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Sat, 08 Sep 2018
Why I never finish my Haskell programs (part 2 of ∞)
Here's something else that often goes wrong when I am writing a Haskell program. It's related to the problem in the previous article but not the same. Let's say I'm building a module for managing polynomials. Say
Now clearly this is going to be a functor, so I define the Functor instance, which is totally straightforward:
Then I ask myself if it is also going to be an Applicative.
Certainly the
But what about
The first argument there is a polynomial whose coefficients are functions. This is not something we normally deal with. That ought to be the end of the matter. But instead I pursue it just a little farther. Suppose we did have such an object. What would it mean to apply a functional polynomial and an ordinary polynomial? Do we apply the functions on the left to the coefficients on the right and then collect like terms? Say for example $$\begin{align} \left((\sqrt\bullet) \cdot x + \left(\frac1\bullet\right) \cdot 1 \right) ⊛ (9x+4) & = \sqrt9 x^2 + \sqrt4 x + \frac19 x + \frac14 \\ & = 3x^2 + \frac{19}{9} x + \frac 14 \end{align}$$ Well, this is kinda interesting. And it would mean that the
Then the ⊛ can be understood to be just like polynomial
multiplication, except that coefficients are combined with function
composition instead of with multiplication. The operation is
associative, as one would hope and expect, and even though the ⊛
operation is not commutative, it has a two-sided identity element,
which is This is different from the failure mode of the previous article because in that example I was going down a Haskell rabbit hole of more and more unnecessary programming. This time the programming is all trivial. Instead, I've discovered a new kind of mathematical operation and I abandon the programming entirely and go off chasing a mathematical wild goose. [Other articles in category /prog/haskell] permanent link |