galoisenne

This is a very nice approach. Unfortunately, Piantadosi's pairing function does not work for acyclic grammars (i.e., finite CFLs). We found a variant that handles enumerating all trees of a certain width from an arbitrary CFG, i.e., L(G) ∩ Σ^n, or whose language is finite [1]: [1]: https://github.com/breandan/galoisenne/blob/master/latex/laf...

During a recent programming adventure, I discovered that finite elementary cellular automata with periodic boundary conditions can be simulated using matrix-vector multiplication with a circulant matrix and a strange kind of algebra, which I call a kernel algebra. [1] I wonder if this same idea could be generalized to higher dimensional automata on other topological surfaces. [1]: https://github.com/breandan/galoisenne/blob/8f0f1e9e4e02062c...

> One person noodling around for a few weeks Well, I've been trying for about three years, but to be fair I'm a pretty slow programmer so you may have better luck. > a language being too obscure to have seen much academic study Doesn't seem too obscure to me. https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=kotl... > T isn't generic in that code, it's just a class. I think it needs to be a generic type in Kotlin for this to work, because otherwise it will dispatch to a single method. It's tricky to get Kotlin to do much compile time computation. Not saying that it's impossible, but LMK when you've actually tried it. Here's some sample code if you want to try encoding a Boolean logic: https://github.com/breandan/galoisenne/blob/2e465e7a753f6341...

> I hope you mean by parallel processing, on a single machine. Not necessarily! Parallelism in the more general sense. We can parallelize matrix multiplication and there is good research on distributing graph algorithms across multiple machines. Spielman discusses this in his recent talk on Algebraic Graph Theory: https://youtu.be/CDMQR422LGM?t=1755 > I guess by alphabets you mean dictionaries. Just unordered collections of symbols. Depending on the dictionary or alphabet implementation, it might have an ordering. > I don't agree that the web is a directed graph. This was intentionally vague, but depending on which types of links you consider it may or may not have directionality. Hyperlinks, for example, are directional. > I wouldn't include formula or code parsing because in such cases they are just a means to an end. Part of my goal is

William Hamilton has an interesting variation of Euler's Seven Bridges puzzle [0] which asks whether there is a path visiting every vertex of a polyhedron exactly once. Strangely, while the complexity of detecting Eulerian paths is linear in the number of edges [1], the complexity of detecting Hamiltonian paths is NP-complete [2]! Hamilton (who was also fond of crossing bridges [3]) developed some interesting algebraic structures to study the polyhedron problem [4]. I recently became interested in Eulerian and Hamiltonian paths after taking a class in Graph Representation Learning [5]. In it, I learned there are many interesting connections between algebra and graph theory [6] and started writing a library called Kaliningraph to study some of those connections [7]. [0]: http://eulerarchive.maa.org/docs/originals/E053.pdf [1]: https://en.w