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Comment by evanb | Hacker News Reader

All of this business about multiway computation and even “rulial space” seems to have been anticipated by Toffoli to some extent in “Action, or the Fungibility of Computation” [0].

The brilliant point of this paper is to point out that in the same way entropy describes our ignorance of the microscopic state of a system, the action (the time integral of the Lagrangian [itself equal to kinetic energy – potential energy]) seems to quantify our ignorance of the microscopic law that governs a system.

It’s not made explicit, but as a lattice gauge theorist it’s an easy analogy to make: the gauge configurations that contribute correspond to different Dirac operators (ie. PDEs) for matter—-gauge symmetry is in some sense a “rulial space” but… you know… discovered 100 years ago.

[0] http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.42....

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Thanks for the link! I'd be curious if you know about other information-theoretic approaches to understanding phenomena even in say classical mechanics (let's put aside statistical mechanics where the connections are already well understood). I'm dimly aware that there are fields like "symbolic dynamics" but not sure of the best entry point into that literature, or where to find the most powerful perspectives it offers without getting stuck in the weeds.

Not sure I have anything concrete to say, except that the “try all rules” rulial space seems to be an interesting way to make Tegmark’s mathematical universe hypothesis [or at least the more limited computable version] into a concrete thing. But I have to admit that when I read Toffili’s paper I found it amazing, exciting, and otherwise hard to see how to advance. I reread it every few years and get something new each time.

Wow, it looks extremely interesting!

You know, it's bizarre, I recently pursued a similar approach, attempting to derive quadratic kinetic energy of an automaton controlled by a computer program (it used a circular tape to decide whether to move left or right). I set up a coarse-groaning, identifying energy with information, well, negentropy of the state of the tape, and arrived at similar results: higher energy = fewer microstates, in other words, there are fewer programs that move at the speed of light than at some slower speed. Which makes sense! There is only one tape that tells the automaton to move right at every step. Toffoli says exactly this: "A low-energy state is 'cheap' because it is 'common', there are so many ways to achieve it". Conservation of energy is exactly microscopic reversibility! What could be cleaner?

That whole approach is on the shelf for now, because I first want to develop a discrete version of contact, and thence symplectic geometry, since that will provide the mathematical formalism to express a kind of "discrete classical physics", rather than just a single toy model.

Anyway, I'm beyond grateful for this reference!

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