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macrocephalopod's profile
macrocephalopod
macrocephalopod
macrocephalopod
@macrocephalopod

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macrocephalopod

@macrocephalopod

Paul Allen, Vice President M&A, Pierce & Pierce (Sie/Hir) | Vegan | Silence is Violence | Women’s Rights Are Human Rights | ACAB (Assigned Cephalopod At Birth)

Joined December 2020

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    1. macrocephalopod‏ @macrocephalopod 2 Mar 2021

      One of my favorite math/programming mashups is that you can take an algebraic data type (like in OCaml, Haskell, F# etc) and convert it into a function, and the series expansion of the function counts the number of instantiations of the data type.

      6 replies 16 retweets 126 likes
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    2. macrocephalopod‏ @macrocephalopod 2 Mar 2021

      For example, a linked list in Haskell has this definition, it's either empty or a node containing a single element and another listpic.twitter.com/BLveY6sGad

      1 reply 0 retweets 11 likes
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    3. macrocephalopod‏ @macrocephalopod 2 Mar 2021

      Interpreting sum types as addition and product types as multiplication (that's why they have those names duh) you can write L(x) = 1 + xL(x) which you can solve to give the function L(x) = 1/(1-x)

      1 reply 0 retweets 13 likes
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    4. macrocephalopod‏ @macrocephalopod 2 Mar 2021

      This has the series expansion L(x) = 1 + x + x^2 + x^3 + ... which tells you that a list is either empty (1) or has one element (x) or two elements (x^3) etc etcpic.twitter.com/YG70F7OfL7

      1 reply 0 retweets 12 likes
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    5. macrocephalopod‏ @macrocephalopod 2 Mar 2021

      Binary trees are more fun. A binary tree is either empty, or a branch consisting of an element and a pair of binary treespic.twitter.com/AoBNXSOgWN

      1 reply 0 retweets 11 likes
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    6. macrocephalopod‏ @macrocephalopod 2 Mar 2021

      Converting it to math gives T(x) = 1 + xT(x)^2 which you can solve with the quadratic formula (what) to give T(x) = (1 - sqrt(1 - 4x))/(2x)

      1 reply 0 retweets 11 likes
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    7. macrocephalopod‏ @macrocephalopod 2 Mar 2021

      The coefficients in the series expansion are telling you that there is one empty binary tree, one binary tree with one element, two binary trees with two elements (the second element could either be on the left or the right branch)pic.twitter.com/dYJaI8Y2Ye

      1 reply 0 retweets 12 likes
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    8. macrocephalopod‏ @macrocephalopod 2 Mar 2021

      It's not immediately obvious that there are five binary trees with three elements, or fourteen binary trees with four elements, but there arepic.twitter.com/y3ksysrXxo

      1 reply 0 retweets 15 likes
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    9. macrocephalopod‏ @macrocephalopod 2 Mar 2021

      The functions you derive from the algebraic data type are called "generating functions" and they are useful for all kinds of counting problems with unlabeled data (ordinary gf) or labeled data (exponential gf) or probability problems (probability generating function)

      1 reply 1 retweet 19 likes
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    10. macrocephalopod‏ @macrocephalopod 2 Mar 2021

      The mind-blowingly cool thing is that you can differentiate the generating function to get a new function, and if you turn that back into an ADT, it is the zipper type associated with the original ADT (that is, the full data structure plus a 'focus' that you can move around)

      1 reply 1 retweet 22 likes
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      macrocephalopod‏ @macrocephalopod 2 Mar 2021

      I first saw this in a Conor McBride paper from about 2000 but I'm not sure if that's the first time it was noticed, as once you have the ADT <--> generating function correspondence, differentiating it seems like quite a natural thing to do 🤷‍♀️http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.176.2720&rep=rep1&type=pdf …

      3:42 AM - 2 Mar 2021
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      • breandan Choubris tetra 💎 (bouge & carge) Matt Teichman . Agustin Lebron Invariant Perspective Emily Rajiv Chauhan
      1 reply 1 retweet 21 likes
        1. macrocephalopod‏ @macrocephalopod 2 Mar 2021

          Anyway I will probably go back to regularly scheduled finance content after this, feel free to ignore and pretend nothing weird has happened if you like.

          5 replies 0 retweets 20 likes
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