Re: [Haskell-cafe] Hit a wall with the type system

On Nov 29, 2007 4:31 AM, Luke Palmer wrote:

On Nov 29, 2007 4:02 AM, Chris Smith wrote:

...

I was talking to a few people about this on #haskell, and it was suggested I ask here. I should say that I'm playing around here; don't mistake this for an urgent request or a serious problem.

Suppose I wanted to implement automatic differentiation of simple functions on real numbers; then I'd take the operations from Num, Fractional, and Floating, and define how to perform them on pairs of values and their differentials, and then I'd write a differentiate function... but finding an appropriate type for that function seems to be a challenge.

Oh, I think I totally missed the point. I missed the word "simple". I think the problem is that a function of type Num a => a -> a can be any function whatsoever, it does not have to be a simple combination of operators (it could, for example, use show, do a string transformation, and then read the result). So while you can do your AD type, I think a function which differentiates (Num a => a -> a) is not possible using this approach. You must resort to numerical methods... Luke

...

We have:

1. Differentiating a function of the most general type (Num a => a -> a) should produce a result of type (Num a => a -> a).

I don't see why this should be true. Int -> Int is an instance of this type, but derivatives require limits, which integers don't have. Do you intend to output the difference sequence of the function in this case?

But then Double -> Double is also an instance of this type. Do you intend to approximate the real derivative when it's specialized to this?

Instead of worrying about the types, first tell us what semantics you want.

Luke