Formulas of basic modal logic involving only $\top$, $\bot$, propositional connectives and modalities

Question

I am given a Kripke model $\mathcal{M}=(W, R, L)$ where $W=\{w_1, w_2, w_3, w_4\}$ $R=\{(w_1, w_2), (w_2, w_3), (w_4, w_1), (w_4, w_3)\}$, and for all $w \in W$, $L(w)=\varnothing$.

For each $w \in W$, I then want to find a basic modal formula that is only true at $w$. The formulas may only involve $\top$,$\bot$, and propositional connectives and modalities. The modalities that I have is necessity $\square$ and and possibility $\Diamond$.

I think that the formula $\square \bot$ is only true at $w_3$ since $w_3$ has no accessible worlds. But I am stuck finding formulas that are only true at the other worlds.

Answer 1

Hint: Continuing your approach, at what world(s) is $\square\square\bot$ true? What about $\square\square\square\bot$?

Now can you take Boolean combinations of these formulas and the one you found to isolate the individual worlds?