\begin{frame}{Decidability} \begin{goal}{} A decision problem $P$ is decidable if \begin{itemize} \item $P$ is semidecidable, and \item $\overline{P}$ is semidecidable. \end{itemize} \smallskip \end{goal} \pause\bigskip The following question is undecidable, but semidecidable: \begin{block}{Halting problem} Does TM $M$ reach a halting state for input $w$? \textcolor{gray}{(Input: $M$ and $w$.)} \end{block} \pause (Semidecidable: execute $M$ on $w$ and wait.) \pause\bigskip The following question not decidable and \alert{not} semidecidable: \begin{block}{Universal halting problem} Does TM $M$ reach a halting state on all $w \in \Sigma^*$? \textcolor{gray}{(Input: $M$.)} \end{block} (The complement is also not semidecidable.) \bigskip \end{frame}