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.
FSMs are sequential models and then there are functional models and concurrent models.
https://www.complexityexplorer.org/courses/185-introduction-...
This free course is based on her book "Complexity: A Guided Tour"
https://www.amazon.com/Complexity-Guided-Tour-Melanie-Mitche...
Other names/references include "The nature of computation" Stuart kauffman. "Nonlinear dynamics and chaos"
Too many stat mech books to possibly recommend, for the idea of emergence you'll want something including phase transitions so... Can't go wrong with 'thermodynamics and an introduction to thermostatistics". And for stat mech with lots of complex systems diversions, Sethna's book is excellent but difficult to learn from. "Entropy, complexity, and order parameters. Names in thermo/complexity: Sethna Goldenfeld Parisi HE stanley
Finally, it's hard to talk about complex systems without talking about networks, "Networks"-Newman is unambiguously the best choice. Names in networks: Mark Newman Reka Albert Avoid barabasi especially more recent stuff.
There may be better lectures/resources out there I don't know of, but I would start with crutchields lectures. Note again I am coming from heavy physics bias.
I actually haven't found many good books for 'Complex systems', but if you want to take a shot anyway, I would look at https://academic.oup.com/book/25504 and https://press.princeton.edu/books/paperback/9780691122045/cr...
I found it pretty accessible from a programmers point of view.
[0] https://www.amazon.com/COMPLEXITY-CRITICALITY-Imperial-Colle...
The blurb: "A path winds its way through a forest. Why does it go the way it goes? Did someone design it? Or was the path made smooth by feet that chose the smoothest path? Maybe some of both? Confluence examines the many ways in which organized, intentional plans (like paths we design) and self-organized, unintentional patterns (like paths that emerge where we walk) intermingle (happen at the same time and place) and interact (influence each other). The book lays out seven “thinking spaces” (like this one) that explore various aspects of the structures and relationships that flow together in our lives."
1. cross-scale interaction
2. downward causation
Would be happy to learn of other terms too.
1) https://en.wikipedia.org/wiki/Complex_system#See_also
Like water through a sieve.
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?
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,
As I'm writing this out and thinking about it, where would fictional objects fit on the micro <--> macro axis?
On the other hand, you could argue that they’re not “real” and therefore not macro
I don't know how much Hofstadter still invests in this idea, but at the time of GEB, he seemed pretty convinced it is/was a central part of how complex systems like minds/brains function.
The researchers suggest we think about emergence as a kind of “software in the natural world.” Just as the software of your laptop runs without having to keep track of all the microscale information about the electrons in the computer circuitry, so emergent phenomena are governed by macroscale rules that seem self-contained, without heed to what the component parts are doing.
Using a mathematical formalism called computational mechanics, the researchers identified criteria for determining which systems have this kind of hierarchical structure."
[...]
>"Indeed, the degrees of freedom, or independent variables, that capture the behavior of these systems at microscopic and macroscopic scales have precisely the relationship that the theory predicts."
Related:
https://en.wikipedia.org/wiki/Hierarchy
https://en.wikipedia.org/wiki/Dimension
https://en.wikipedia.org/wiki/Degrees_of_freedom
https://en.wikipedia.org/wiki/Phase_space
https://en.wikipedia.org/wiki/Computational_mechanics
https://en.wikipedia.org/wiki/Partial_differential_equation
https://en.wikipedia.org/wiki/Self-organization (a.k.a. "Self Organizing System(s)")
https://en.wikipedia.org/wiki/Consciousness
https://news.ycombinator.com/item?id=39860388
https://en.wikipedia.org/wiki/Emergence
And they're all related!
Magic!