\begin{frame} \frametitle{Complexity of Local Confluence} \small \begin{definition} For a Turing machine $M$ we define $H_M$ to be $\atrs_M$ extended with: \vspace{-1ex} \begin{align*} \bfunap{\astate}{x}{\funap{f}{y}} &\to \tmT && \mbox{for every $f \in \Sigma, \astate \in \tmstates$ with $\tmtrans{\astate}{f}$ is undefined} \\ \bfunap{\astate}{x}{\tmiblank} &\to \tmT && \mbox{for every $\astate \in \tmstates$ with $\tmtrans{\astate}{\stmblank}$ is undefined} \end{align*} \end{definition} \pause \begin{theorem} $WCR(\atrs)$ and $WCR(\atrs,t)$ are $\csig{0}{1}$-complete. \end{theorem} \pause \begin{proof} We extend $H_M$ with: \vspace{-1.5ex} \begin{align*} S &\to T & S &\to q_0(\tmiblank,\tmiblank) \end{align*} \vspace{-4.5ex} \noindent\pause Then $\langle T, q_0(\tmiblank,\tmiblank) \rangle$ and $\langle q_0(\tmiblank,\tmiblank),T \rangle$ are the only critical pairs.\\ \pause\smallskip Hence $WCR(\atrs)$ and $WCR(\atrs,S)$ hold\\ \quad $\Longleftrightarrow$ $q_0(\tmiblank,\tmiblank) \to^* T$\\ \quad $\Longleftrightarrow$ $M$ halts on the empty tape. \end{proof} \end{frame}