]]>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.
To not bury the lede, we finally have our first new value for the OEIS sequence A000769:
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 record-breaking sizes. The largest at the time I’m writing this is 64×64, shown below, with hopefully more on the way.
[diagram]
]]>I finally finished all the available puzzles and so Tametsi has released its grip on me. I’ve been trying to figure out what made it so compelling, and here’s the best I could come up with.
]]>Finding syntheses for small still lifes is easy, but at some unknown point below 154 it becomes impossible. Today, we completed our collaborative project to notch the lower bound up from 22 to 23, by giving explicit syntheses for all 1,646,147 (strict) still lifes with population 23.
]]>To my eye, the configurations that work can be quite balanced and attractive. Here are some symmetrical ones for (though in general the solutions are not necessarily symmetric in this way):
[diagram]
The most extensive searches for configurations so far have been done by Achim Flammenkamp and later by Ben Chaffin. I’ve written some CUDA code of my own with the goal of pushing things a little further, and I’ll explain it in the rest of this post.
]]>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 to me so obvious that it must be written down somewhere already. In the literature there is work that gets close (and I’ll do some comparisons at the end), but none are quite what I want. Please let me know if I’ve missed something.
Edit: It’s been pointed out to me that the type theory I’ve put together is almost exactly the Enriched Effect Calculus, though I still haven’t seen the semantics in ordinary vector spaces discussed anywhere.
]]>Here’s what we’ve watched so far, together with my ratings using almost the simplest possible system.
]]>We’ve spent years fighting the vandals who want to destroy the heritage of our neighbourhood. Why should we make an exception for this? I see no other option: we must forbid residents from bringing any new cars into the area. This is already a compromise, as if I had my way we would ban every car newer than the house it is parked in front of. (Of course, some especially old houses would need hitching posts to be reinstalled.)
]]>