 One can understand the importance of the notion of reachability in the following way: If \(\omega '\) is reachable from \(\omega\), then if \(\omega\) obtains then some agent can reason that some other agent thinks that \(\omega '\) is possible. Looking at Figure 2.1d, if \(\omega = \omega_1\) occurs, then Cathy (who knows only that \(\{\omega_1, \omega_2\}\) has occurred) knows that Jennifer thinks that \(\omega_5\) might have occurred (even though Cathy knows that \(\omega_5\) did not occur). So Cathy cannot rule out the possibility that Jennifer thinks that Mark thinks that that \(\omega_8\) might have occurred. And Cathy cannot rule out the possibility that Jennifer thinks that Mark thinks that Cathy believes that \(\omega_7\) is possible. In this sense, \(\omega_7\) is reachable from \(\omega_1\). The chain of states which establishes this is \(\omega_1, \omega_2, \omega_5, \omega_8, \omega_7\), since \(\mathcal{H}_1 (\omega_1) = \mathcal{H}_1 (\omega_2),\) \(\mathcal{H}_2 (\omega_2) = \mathcal{H}_2 (\omega_5),\) \(\mathcal{H}_3 (\omega_5) = \mathcal{H}_3 (\omega_8),\) and \(\mathcal{H}_1 (\omega_8) = \mathcal{H}_1 (\omega_7)\). Note that one can show similarly that in this example any state is reachable from any other state. This example also illustrates the following immediate result: