[Haskell-cafe] foldl in terms of foldr

Dan Weston westondan at imageworks.com
Tue Jan 26 18:10:48 EST 2010

``` > f :: a -> b -> c is a function that takes an a, a b, and returns a c.
>
> g :: (a -> b) -> c takes one argument, which is expected to be a
> function from a to b.  g returns a c.
>
> That stuff I mentioned before about variable binding and function
> application still applies.  We can show that f and g have "isomorphic"
> types.  But they are conceptually different types.

Except that f and g are not isomorphic. In fact, there exists no defined
fuction g :: (a -> b) -> c
(what type would (g id) be?)

Perhaps you meant g :: a -> (b -> c)?

Alexander Solla wrote:
> On Jan 26, 2010, at 8:11 AM, Xingzhi Pan wrote:
>
>> I'm a little confused with the type of step here.  Can it be
>> considered as taking 2 or 3 arguments and then the compiler has to
>> infer to decide?
>
> The compiler is going to build up a sequence of functions.  Consider
> the following (mathematical) function:
>
> f(x, y, z) = x^2 + y^2 + z^2.
>
> This is a function in three arguments.  But if you bind any of them
> (say, x) to a value x', you end up with a function g(y,z) =
> f(x',y,z).  This is a function in two arguments.  Bind another
> variable, and you get another function, with one less argument.  You
> need to bind all the variables in order to compute f for a point.
>
>>  Say if I, as a code reader, meet such a function
>> defined with three formal parameters, how can I draw the conclusion of
>> its type (and it takes 2 arguments actually)?
>
>
> You can derive this from the syntactic properties of types.  Count the
> number of arrows that are not "in parentheses".  That's how many
> arguments the function takes.
>
> f :: a -> b -> c is a function that takes an a, a b, and returns a c.
>
> g :: (a -> b) -> c takes one argument, which is expected to be a
> function from a to b.  g returns a c.
>
> That stuff I mentioned before about variable binding and function
> application still applies.  We can show that f and g have "isomorphic"
> types.  But they are conceptually different types.
>
> _______________________________________________