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?
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.
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.
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.