Formula or Operator not definable proof with Bisimulation in Modal Logic

Question

I have been searching for literature about showing that a property is not modally definable.

I found these two examples on the Web where the author uses the disjoint union for constructing two models that are bisimilar for showing that the operator global box is not definable. Global Box

But later on the author also shows that asymmetry is not definable by two models that are bisimilar again but this time without the disjoint union. Asymmetry

My question is how to define two models that are bisimilar in order to prove that a property is not definable. I know that the property must hold in one model but not the other, but how to achieve this?

Answer 1

There is no general principle of construction such models. In every particular situation you have to use creativity. But some facts can help you. Namely, you should keep in mind that aforementioned disjoint unions, upward-closed submodels (sometimes called generated submodels) and p-morphisms preserve modal satisfability (by means of bisimulation). As you pointed, in the former example the disjoint union was used, while in the later a p-morphism was constructed. In general, it's not enough to use only these three things, but for textbooks examples it suffices.