The new math of how large-scale order emerges

128 points16
jsenn13 hours ago

Very cool to see an article that discusses Crutchfield's Epsilon machine formalism. It's one of those rare theories that is conceptually powerful but also simple and concrete enough that it can be implemented in a couple hundred lines of code.

For those interested, [1] is a readable (if quirky) introduction to the theory. The paper discussed in this article seems to discuss a way of "stacking" epsilon machines, so that you have a machine that describes the state transitions of a machine that describes a data set. I wonder if this gets around the main weakness of the e-machine formalism, namely that for a process with non-finite memory, there's no obvious next class of automata to try after finite state machines. In a sense FSMs are the only non-arbitrary model of computation; everything else basically boils down to augmenting a finite-state control with a gadget for storing data, like a stack (pushdown automata), register (counter/register machines), random-access tape (Turing machines), random-access tape but you're only allowed a tape the size of the input (linear-bounded automata) etc. You can constrain that gadget in pretty much arbitrary ways, which makes it difficult to choose a computational model for a non-finite process.


gnulinux8 hours ago

Amazing paper, thank you for sharing!

habitue9 hours ago

This is really cool. Thanks for the link.

rramadass7 hours ago

What are some good resources (books/papers/videos/etc.) to get started on "Complex Systems", "Emergence", "Self-Organization" and related topics?

JPC215 hours ago

Personally I would read a book on statistical physics instead, since this is where the study of 'emergence' began. The Feynman lectures would be good for this.

I actually haven't found many good books for 'Complex systems', but if you want to take a shot anyway, I would look at and

rramadass3 hours ago

Great. The OUP book linked above viz. Introduction to the Theory of Complex Systems by Stefan Thurner et al. looks particularly good since the ToC lists all the needed interdisciplinary topics for a study of this subject.

epsilonic7 hours ago

Anything by John H. Holland is good.

rramadass6 hours ago

Thanks. I have his Hidden Order book and also Per Bak's How Nature Works. Any others you would recommend by these or other authors?

danyala6 hours ago

I can't recommend books but here are a couple of other good phrases to search with, or just to discuss with one of the sparkier LLMs, which tend to find this topic quite interesting:

1. cross-scale interaction

2. downward causation

Would be happy to learn of other terms too.

rramadass3 hours ago

I generally find Wikipedia a very good starting point to get an overview and a gateway to further research/study; their "See Also" sections list plenty of terms/phrases to look up;




uoaei6 hours ago

Prerequisites are heavily aligned with physics / applied math, particularly: thermodynamics, dynamical systems, network theory, and a strong emphasis on interdisciplinary study. All should be approached from a perspective rooted mostly in rigorous information theory.

rramadass3 hours ago

Well said. This recommendation from user JPC21 seems to cover everything you list -

fiforpg14 hours ago

This Quanta trickled through my head

Like water through a sieve.

nico9 hours ago

> exemplifying the way large-scale patterns and organization can arise from innumerable microscopic interactions

Is it proven that the flow of emergence is from micro to macro?

ie. Can emergence go the other way? What’s the starting point of the process? Can a macro process cause micro processes? Or is it always the other way around? Does causality always run in one direction?

naasking8 hours ago

What would be an example of a macro process causing a micro without going through micro processes?

I would think the best you can do is something like fractal geometry, where self-similarity appears at all scales. In some sense, the rules are both micro and macroscopic. An example where this might have real-world implications is Palmer's Invariant Set Theory, which suggests that this fractal structure shapes both cosmological structures and what we see in quantum theory, eg. like violations of Bell's inequality,

boxfire7 hours ago

The Mars global dust storm is caused by coupling of angular momentum of the (solar) system, a global a effect. The Mars system itself down to the dust does not create sufficient conditions

Erem6 hours ago

Would lunar gravity -> tides -> waves -> erosion of particles from a rock count?

tines6 hours ago

"Lunar gravity" is the effect of innumerable individual particles exerting a gravitational force, so that's not really macro.

Blahah3 hours ago


danyala8 hours ago

I wish Descartes and his peers could have lived to see this. It must have been misery.