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7.2

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1 parent 4c97752 commit 5ea37f4587e46318a669ebce1597405df82a9e28 @tianyicui committed Dec 18, 2011
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82 EOTTOC.lyx
@@ -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
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