 Here we recall that \(R[G!]\) is the function that obtains after the public announcement of G; that is, we have \(xR[G!]_ay\) if and only if x and y are in the model after the announcement of G (i.e., \(M,w\models G\) and \(M,y\models G\)) and there is an a-arrow from x to y in the original model (i.e., \(xR_ay\)). The relation \(R[G!]_B\) is then the union of the relations for those agents in B; that is, we have \(xR[G!]_By\) if and only if there is an \(a\in B\) with \(xR[G!]_ay\). Finally, \((R[G!]_B)^*\) is the reflexive-transitive closure of the relation \(R[G!]_B\); that is, we have \(x(R[G!]_B)^*y\) if and only if \(x=y\) or there is a finite sequence 