Seven Trees in One · HackerPapers discussions

Seven Trees in One — mentioned by cvoss

Mon, 20 May 2024 18:57:25 GMT

The connection between type isomorphisms and algebraic equalities is known and explored in a variety of interesting ways. I don't know the precise history of the development of the notation and vocabulary, but there's a reason types like A+B are notated with a plus symbol and called "sums". Similarly for products. And it may surprise you (or not) to learn that people sometimes call A -> B an exponential type, and even notate it with superscript notation. One of my favorites is the result that the binary tree type is isomorphic to septuples of binary trees, which you can arrive at by using the…

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Seven Trees in One — mentioned by enricozb

Thu, 12 Jan 2023 19:53:32 GMT

Some related links (specifically that they perform "high school" algebra on types): - Seven trees in one [0], a paper showing a bijection between binary trees and seven-tuples of binary trees. - The algebra of algebraic data types [1] discusses Taylor expansions of data types. - Differentiation of higher-order types [2] is commenting on OP's paper. I used to be really interested in type theory, but haven't had as much time to explore it, but over the years I've saved these links (and the one posted). [0]: https://arxiv.org/pdf/math/9405205.pdf [1]…

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Seven Trees in One — mentioned by svat

Mon, 10 Oct 2022 16:34:27 GMT

A very nice fact that I coincidentally learned just yesterday, which is not in the linked answer but is touched on in one of the comments and in an edit on the accepted answer: there is an explicit bijection that maps a binary tree to seven binary trees (and vice-versa, by the inverse function). Consider the definition of a binary tree: "a binary tree is either empty, or is a root with a left tree and a right tree". This we can write as T = 1 + T² Solving the analogous equation z = 1 + z² in the complex numbers gives z to be a primitive sixth-root of unity…

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Seven Trees in One — mentioned by mathgenius

Sat, 20 Jun 2020 23:12:28 GMT

If you like that then you are going to love this: "Seven Trees in One" by Andreas Blass. https://arxiv.org/abs/math/9405205

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Seven Trees in One — mentioned by gjm11

Thu, 25 Jun 2015 11:27:33 GMT

A few remarks on this stuff. 1. Very closely related is the pure-mathematical technique of generating functions, where you take a sequence (let's say the Fibonacci sequence 0,1,1,2,3,5,8,13,...) and use its elements as coefficients in a power series (0+1x+1x^2+2x^3+3x^4+5x^5+8x^6+13x^7+...) and then notice that (e.g.) shifting the sequence by 1 is the same as multiplying by x, so if F is the series for (f(n)) then x.F is the series for (f(n-1)) and x^2.F is the series for (f(n-2)) apart from some corrections for very small n, which gets you F=(x+x^2)F+x or F = x/(1-x-x^2), and now you can…

Paper and discussion history. A mention is not an endorsement.