (Proxy :: Proxy 1) of
Just prf -> gcastWith prf one'
Nothing -> undefined -- unsafeCoerce (val (Proxy :: Proxy a))
-- where
--   val :: (Field Add Mul b) => Proxy b -> b
--   val _ = one + undefined -- fromIntegerP (Proxy :: Proxy (n - 1))


The original core of the project is a short implementation of elliptic curve addition over Q, which I've put on hold temporarily as I try to work out the issues outlined above first. This part uses a Protolude "fork" called Lemmata that I expect will evolve over time.