2232kdc-0006kdc-0006.xmlNondeterministic dynamical systems and crossing boundaries2024413Kevin CarlsonManuel observed that the two kinds of subobject of discrete possibilitic dynamical systems (dpds, )
have some semantic content directly relevant to boundaries. To wit, a subsystem in the tight
category of dpds being a subset of states which can never be escaped, in cases where is like an agent in a world,
this sounds like a purely impermeable boundary. But can we make sense of the semantics here? We've been imagining
has a set (or whatever) of states, so the boundary of seems, here, like a boundary the world can never cross. Could I think about this as if, if I'm , then I can never reach beyond by own boundary into parts of the world outside of me, and vice versa if I'm , a lax subobject, then I have the option to stay within my own boundary but the option (if I'm a proper subobject) to reach outside as well, in certain states?OK, let's consider as the states of the world in which I most certainly continue to exist. It seems like we can produce an ultrametric on by counting how long it would take the world to reach a state from . Thus states not in but reachable directly from could be thought of as states in which I've been just moderately deformed--perhaps I've lost a finger, or converted to a different religion. A step further away, perhaps I'm dedicating my life to advocacy for those with less than ten fingers, and I'm further away from existing, in some sense, as the self I previously was. This is a somewhat satisfying story! I wonder if it could be useful for a multi-agent case.I guess one case you could consider: two lax subobjects ,, and a third lax subobject representing those states of the world in which a certain dyad of agents certainly continues to exist. If the dyad is, say, a happy marriage, then states one step away from might include those in which the marriage is strained, and states two steps away might include those in which the marriage has ended, all of which still lie within . I'm interested in the potential of this story to incorporate the creation and destruction of things-with-boundaries. It seems to handle a notion of boundary crossing as an agent modifying itself or being modified, but there's not enough structure yet to see two agents interacting in a way that can be described in detail. That seems workable, though: if we define the agents in more detail then we'll be able to construct the state spaces discussed above as a result of properties of those agents.Related2234kdc-0005kdc-0005.xmlSubobjects of possibilistic dynamical systems2024413Kevin CarlsonA discrete possibilistic dynamical system (dpds) is a Kleisli morphism for the nonempty powerset monad , that is, a morphism
where is the set of nonempty subsets of . Note that is at least a topos, here--indeed, for the nonempty powerset monad to be well behaved we may need to be in a Boolean topos. I'm not sure how necessary of a restriction this is. For possibilistic systems, think of something like one agent's model of the behavior of another agent with a comparable complexity. Especially if I don't know you personally, there's a range of things I imagine you might do while reading this paragraph: get bored and give up, go make a cup of tea, look at an email on your phone, or keep reading, and while in some sense I could probably assign probabilities to these outcomes, I think it is actually bad modeling to insist on pretending I'm doing so, when that's just not actually how I think about you: there are plausible actions, as well as things I'm substantially sure you won't do, and that's it.Strict morphisms of dpds are the obvious thing: a map of state spaces such that A key point I have to make is what this means for subobjects: they're just the maps of dpds whose underlying map is mono in , and what this means for the dynamics is a subsystem of a dpds, in the most obvious sense, is a subobject of the state space from which it is impossible to ever escape.This question of subsystems arose at the Boundaries workshop in the context of Nathaniel Osgood, Manuel Baltieri, Martin Biehl, and Matteo Capucci's work on a control theoretic approach to agent-environment boundaries. Here there is a specified "good" subspace of state space, which might be the subspace where the agent stays alive, and we look for a subobject of the system contained in this good space--this means a region of state space within which, having once entered, the agent will always remain. In generalizing from deterministic dynamical systems to possibilistic ones, then, the most natural move produces the notion of a set of states from which the agent can never depart, no matter what decision it makes. This is a natural and useful concept, but it's certainly not the only one we might propose. If we think the agent is smart and makes good decisions (hopefully), then it's enough for survival if the agent can always stay in the viable region.To this end, consider a less obvious category whose objects are still dpds. For this, it's critical that the algebras for actually form a 2-category, because the powerset is of course a poset, not just a set. Therefore we could just as well define a lax morphism of dpds using the square below; since powersets are just posets, there is a unique choice of the arrow filling the square, and all it says in elements is that for all .
This condition on says that is equipped with its own dynamics, and that every state reachable from in is also so reachable in . In particular, since we're using , there is always a move from to another point in , which means we've successfully formalized a notion of a possibilistic subsystem in which it's possible, rather than unavoidable, to remain permanently. Note that we've mildly modified the intuitive notion in that comes with its own dynamics that might not be maximal with respect to making a lax subobject--there is always a maximal such choice, if it's nonempty, given by intersecting each with . This is a kind of "induced" subobject in analogy with the concept of induced subgraph; in categorical terms these are the extremal monomorphisms of the category of dpds and lax maps. You might also note that this category of dpds and lax maps looks like it should be a 2-category; but I don't think there's any reasonable way to add 2-morphisms (which would be modifications) since the state spaces are discrete.