It’s great to see someone working on this subject. I’d like to point you to Jim Crutchfield’s work, in case you aren’t familiar with it, where he proposes a “calculii of emergence” wherein you start with a dynamical system and via a procedure of teasing out the equivalence classes of how the past constrains the future, can show that you get the “computational structure” or “causal structure” or “abstract structure” (all loaded terms, I know, but there’s math behind it), of the system. It’s a compressed symbolic representation of what the dynamical system is “computing” and furthermore you can show that it is optimal in that this representation preserves exactly the information-theory metrics associated with the dynamical system, e.g. metric entropy. Ultimately, the work describes a heirarchy of systems of increasing computational power (a kind of generalization of the Chomsky heirarchy, where a source of entropy is included), wherein more compressed and more abstract representations of the computational structure of the original dynamical system can be found (up to a point, very much depending on the system). https://www.sciencedirect.com/science/article/pii/0167278994902739
The reason I think you might be interested in this is because it gives a natural notion of just how compressible (read: abstractable) a continous dynamical system is, and has the mathematical machinery to describe in what ways exactly the system is abstractable. There are some important differences to the approach taken here, but I think sufficient overlap that you might find it interesting/inspiring.
There’s also potentially much of interest to you in Cosma Shalizi’s thesis (Crutchfield was his advisor): http://bactra.org/thesis/
The general topic is one of my favorites, so hopefully I will find some time later to say more! Thanks for your interesting and though provoking work.
It’s great to see someone working on this subject. I’d like to point you to Jim Crutchfield’s work, in case you aren’t familiar with it, where he proposes a “calculii of emergence” wherein you start with a dynamical system and via a procedure of teasing out the equivalence classes of how the past constrains the future, can show that you get the “computational structure” or “causal structure” or “abstract structure” (all loaded terms, I know, but there’s math behind it), of the system. It’s a compressed symbolic representation of what the dynamical system is “computing” and furthermore you can show that it is optimal in that this representation preserves exactly the information-theory metrics associated with the dynamical system, e.g. metric entropy. Ultimately, the work describes a heirarchy of systems of increasing computational power (a kind of generalization of the Chomsky heirarchy, where a source of entropy is included), wherein more compressed and more abstract representations of the computational structure of the original dynamical system can be found (up to a point, very much depending on the system). https://www.sciencedirect.com/science/article/pii/0167278994902739
The reason I think you might be interested in this is because it gives a natural notion of just how compressible (read: abstractable) a continous dynamical system is, and has the mathematical machinery to describe in what ways exactly the system is abstractable. There are some important differences to the approach taken here, but I think sufficient overlap that you might find it interesting/inspiring.
There’s also potentially much of interest to you in Cosma Shalizi’s thesis (Crutchfield was his advisor): http://bactra.org/thesis/
The general topic is one of my favorites, so hopefully I will find some time later to say more! Thanks for your interesting and though provoking work.