Hans Aberg commenting Wolfram Kahl:
> For expressing algebraic relations, I think one can use universal algebra
> by factoring through the free universal algebra of a particular set of
> relations. For example, if one wants to state that a binary operator is
> commutative, one can say that is should be defined on the symmetric set of
> order two, that is the set of pairs (x, y) where (x, y) is equivalent to
> (y, x). If one wants to say that an operator is associative (with
> identity), then it factors through the free monoid, or list of the algebra
> itself, so this amount to saying that it is defined on the set of lists.
...
> Otherwise, I think that Haskell need a better way to more exactly describe
> what the object are. For example, a group G is formally a quadruple (G, e,
> *, -1; R), where e is an identity, * multiplication, -1 an inverse, and R
> some relations. Those details are wholly inexpressible in Haskell as
> matters now stand.
...
Perhaps, (no, certainly), I have different mathematic sensibility than
Hans Aberg, but what I *really need* from Haskell is the possibility to
*code* something efficiently, and not just to represent it.
Saying that an operator is commutative does not help me, unless I have
some ways to change the argument order to optimize the evaluation of
a complex expression. This might be a dynamic problem, for example a
Smalltalk object is (might be) able to resend automatically the message
to its argument, with itself as the argument (switching:
receiver<->arg);
this enhances the possibility of forcing the type conversion.
The associativity might be used to automatically optimize some recursive
evaluations through embedding, for example transforming the recursive
n! = n*(n-1)! into tail recursion. I know how to do that by hand.
The monadic laws and some other theorems readable from the commutativity
of some categorical diagrams were for me always a way to restructure the
code in order to optimize it, for example (map f) . (map g) --=>
map (f . g) is a kind of deforestation. Anyway saying that the
associativity
means that the operator is defined on a set of lists doesn't make
me happy at all. But I am a mathematical troglodyte, and a
non-constructive
elegance in programming has no sex-appeal for me...
Jerzy Karczmarczuk
Caen, France
