# How should we interpret "common coarsening" in this proof of the uniqueness of coarsest bisimulation?

On page 4 of this pdf in a theoretical RL course, we have a proof of the uniquness of the coarsest bisimulation.

A bisimulation $$\phi$$ is a mapping from states $$s \in\mathcal{S}$$ to abstract states $$\phi(S)$$ such that the following property holds: $$\forall s_1, s_2 \in \mathcal{S}, \forall a \in \mathcal{A}, \forall x' \in \phi(\mathcal{S})$$, $$\phi(s_1)=\phi(s_2) \implies R(s_1,a)=R(s_2,a) \land P(x'|s_1,a) = P(x'|s_2,a)$$.

In other words, a bisimulation groups states together such that the grouping preserves the reward and transition probabilities in the fashion above.

The definition of a common coarsening is given as follows:

$$\phi_{12}(s_1) = \phi_{12}(s_2)$$ if and only if the two states are equivalent under $$\phi_1$$ or $$\phi_2$$.

This definition does not make sense to me. Does anyone know how to interpret it correctly?

For example, suppose I have five states $$s_1, s_2, s_3, s_4, s_5$$. If $$\phi_1$$ groups $$s_1, s_2, s_3$$ as $$g_1$$, and $$s_4, s_5$$ as $$g_2$$, while $$\phi_2$$ groups $$s_1,s_2$$ as $$g_3$$ and $$s_3, s_4, s_5$$ as $$g_4$$, then it's unclear to me what the common coarsening does.

It seems like the definition invokes a contradiction. $$s_3$$ and $$s_4$$ should be in one group under $$\phi_{12}$$ due to $$\phi_2$$, and $$s_2$$ and $$s_3$$ should be in the same group under $$\phi_{12}$$ due to $$\phi_1$$. Then $$s_2, s_3, s_4$$ should be in the same group under $$\phi_{12}$$ by transitivity. But if we examine $$s_2$$ and $$s_4$$, neither $$\phi_1$$ nor $$\phi_2$$ sees them as equivalent, so by the under the definition quoted above, they should not be in the same group.

Can someone clarify how this definition works?