On Sat, Feb 9, 2013 at 6:35 PM, Robert Goss wrote:

Dear all,

Looking to learn a little haskell I tried to program up a little group theory but feel I am stuck on what the types should be. At first it seemed obvious (and the algebra package does this) that the type of a group should be given by:

class Group g where mul :: g -> g -> g inv :: g -> g unit :: g

My problem is this seems to assume that the type of group you are in is encoded by the type system.

I first ran into problems with this when I wanted to define a cyclic group. The only way I could define the type was either to define each cyclic group separately so have C2, C3, C4, ... or parametrise it over the class Nat. So a cyclic group would have the type Cyclic (Succ(Succ( ... Succ(Zero)) ... )) which would consistently define all cyclic groups but is hardly any better. For example a computation mod a large prime p is not going to be pleasant.

I came up with a partial solution by realising that a group is defined as a set X and some operations on it to get

class Group g x where mul :: g -> x -> x -> x inv :: g-> x -> x unit :: g -> x

Making it easy to define cyclic groups and all my old groups carry over. But now to evaluate an expression I need to hang onto the group i am in and pass it around. As i am newish to haskell I wanted to know if I have missed a simpler more obvious way of doing things?

All the best,

Robert Goss

Taking your first approach I could do this much: --------------------------------------------------- class Group g where mul :: g -> g -> g inv :: g -> g unit :: g class Group g => CycGroup g where gen :: g class CycGroup g => FiniteCycGroup g where baseSet :: [g] -- I would have preferred to have order :: Int, to baseSet -- but ghc does not like that for some reason data G2 = A|B instance Group G2 where unit = A inv A = A inv B = B mul A A = A mul A B = B mul B A = B mul B B = A instance CycGroup G2 where gen = B instance FiniteCycGroup G2 where baseSet = [A, B] ------------------------------------------------------ -- http://www.the-magus.in http://blog.languager.org