I have a new post on the Topos blog. A topos can be specified by the geometric theory that it classifies. Though the sequents of a theory are described formally and syntactically, its interaction with the world of sets (through set-indexed disjunctions and axiom schemas) is often a little hand-wavy. In this post I describe two ways of doing this more precisely using type theory. > Read Post
The values of the impartial game Officers are probably periodic eventually. Can we say anything about what the period could be? > Read Post
( Previously , previously ) To not bury the lede, we finally have our first new value for the OEIS sequence A000769 : a ( 19 ) = 32577 a(19) = 32577 a ( 19 ) = 32577 This was calculated in 255 GPU-hours using eight RTX 4090s rented from Runpod . Not only that! With help from Thomas Prellberg and the GPU cluster at Queen Mary University of London, we’ve been able to find some solutions with…
For the last month I’ve been completely addicted to the logic puzzle game Tametsi . Put briefly, it’s ordinary Minesweeper but with a set of 160 hand-crafted puzzles rather than a randomly generated starting board. Beyond the Minesweeper rules that we’re all familiar with, there are only a couple of additions: non-square/irregular grids for many of the puzzles, and “global” constraints, such as a…
In the Game of Life, which still lifes can be produced by crashing gliders together? We’ve known for a few years that the answer cannot be “all of them”, because in 2022 Ilkka Törmä and Ville Salo found a patch of still life that, if it exists in the universe, must have existed since the beginning of time. And so, there is no way we could have produced it out of empty space through glider…
The code in the previous post is fairly quick, but of course we’d always prefer it to be quicker. I’ll keep a record here of things I’ve tried and whether they worked. > Read Post
The No-three-in-line problem asks how many points can be placed on a n × n n \times n n × n grid so that no three points are on the same line, where the lines considered are of any slope and not just orthogonal and diagonal. Each row/column can contain at most 2 points, so clearly the answer is at most 2 n 2n 2 n . The real question is, can we actually achieve 2 n 2n 2 n for every grid size? It’s…
Benton’s Linear/Nonlinear Logic has models in (lax) monoidal adjunctions between monoidal and cartesian categories. [diagram] The tasks performed in GPU kernels often deal with linear and nonlinear maps between real vector spaces. [diagram] I want to take this and run with it: let’s see what it looks like to define some mixed linear/nonlinear functions in a linear/nonlinear type theory. This feels…
For years now I’ve been watching a movie in-person with friends every weekend that I’ve been able to. The movie selection system is perfect, crystalline, sacred, with no downsides or exploitable vulnerabilities. The nominator is the person who was previously nominator the longest ago. The nominator chooses two movies that haven’t been nominated before. Everybody else votes for which to watch. If…
When I walk around my neighbourhood, I am struck by the beautiful character of the houses here: the draughty Queenslanders and timber fences, the majestic external staircases, I could go on forever. But something always felt off, and I finally realise what it is. How can I admire a house with elegant stilts and sweeping verandahs when there is a bright red 2019 Ford Puma sitting in the driveway?…