new pages of SCFG

Section04-Phrasal-and-Syntactic-Models/section04-test.tex

@@ -124,142 +124,57 @@
 \section{基于层次短语的模型}
 \section{基于层次短语的模型}
 
 
 %%%------------------------------------------------------------------------------------------------------------
 %%%------------------------------------------------------------------------------------------------------------
-%%%  使用翻译规则
-\begin{frame}{使用翻译规则描述双语句子生成过程}
+%%%  上下文无关文法
+\begin{frame}{同步上下文无关文法}
 \begin{itemize}
 \begin{itemize}
-\item 翻译过程可以用上述规则描述
-    \begin{itemize}
-    \item 每个变量在源语和目标语端可以被同步替换
-    \end{itemize}
+\item 以上这种对翻译的描述方式，可以用同步上下文无关文法来定义，记Synchronous Context-Free Grammar(SCFG)
 \end{itemize}
 \end{itemize}
 
 
-\begin{center}
-\begin{tikzpicture}
-\begin{scope}
-
-\visible<3->{
-% rule 1 (source)
-\node [anchor=west] (rule1s1) at (0,0) {与};
-\node [anchor=west,inner sep=2pt,fill=black] (rule1s2) at ([xshift=0.5em]rule1s1.east) {\scriptsize{{\color{white} $X_1$}}};
-\node [anchor=west] (rule1s3) at ([xshift=0.5em]rule1s2.east) {有};
-\node [anchor=west,inner sep=2pt,fill=black] (rule1s4) at ([xshift=0.5em]rule1s3.east) {\scriptsize{{\color{white} $X_2$}}};
-
-% rule 1 (target)
-\node [anchor=west] (rule1t1) at ([xshift=0.8in]rule1s4.east) {have};
-\node [anchor=west,inner sep=2pt,fill=black] (rule1t2) at ([xshift=0.5em]rule1t1.east) {\scriptsize{{\color{white} $X_2$}}};
-\node [anchor=west] (rule1t3) at ([xshift=0.5em]rule1t2.east) {with};
-\node [anchor=west,inner sep=2pt,fill=black] (rule1t4) at ([xshift=0.5em]rule1t3.east) {\scriptsize{{\color{white} $X_1$}}};
+\begin{beamerboxesrounded}[upper=uppercolblue,lower=lowercolblue,shadow=true]{定义 - 同步上下文无关文法}
+{\small
+一个同步上下文无关文法由五部分构成$(N, T_s, T_t, I, R)$，其中 \\
+1. $N$是非终结符集合 \\
+2. $T_s$和$T_t$分别是源语言和目标语终结符集合\\
+3. $I \subseteq N$起始非终结符集合\\
+4. $R$是规则集合，每条规则$r \in R$有如下形式
+
+\begin{displaymath}
+\textrm{LHS} \to <\alpha, \beta, \sim>
+\end{displaymath}
+其中，$\textrm{LHS} \in N$表示规则的左部，它是一个非终结符；规则右部由三部分组成，$\alpha \in (N \bigcup T_s)^{*}$表示由源语言终结符和非终结符组成的串；$\beta \in (N \bigcup T_t)^{*}$ 表示由目标语言终结符和非终结符组成的串；$\sim$表示$\alpha$和$\beta$中终结符的1-1对应关系
 }
 }
+\end{beamerboxesrounded}
 
 
-\visible<1->{
-% phrase 1 (source and target)
-\node [anchor=north] (phrase1s1) at ([yshift=-1em]rule1s2.south) {北韩};
-\node [anchor=north] (phrase1t1) at ([yshift=-1em]rule1t4.south) {North Korea};
-}
-
-\visible<2->{
-% phrase 2 (source and target)
-\node [anchor=north] (phrase2s1) at ([yshift=-3em]rule1s4.south) {邦交};
-\node [anchor=north] (phrase2t1) at ([yshift=-3em]rule1t2.south) {diplomatic relations};
-}
-
-\begin{pgfonlayer}{background}
-\visible<3->{
-\node [rectangle,draw=red,inner sep=1pt,thick,fill=white,drop shadow] [fit = (rule1s1) (rule1s4)] (rule1s) {};
-\node [rectangle,draw=red,inner sep=2pt,thick,fill=white,drop shadow] [fit = (rule1t1) (rule1t4)] (rule1t) {};
-}
-\end{pgfonlayer}
-
-\visible<3->{
-% edges (phrases 1-2 to rule1)
-\draw [->] (phrase1s1.north) -- ([yshift=-0.1em]rule1s2.south);
-\draw [->] (phrase1t1.north) -- ([yshift=-0.1em]rule1t4.south);
-\draw [->] (phrase2s1.north) -- ([yshift=-0.1em]rule1s4.south);
-\draw [->] (phrase2t1.north) -- ([yshift=-0.1em]rule1t2.south);
-}
-
-\visible<4->{
-% rule 2 (source)
-\node [anchor=west,inner sep=2pt,fill=black] (rule2s1) at ([yshift=3.5em,xshift=-0.5em]rule1s1.north west) {\scriptsize{{\color{white} $X_1$}}};
-\node [anchor=west] (rule2s2) at ([xshift=0.5em]rule2s1.east) {的};
-\node [anchor=west,inner sep=2pt,fill=black] (rule2s3) at ([xshift=0.5em]rule2s2.east) {\scriptsize{{\color{white} $X_2$}}};
-
-% rule 2 (target)
-\node [anchor=west,inner sep=2pt,fill=black] (rule2t1) at ([yshift=3.5em,xshift=4em]rule1t1.north west) {\scriptsize{{\color{white} $X_2$}}};
-\node [anchor=west] (rule2t2) at ([xshift=0.5em]rule2t1.east) {that};
-\node [anchor=west,inner sep=2pt,fill=black] (rule2t3) at ([xshift=0.5em]rule2t2.east) {\scriptsize{{\color{white} $X_1$}}};
-
-% phrase 3 (source and target)
-\node [anchor=north] (phrase3s1) at ([yshift=-0.8em]rule2s3.south) {\footnotesize{少数 国家}};
-\node [anchor=north] (phrase3t1) at ([yshift=-0.8em]rule2t1.south) {\footnotesize{the few countries}};
-
-% edges (phrase 3 to rule 2 and rule1 to rule2)
-\draw [->] (phrase3s1.north) -- ([yshift=-0.1em]rule2s3.south);
-\draw [->] (phrase3t1.north) -- ([yshift=-0.1em]rule2t1.south);
-\draw [->] ([xshift=2em]rule1s.north west) ..controls +(north:1.5em) and +(south:1.5em).. ([yshift=-0.1em]rule2s1.south);
-\draw [->] ([xshift=-2em]rule1t.north east) ..controls +(north:1.5em) and +(south:1.5em).. ([yshift=-0.1em]rule2t3.south);
-}
-
-\begin{pgfonlayer}{background}
-\visible<4->{
-\node [rectangle,draw=blue,inner sep=4pt,thick,fill=white,drop shadow] [fit = (rule2s1) (rule2s3)] (rule2s) {};
-\node [rectangle,draw=blue,inner sep=4pt,thick,fill=white,drop shadow] [fit = (rule2t1) (rule2t3)] (rule2t) {};
-}
-\end{pgfonlayer}
-
-\visible<5->{
-% rule 3 (source)
-\node [anchor=west,inner sep=2pt,fill=black] (rule3s1) at ([yshift=2.5em,xshift=4em]rule2s1.north west) {\scriptsize{{\color{white} $X_1$}}};
-\node [anchor=west] (rule3s2) at ([xshift=0.5em]rule3s1.east) {之一};
-
-% rule 3 (target)
-\node [anchor=west] (rule3t1) at ([yshift=2.5em,xshift=-4em]rule2t1.north west) {one of};
-\node [anchor=west,inner sep=2pt,fill=black] (rule3t2) at ([xshift=0.5em]rule3t1.east) {\scriptsize{{\color{white} $X_1$}}};
+\end{frame}
 
 
-% edges: rule 2 to rule 3
-\draw [->] ([xshift=-1em]rule2s.north) ..controls +(north:1.2em) and +(south:1.2em).. ([yshift=-0.1em]rule3s1.south);
-\draw [->] ([xshift=1em]rule2t.north) ..controls +(north:1.2em) and +(south:1.2em).. ([yshift=-0.1em]rule3t2.south);
-}
+%%%------------------------------------------------------------------------------------------------------------
+%%%  上下文无关文法 - 一些实例
+\begin{frame}{同步上下文无关文法 - 实例}
+\begin{itemize}
+\item SCFG可以被看做是对CFG的扩展，相当于把单语的CFG扩展到双语，如下是一些SCFG规则，其中每个规则非终结符的对应用非终结符的标号表示
+\begin{eqnarray}
+\textrm{S} & \to & <\textrm{NP}_1\ \textrm{希望}\ \textrm{VP}_2,\ \textrm{NP}_1\ \textrm{wish to}\ \textrm{VP}_2> \nonumber \\
+\textrm{VP}& \to & <\textrm{对}\ \textrm{NP}_1\ \textrm{感到}\ \textrm{VP}_2,\ \textrm{be}\ \textrm{VP}_2\ \textrm{with}\ \textrm{NP}_1> \nonumber \\
+\textrm{NN} & \to & <\textrm{强大},\ \textrm{strong}> \nonumber
+\end{eqnarray}
+
+\item<2-> 这里NP、VP等是有语言学意义的非终结符。当然，在机器翻译中这些并不是必要的，可以使用更简单的文法，只包含一种非终结符
+\begin{eqnarray}
+\textrm{X} & \to & <\textrm{X}_1\ \textrm{希望}\ \textrm{X}_2,\ \textrm{X}_1\ \textrm{wish to}\ \textrm{X}_2> \nonumber \\
+\textrm{X}& \to & <\textrm{对}\ \textrm{X}_1\ \textrm{感到}\ \textrm{X}_2,\ \textrm{be}\ \textrm{X}_2\ \textrm{with}\ \textrm{X}_1> \nonumber \\
+\textrm{X} & \to & <\textrm{强大},\ \textrm{strong}> \nonumber
+\end{eqnarray}
 
 
-\begin{pgfonlayer}{background}
-\visible<5->{
-\node [rectangle,draw=ugreen,inner sep=1pt,thick,fill=white,drop shadow] [fit = (rule3s1) (rule3s2)] (rule3s) {};
-\node [rectangle,draw=ugreen,inner sep=2pt,thick,fill=white,drop shadow] [fit = (rule3t1) (rule3t2)] (rule3t) {};
-}
-\end{pgfonlayer}
-
-\visible<6->{
-% rule 4 (source)
-\node [anchor=west,inner sep=2pt,fill=black] (rule4s1) at ([yshift=3.5em,xshift=-3.5em]rule3s1.north west) {\scriptsize{{\color{white} $X_1$}}};
-\node [anchor=west] (rule4s2) at ([xshift=0.5em]rule4s1.east) {是};
-\node [anchor=west,inner sep=2pt,fill=black] (rule4s3) at ([xshift=0.5em]rule4s2.east) {\scriptsize{{\color{white} $X_2$}}};
-
-% rule 2 (target)
-\node [anchor=west,inner sep=2pt,fill=black] (rule4t1) at ([yshift=3.5em,xshift=2em]rule3t1.north west) {\scriptsize{{\color{white} $X_1$}}};
-\node [anchor=west] (rule4t2) at ([xshift=0.5em]rule4t1.east) {is};
-\node [anchor=west,inner sep=2pt,fill=black] (rule4t3) at ([xshift=0.5em]rule4t2.east) {\scriptsize{{\color{white} $X_2$}}};
-
-% phrase 4 (source and target)
-\node [anchor=north] (phrase4s1) at ([yshift=-0.8em]rule4s1.south) {\footnotesize{澳洲}};
-\node [anchor=north] (phrase4t1) at ([yshift=-0.8em]rule4t1.south) {\footnotesize{Australia}};
-
-% edges (phrase 4 to rule 4 and rule3 to rule4)
-\draw [->] (phrase4s1.north) -- ([yshift=-0.1em]rule4s1.south);
-\draw [->] (phrase4t1.north) -- ([yshift=-0.1em]rule4t1.south);
-\draw [->] ([xshift=1em]rule3s.north) ..controls +(north:1.5em) and +(south:1.5em).. ([yshift=-0.1em]rule4s3.south);
-\draw [->] ([xshift=-1em]rule3t.north east) ..controls +(north:1.5em) and +(south:1.5em).. ([yshift=-0.1em]rule4t3.south);
-}
+\end{itemize}
 
 
-\begin{pgfonlayer}{background}
-\visible<6->{
-\node [rectangle,draw=orange,inner sep=4pt,thick,fill=white,drop shadow] [fit = (rule4s1) (rule4s3)] (rule4s) {};
-\node [rectangle,draw=orange,inner sep=4pt,thick,fill=white,drop shadow] [fit = (rule4t1) (rule4t3)] (rule4t) {};
-}
-\end{pgfonlayer}
+\end{frame}
 
 
-\end{scope}
-\end{tikzpicture}
-\end{center}
+%%%------------------------------------------------------------------------------------------------------------
+%%%  上下文无关文法 - 一些实例
+\begin{frame}{同步上下文无关文法 - 推导}
+\begin{itemize}
+\item 可以使用SCFG规则同步生成源语言和目标语言句对，这个过程和CFG的使用非常类似，只是需要同时生成两个语言的句子。每次规则的使用都会同步替换源语言和目标语言串中的一个非终结符，如下是一个例子
+\end{itemize}
 \end{frame}
 \end{frame}
 
 
 %%%------------------------------------------------------------------------------------------------------------
 %%%------------------------------------------------------------------------------------------------------------

Section04-Phrasal-and-Syntactic-Models/section04.tex

@@ -1507,6 +1507,7 @@ $d$是一个$(\textbf{s},\textbf{t})$上基于短语的翻译推导，$\textrm{P
 \visible<3->{
 \visible<3->{
 \node [rectangle,draw=red,inner sep=1pt,thick,fill=white,drop shadow] [fit = (rule1s1) (rule1s4)] (rule1s) {};
 \node [rectangle,draw=red,inner sep=1pt,thick,fill=white,drop shadow] [fit = (rule1s1) (rule1s4)] (rule1s) {};
 \node [rectangle,draw=red,inner sep=2pt,thick,fill=white,drop shadow] [fit = (rule1t1) (rule1t4)] (rule1t) {};
 \node [rectangle,draw=red,inner sep=2pt,thick,fill=white,drop shadow] [fit = (rule1t1) (rule1t4)] (rule1t) {};
+\draw [<->,dotted,thick,red] ([xshift=0.1em]rule1s.east) -- ([xshift=-0.1em]rule1t.west);
 }
 }
 \end{pgfonlayer}
 \end{pgfonlayer}
 
 
@@ -1525,7 +1526,7 @@ $d$是一个$(\textbf{s},\textbf{t})$上基于短语的翻译推导，$\textrm{P
 \node [anchor=west,inner sep=2pt,fill=black] (rule2s3) at ([xshift=0.5em]rule2s2.east) {\scriptsize{{\color{white} $X_2$}}};
 \node [anchor=west,inner sep=2pt,fill=black] (rule2s3) at ([xshift=0.5em]rule2s2.east) {\scriptsize{{\color{white} $X_2$}}};
 
 
 % rule 2 (target)
 % rule 2 (target)
-\node [anchor=west,inner sep=2pt,fill=black] (rule2t1) at ([yshift=3.5em,xshift=4em]rule1t1.north west) {\scriptsize{{\color{white} $X_2$}}};
+\node [anchor=west,inner sep=2pt,fill=black] (rule2t1) at ([xshift=1.8in]rule2s3.east) {\scriptsize{{\color{white} $X_2$}}};
 \node [anchor=west] (rule2t2) at ([xshift=0.5em]rule2t1.east) {that};
 \node [anchor=west] (rule2t2) at ([xshift=0.5em]rule2t1.east) {that};
 \node [anchor=west,inner sep=2pt,fill=black] (rule2t3) at ([xshift=0.5em]rule2t2.east) {\scriptsize{{\color{white} $X_1$}}};
 \node [anchor=west,inner sep=2pt,fill=black] (rule2t3) at ([xshift=0.5em]rule2t2.east) {\scriptsize{{\color{white} $X_1$}}};
 
 
@@ -1544,6 +1545,7 @@ $d$是一个$(\textbf{s},\textbf{t})$上基于短语的翻译推导，$\textrm{P
 \visible<4->{
 \visible<4->{
 \node [rectangle,draw=blue,inner sep=4pt,thick,fill=white,drop shadow] [fit = (rule2s1) (rule2s3)] (rule2s) {};
 \node [rectangle,draw=blue,inner sep=4pt,thick,fill=white,drop shadow] [fit = (rule2s1) (rule2s3)] (rule2s) {};
 \node [rectangle,draw=blue,inner sep=4pt,thick,fill=white,drop shadow] [fit = (rule2t1) (rule2t3)] (rule2t) {};
 \node [rectangle,draw=blue,inner sep=4pt,thick,fill=white,drop shadow] [fit = (rule2t1) (rule2t3)] (rule2t) {};
+\draw [<->,dotted,thick,blue] ([xshift=0.1em]rule2s.east) -- ([xshift=-0.1em]rule2t.west);
 }
 }
 \end{pgfonlayer}
 \end{pgfonlayer}
 
 
@@ -1553,7 +1555,7 @@ $d$是一个$(\textbf{s},\textbf{t})$上基于短语的翻译推导，$\textrm{P
 \node [anchor=west] (rule3s2) at ([xshift=0.5em]rule3s1.east) {之一};
 \node [anchor=west] (rule3s2) at ([xshift=0.5em]rule3s1.east) {之一};
 
 
 % rule 3 (target)
 % rule 3 (target)
-\node [anchor=west] (rule3t1) at ([yshift=2.5em,xshift=-4em]rule2t1.north west) {one of};
+\node [anchor=west] (rule3t1) at ([xshift=1.0in]rule3s2.east) {one of};
 \node [anchor=west,inner sep=2pt,fill=black] (rule3t2) at ([xshift=0.5em]rule3t1.east) {\scriptsize{{\color{white} $X_1$}}};
 \node [anchor=west,inner sep=2pt,fill=black] (rule3t2) at ([xshift=0.5em]rule3t1.east) {\scriptsize{{\color{white} $X_1$}}};
 
 
 % edges: rule 2 to rule 3
 % edges: rule 2 to rule 3
@@ -1565,6 +1567,7 @@ $d$是一个$(\textbf{s},\textbf{t})$上基于短语的翻译推导，$\textrm{P
 \visible<5->{
 \visible<5->{
 \node [rectangle,draw=ugreen,inner sep=1pt,thick,fill=white,drop shadow] [fit = (rule3s1) (rule3s2)] (rule3s) {};
 \node [rectangle,draw=ugreen,inner sep=1pt,thick,fill=white,drop shadow] [fit = (rule3s1) (rule3s2)] (rule3s) {};
 \node [rectangle,draw=ugreen,inner sep=2pt,thick,fill=white,drop shadow] [fit = (rule3t1) (rule3t2)] (rule3t) {};
 \node [rectangle,draw=ugreen,inner sep=2pt,thick,fill=white,drop shadow] [fit = (rule3t1) (rule3t2)] (rule3t) {};
+\draw [<->,dotted,thick,ugreen] ([xshift=0.1em]rule3s.east) -- ([xshift=-0.1em]rule3t.west);
 }
 }
 \end{pgfonlayer}
 \end{pgfonlayer}
 
 
@@ -1575,7 +1578,7 @@ $d$是一个$(\textbf{s},\textbf{t})$上基于短语的翻译推导，$\textrm{P
 \node [anchor=west,inner sep=2pt,fill=black] (rule4s3) at ([xshift=0.5em]rule4s2.east) {\scriptsize{{\color{white} $X_2$}}};
 \node [anchor=west,inner sep=2pt,fill=black] (rule4s3) at ([xshift=0.5em]rule4s2.east) {\scriptsize{{\color{white} $X_2$}}};
 
 
 % rule 2 (target)
 % rule 2 (target)
-\node [anchor=west,inner sep=2pt,fill=black] (rule4t1) at ([yshift=3.5em,xshift=2em]rule3t1.north west) {\scriptsize{{\color{white} $X_1$}}};
+\node [anchor=west,inner sep=2pt,fill=black] (rule4t1) at ([xshift=2.0in]rule4s2.east) {\scriptsize{{\color{white} $X_1$}}};
 \node [anchor=west] (rule4t2) at ([xshift=0.5em]rule4t1.east) {is};
 \node [anchor=west] (rule4t2) at ([xshift=0.5em]rule4t1.east) {is};
 \node [anchor=west,inner sep=2pt,fill=black] (rule4t3) at ([xshift=0.5em]rule4t2.east) {\scriptsize{{\color{white} $X_2$}}};
 \node [anchor=west,inner sep=2pt,fill=black] (rule4t3) at ([xshift=0.5em]rule4t2.east) {\scriptsize{{\color{white} $X_2$}}};
 
 
@@ -1594,6 +1597,7 @@ $d$是一个$(\textbf{s},\textbf{t})$上基于短语的翻译推导，$\textrm{P
 \visible<6->{
 \visible<6->{
 \node [rectangle,draw=orange,inner sep=4pt,thick,fill=white,drop shadow] [fit = (rule4s1) (rule4s3)] (rule4s) {};
 \node [rectangle,draw=orange,inner sep=4pt,thick,fill=white,drop shadow] [fit = (rule4s1) (rule4s3)] (rule4s) {};
 \node [rectangle,draw=orange,inner sep=4pt,thick,fill=white,drop shadow] [fit = (rule4t1) (rule4t3)] (rule4t) {};
 \node [rectangle,draw=orange,inner sep=4pt,thick,fill=white,drop shadow] [fit = (rule4t1) (rule4t3)] (rule4t) {};
+\draw [<->,dotted,thick,orange] ([xshift=0.1em]rule4s.east) -- ([xshift=-0.1em]rule4t.west);
 }
 }
 \end{pgfonlayer}
 \end{pgfonlayer}
 
 
@@ -1605,10 +1609,50 @@ $d$是一个$(\textbf{s},\textbf{t})$上基于短语的翻译推导，$\textrm{P
 %%%------------------------------------------------------------------------------------------------------------
 %%%------------------------------------------------------------------------------------------------------------
 \subsection{同步上下文无关文法}
 \subsection{同步上下文无关文法}
 
 
+
 %%%------------------------------------------------------------------------------------------------------------
 %%%------------------------------------------------------------------------------------------------------------
 %%%  上下文无关文法
 %%%  上下文无关文法
 \begin{frame}{同步上下文无关文法}
 \begin{frame}{同步上下文无关文法}
-% 形式定义
+\begin{itemize}
+\item 以上这种对翻译的描述方式，可以用同步上下文无关文法来定义，记Synchronous Context-Free Grammar(SCFG)
+\end{itemize}
+
+\begin{beamerboxesrounded}[upper=uppercolblue,lower=lowercolblue,shadow=true]{定义 - 同步上下文无关文法}
+{\small
+一个同步上下文无关文法由五部分构成$(N, T_s, T_t, I, R)$，其中 \\
+1. $N$是非终结符集合 \\
+2. $T_s$和$T_t$分别是源语言和目标语终结符集合\\
+3. $I \subseteq N$起始非终结符集合\\
+4. $R$是规则集合，每条规则$r \in R$有如下形式
+
+\begin{displaymath}
+\textrm{LHS} \to <\alpha, \beta, \sim>
+\end{displaymath}
+其中，$\textrm{LHS} \in N$表示规则的左部，它是一个非终结符；规则右部由三部分组成，$\alpha \in (N \bigcup T_s)^{*}$表示由源语言终结符和非终结符组成的串；$\beta \in (N \bigcup T_t)^{*}$ 表示由目标语言终结符和非终结符组成的串；$\sim$表示$\alpha$和$\beta$中终结符的1-1对应关系
+}
+\end{beamerboxesrounded}
+
+\end{frame}
+
+%%%------------------------------------------------------------------------------------------------------------
+%%%  上下文无关文法 - 一些实例
+\begin{frame}{同步上下文无关文法 - 实例}
+\begin{itemize}
+\item SCFG可以被看做是对CFG的扩展，相当于把单语的CFG扩展到双语，如下是一些SCFG规则，其中每个规则非终结符的对应用非终结符的标号表示
+\begin{eqnarray}
+\textrm{S} & \to & <\textrm{NP}_1\ \textrm{希望}\ \textrm{VP}_2,\ \textrm{NP}_1\ \textrm{wish to}\ \textrm{VP}_2> \nonumber \\
+\textrm{VP}& \to & <\textrm{对}\ \textrm{NP}_1\ \textrm{感到}\ \textrm{VP}_2,\ \textrm{be}\ \textrm{VP}_2\ \textrm{with}\ \textrm{NP}_1> \nonumber \\
+\textrm{NN} & \to & <\textrm{强大},\ \textrm{strong}> \nonumber
+\end{eqnarray}
+
+\item<2-> 这里NP、VP等是有语言学意义的非终结符。当然，在机器翻译中这些并不是必要的，可以使用更简单的文法，只包含一种非终结符
+\begin{eqnarray}
+\textrm{X} & \to & <\textrm{X}_1\ \textrm{希望}\ \textrm{X}_2,\ \textrm{X}_1\ \textrm{wish to}\ \textrm{X}_2> \nonumber \\
+\textrm{X}& \to & <\textrm{对}\ \textrm{X}_1\ \textrm{感到}\ \textrm{X}_2,\ \textrm{be}\ \textrm{X}_2\ \textrm{with}\ \textrm{X}_1> \nonumber \\
+\textrm{X} & \to & <\textrm{强大},\ \textrm{strong}> \nonumber
+\end{eqnarray}
+
+\end{itemize}
 \end{frame}
 \end{frame}
 
 
 %%%------------------------------------------------------------------------------------------------------------
 %%%------------------------------------------------------------------------------------------------------------