Incorporating history into ACT representations of dynamical systems

Simon Frost

Mon Feb 03 2025

Hi All,

I’ve been going through David Jaz Myers Dynamical Systems book (which I’m really enjoying), and I was wondering where systems that require history fit in. Examples include delay differential equations (such as my example here for an SIR model with a fixed duration of infectiousness), as well as systems where the observables of the system require a knowledge of the history (see this example). Is representing these systems in David’s framework just thinking about the joint evolution of the states, X, as well as the history, H (which would include, in the most general case, the history of X as well as the inputs I)? How would one put this into a lens-based perspective?

Paul Wang

Tue Feb 04 2025

Hi,

Interesting coincidence, I am currently working on related questions ! (Not in the differential case though)
I think the short answer is: yes, you need to replace state spaces (possibly input/output spaces as well) with “thicker” ones that contain all the history, and probably use the “dependent lenses” version, since some inputs might not make sense given some histories.

Simon Frost

Tue Feb 04 2025

Thanks @paulwang! Any suggestions of a placeholder syntax I could use? I’m not bound to differential equations - I’m more broadly interested in different modeling paradigms (ODEs/SDEs/Markov models/function maps/agent-based models), and having a high-level representation for these models would be very useful.

Paul Wang

Tue Feb 11 2025

As far as I know, there is no high-level framework capturing all these examples, except possibly behavioural settings, that use more of an external point of view (See for instance Chapter 6 of David Jaz Myers’ book), and might no be what you are looking for.
From what I understood, several people, for instance in the Safeguarded AI project, are working on extending categorical systems theory to nondeterministic cases.