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The algebra and calculus of algebraic data types (2015)
!) of algebraic data types Note: This article assumes some introductory Haskell knowledge. Introduction Just as algebra is fundamental to the whole of mathematics, algebraic data types (ADTs) are fundamental to many common functional programming languages.
They’re the primitives upon which all of our richer data structures are built, including everything from sets, maps, and queues, to bloom filters and neural networks. Some research has been done on deciphering some meaning from this mess, but I can’t use negative and fractional types without adopting the additional language semantics they propose. There are no holes in a constant∂ac=0\partial_a c = 0∂ac=0Sums are straightforward∂∂af(a)+g(a)=∂∂af(a)+∂∂ag(a)\frac{\partial}{\partial a} f(a) + g(a) = \frac{\partial}{\partial a} f(a) + \frac{\partial}{\partial a} g(a)∂a∂f(a)+g(a)=∂a∂f(a)+∂a∂g(a)As are products∂∂af(a)×g(a)=∂∂af(a)×g(a)+f(a)×∂∂ag(a)\frac{\partial}{\partial a} f(a) \times g(a) = \frac{\partial}{\partial a} f(a) \times g(a) + f(a) \times \frac{\partial}{\partial a} g(a)∂a∂f(a)×g(a)=∂a∂f(a)×g(a)+f(a)×∂a∂g(a)The chain rule still applies for composition∂∂a(f∘g)=(∂∂af∘g)×∂∂ag\frac{\partial}{\partial a} \left(f \circ g\right) = \left(\frac{\partial}{\partial a} f \circ g\right) \times \frac{\partial}{\partial a} g∂a∂(f∘g)=(∂a∂f∘g)×∂a∂gLet’s take a moment to marvel at the fact that differentiation, a tool developed by Newton and Leibniz over 300 years ago for physics, has made an unexpected appearance in the most discrete of settings.
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