\begin{frame}{Operations on Words} \begin{block}{Power} The power \alert{$v^k$} consists of $k$ concatenations of $v$'s: \begin{talign} v^0 &= \lambda & v^{k+1} &= v^k v \end{talign} \end{block} \begin{exampleblock}{} Let $w = aba$. Then\vspace{-1ex} \begin{align*} w^0 &= \lambda & w^1 &= aba & w^2 &= abaaba & w^3 &= abaabaaba \end{align*} \end{exampleblock} \pause\smallskip \begin{block}{Reverse} The reverse of $a_1 \cdots a_n$ is \begin{talign} \alert{(a_1 \cdots a_n)^R}=a_n \cdots a_1 \end{talign} The reverse can be inductively defined \begin{talign} \lambda^R &= \lambda & (va)^R &= a(v^R) \end{talign} \end{block} \begin{exampleblock}{} The reverse of $abcb$ is $bcba$. \end{exampleblock} \end{frame}