Consider the following world: a set $P$ of past nodes, a set of environment nodes $E$ a set of actor/agent nodes $A$ , and a set of future nodes $F$ . Consider our previously defined function $O p (A; p, f)$ for $p \in P, f \in F$ , which represents how much $A$ is optimizing the value of $f$ with respect to $p$ . For some choices of $A$ and $f$ , we might find that the values $O p (A; p, f)$ allow us to split the members of $P$ into two categories: those for which $O p ≫ 0$ , those with $O p \approx 0$ , and those with $O p ≪ 0$ .

Picking P Apart

We can define $V_{i} = {p \in P ∣ O p (A; p, f_{i}) ≫ 0}$ . In other words, $V_{i}$ is the subset of $P$ which has no local influence on the future node $f_{i}$ whatsoever. If you want to be cute you can think of $V_{i}$ as the victim of $A$ with respect to $f_{i}$ , because it is causally erased from existence.

Remember that $O p$ is high when "freezing" the information going out of $A$ means that the value of $f_{i}$ depends much more on the value $p$ . What does it mean when "freezing" the information of $A$ does the opposite?

Now lets define $U_{i} = {p \in P ∣ O p (A; p, f_{i}) ≪ 0}$ . This means that when we freeze $A$ , these nodes have much less influence on the future. Their influence flows through $A$ . We can call $U_{i}$ the utility-function-like region of $P$ with respect to $f_{i}$ . We might also think of the actions of $A$ as amplifying the influence of $U_{i}$ on $f_{i}$ .

For completeness we'll define $X_{i} = {p \in P ∣ O p (A; p, f_{i}) \approx 0} \equiv P - U - V$ .

Let's now define $U = U_{0} \cup . . . \cup U_{n}$ i.e. the set of points which are utility-like for any future point. $V = V_{0} \cup . . . \cup V_{m}$ will be the set of points which are victim-like for any future points. We might want to define $V^{'} = V_{0} \cap . . . \cap V_{m}$ as the set of "total" victims and $U^{'} = U_{0} \cap . . . \cap U_{m}$ similarly. $X = P - U - V$ meaning $X$ only contains nodes which really don't interact with $A$ at all in an optimizing/amplifying capacity.

We can also think of $U$ and $V$ as the sets of past nodes which have an outsizedly large and small influence on the future as a result of the actions of $A$ , respectively.

Describing D

$D = {d \in E \cup F ∣ \exists v \in V ∣ O p (A; v, d) ≫ 0}$ , in other words $D$ ...

Independent AI Research

Independent AI Research