# tianyicui/EOTTOC

7.2

 @@ -3738,5 +3738,87 @@ Let . \end_layout +\begin_layout Section +Cook's Theorem +\end_layout + +\begin_layout Standard +Bounded Tiling problem: Given a tiling system +\begin_inset Formula $\mathcal{D}$ +\end_inset + +, an integer +\begin_inset Formula $s$ +\end_inset + +, and a function +\begin_inset Formula $f_{0}:\{0,\ldots,s-1\}\mapsto D$ +\end_inset + +, represented by its sequence of values +\begin_inset Formula $(f_{0}(0),\ldots,f_{0}(s-1))$ +\end_inset + +, is there an +\begin_inset Formula $s\times s$ +\end_inset + + tiling by +\begin_inset Formula $\mathcal{D}$ +\end_inset + + extending +\begin_inset Formula $f_{0}$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Bounded TIling is +\begin_inset Formula $\mathcal{NP}$ +\end_inset + +-complete. +\end_layout + +\begin_layout Standard +Its certificate is polynomial, hence in +\begin_inset Formula $\mathcal{NP}$ +\end_inset + +. + Then encoding nondeterministic Turing machine into tiling system. +\end_layout + +\begin_layout Standard +Cook's Theorem: Satisfiability is +\begin_inset Formula $\mathcal{NP}$ +\end_inset + +-complete. + We can reduce Bounded Tiling to Satisfiability. +\end_layout + +\begin_layout Standard +3-Satisfiability is +\begin_inset Formula $\mathcal{NP}$ +\end_inset + +-complete. +\end_layout + +\begin_layout Standard +Max Sat is a +\emph on +generalization +\emph default + of Satisfiability, hence +\begin_inset Formula $\mathcal{NP}$ +\end_inset + +-complete. +\end_layout + \end_body \end_document