Also, it is weird that an agent that can only take one action vacuously knows all things.
You could want to define not “agent A knows fact F”, but “agent A can counterfactually demonstrate that it knows fact F”. So the agent with a single action can’t demonstrate anything.
All that we’d need to add to the definition is the fact that there exists policies in P that distinguish elements of F, ie that for all fi,fj∈F, with fi≠fj, there exists ei∈ϕ(fi) and ej∈ϕ(fj) and a p∈P with p(ei)≠p(ej).
You could want to define not “agent A knows fact F”, but “agent A can counterfactually demonstrate that it knows fact F”. So the agent with a single action can’t demonstrate anything.
All that we’d need to add to the definition is the fact that there exists policies in P that distinguish elements of F, ie that for all fi,fj∈F, with fi≠fj, there exists ei∈ϕ(fi) and ej∈ϕ(fj) and a p∈P with p(ei)≠p(ej).