 These validities, together with the previous one characterizing \([\chi{!}]K\varphi\), are known as the reduction axioms. Here is our first twist: a careful look at these formulas reveals that each one of them characterizes the truth of an announcement formula \([\chi{!}]\varphi\) (the left-hand side of \(\leftrightarrow\)) in terms of formulas (the right-hand side of \(\leftrightarrow\)) whose sub-formulas appearing under the scope of \([\chi{!}]\) are less complex. Moreover: the formula dealing with atoms eliminates \([\chi{!}]\). Thus, given any concrete formula in \(\cL_{\left\{ K, {!} \right\}}\), successive applications of these axioms will eventually produce a semantically equivalent formula where no \([\chi{!}]\) modality appears. This indicates that, expressivity-wise, the public announcement modalities \([\chi{!}]\) are not really needed: anything that can be expressed with them can be also expressed by a formula without them. More precisely, for any formula \(\varphi\) in \(\cL_{\left\{ K, {!} \right\}}\), there is a formula \({\operatorname{tr}(\varphi)}\) in \(\cL_{\left\{ K \right\}}\) such that, for any \((M, w)\),