--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/parser.scala	Wed Oct 31 02:05:12 2012 +0000
@@ -0,0 +1,182 @@
+
+// regular expressions including NOT
+abstract class Rexp
+
+case object NULL extends Rexp
+case object EMPTY extends Rexp
+case class CHAR(c: Char) extends Rexp
+case class ALT(r1: Rexp, r2: Rexp) extends Rexp
+case class SEQ(r1: Rexp, r2: Rexp) extends Rexp
+case class STAR(r: Rexp) extends Rexp
+case class NOT(r: Rexp) extends Rexp
+
+
+// some convenience for typing in regular expressions
+def charlist2rexp(s : List[Char]) : Rexp = s match {
+  case Nil => EMPTY
+  case c::Nil => CHAR(c)
+  case c::s => SEQ(CHAR(c), charlist2rexp(s))
+}
+implicit def string2rexp(s : String) : Rexp = charlist2rexp(s.toList)
+
+
+// nullable function: tests whether the regular 
+// expression can recognise the empty string
+def nullable (r: Rexp) : Boolean = r match {
+  case NULL => false
+  case EMPTY => true
+  case CHAR(_) => false
+  case ALT(r1, r2) => nullable(r1) || nullable(r2)
+  case SEQ(r1, r2) => nullable(r1) && nullable(r2)
+  case STAR(_) => true
+  case NOT(r) => !(nullable(r))
+}
+
+// tests whether a regular expression 
+// cannot recognise more
+def no_more (r: Rexp) : Boolean = r match {
+  case NULL => true
+  case EMPTY => false
+  case CHAR(_) => false
+  case ALT(r1, r2) => no_more(r1) && no_more(r2)
+  case SEQ(r1, r2) => if (nullable(r1)) (no_more(r1) && no_more(r2)) else no_more(r1)
+  case STAR(_) => false
+  case NOT(r) => !(no_more(r))
+}
+
+
+// derivative of a regular expression w.r.t. a character
+def der (c: Char, r: Rexp) : Rexp = r match {
+  case NULL => NULL
+  case EMPTY => NULL  case CHAR(d) => if (c == d) EMPTY else NULL
+  case ALT(r1, r2) => ALT(der(c, r1), der(c, r2))
+  case SEQ(r1, r2) => 
+    if (nullable(r1)) ALT(SEQ(der(c, r1), r2), der(c, r2))
+    else SEQ(der(c, r1), r2)
+  case STAR(r) => SEQ(der(c, r), STAR(r))
+  case NOT(r) => NOT(der (c, r))
+}
+
+// regular expression for specifying 
+// ranges of characters
+def RANGE(s : List[Char]) : Rexp = s match {
+  case Nil => NULL
+  case c::Nil => CHAR(c)
+  case c::s => ALT(CHAR(c), RANGE(s))
+}
+
+// one or more
+def PLUS(r: Rexp) = SEQ(r, STAR(r))
+
+// some regular expressions
+val DIGIT = RANGE("0123456789".toList)
+val NONZERODIGIT = RANGE("123456789".toList)
+
+val NUMBER = ALT(SEQ(NONZERODIGIT, STAR(DIGIT)), "0")
+val LPAREN = CHAR('(')
+val RPAREN = CHAR(')')
+val WHITESPACE = PLUS(RANGE(" \n".toList))
+val OPS = RANGE("+-*".toList)
+
+// for classifying the strings that have been recognised
+abstract class Token
+
+case object T_WHITESPACE extends Token
+case object T_NUM extends Token
+case class T_OP(s: String) extends Token
+case object T_LPAREN extends Token
+case object T_RPAREN extends Token
+case class T_NT(s: String) extends Token
+
+type Rule = (Rexp, List[Char] => Token)
+
+def error (s: String) = throw new IllegalArgumentException ("Cannot tokenize: " + s)
+
+def munch(r: Rexp, action: List[Char] => Token, s: List[Char], t: List[Char]) : Option[(List[Char], Token)] = 
+  s match {
+    case Nil if (nullable(r)) => Some(Nil, action(t))
+    case Nil => None
+    case c::s if (no_more(der (c, r)) && nullable(r)) => Some(c::s, action(t))
+    case c::s if (no_more(der (c, r))) => None
+    case c::s => munch(der (c, r), action, s, t ::: List(c))
+  }
+
+def one_token (rs: List[Rule], s: List[Char]) : (List[Char], Token) = {
+ val somes = rs.map { (r) => munch(r._1, r._2, s, Nil) } .flatten
+ if (somes == Nil) error(s.mkString) else (somes sortBy (_._1.length) head)
+}
+
+def tokenize (rs: List[Rule], s: List[Char]) : List[Token] = s match {
+  case Nil => Nil
+  case _ => one_token(rs, s) match {
+    case (rest, token) => token :: tokenize(rs, rest) 
+  }
+}
+
+def tokenizer(rs: List[Rule], s: String) : List[Token] = 
+  tokenize(rs, s.toList).filterNot(_ match {
+    case T_WHITESPACE => true
+    case _ => false
+  })
+
+
+
+// lexing rules for arithmetic expressions
+val lexing_rules: List[Rule]= 
+  List((NUMBER, (s) => T_NUM),
+       (WHITESPACE, (s) => T_WHITESPACE),
+       (LPAREN, (s) => T_LPAREN),
+       (RPAREN, (s) => T_RPAREN),
+       (OPS, (s) => T_OP(s.mkString)))
+
+
+// examples
+println(tokenizer(lexing_rules, "2 + 3 * 4 + 1"))
+println(tokenizer(lexing_rules, "(2 + 3) * (4 + 1)"))
+
+
+type Grammar = List[(String, List[Token])]
+
+// grammar for arithmetic expressions
+val grammar = 
+  List ("E" -> List(T_NUM),
+        "E" -> List(T_NT("E"), T_OP("+"), T_NT("E")),
+        "E" -> List(T_NT("E"), T_OP("-"), T_NT("E")),
+        "E" -> List(T_NT("E"), T_OP("*"), T_NT("E")),    
+        "E" -> List(T_LPAREN, T_NT("E"), T_RPAREN))
+
+
+def chop[A](ts1: List[A], prefix: List[A], ts2: List[A]) : Option[(List[A], List[A])] = 
+  ts1 match {
+    case Nil => None
+    case t::ts => 
+      if (ts1.startsWith(prefix)) Some(ts2.reverse, ts1.drop(prefix.length))
+      else chop(ts, prefix, t::ts2)
+  }
+
+// examples
+chop(List(1,2,3,4,5,6,7,8,9), List(4,5), Nil)  
+chop(List(1,2,3,4,5,6,7,8,9), List(3,5), Nil)  
+
+def replace[A](ts: List[A], out: List[A], in: List [A]) = 
+  chop(ts, out, Nil) match {
+    case None => None
+    case Some((before, after)) => Some(before ::: in ::: after)
+  }  
+
+def parse1(g: Grammar, ts: List[Token]) : Boolean = {
+  //println(ts)
+  if (ts == List(T_NT("E"))) true
+  else {
+    val tss = for ((lhs, rhs) <- g) yield replace(ts, rhs, List(T_NT(lhs)))
+    tss.flatten.exists(parse1(g, _))
+  }
+}
+ 
+
+println() ; parse1(grammar, tokenizer(lexing_rules, "2 + 3 * 4 + 1"))
+println() ; parse1(grammar, tokenizer(lexing_rules, "(2 + 3) * (4 + 1)"))
+println() ; parse1(grammar, tokenizer(lexing_rules, "(2 + 3) * 4 (4 + 1)"))
+
+
+ 

Binary file slides06.pdf has changed

--- a/slides06.tex	Mon Oct 29 12:31:31 2012 +0000
+++ b/slides06.tex	Wed Oct 31 02:05:12 2012 +0000
@@ -219,6 +219,26 @@
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \mode<presentation>{
 \begin{frame}[c]
+
+``I hate coding. I do not want to look at code.''
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+
+``I am appalled. You do not show code anymore.''
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
 \frametitle{\begin{tabular}{c}ReDoS\end{tabular}}
 
 \begin{itemize}
@@ -348,134 +368,33 @@
 \end{frame}}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
 
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\mode<presentation>{
-\begin{frame}[c]
-\frametitle{\begin{tabular}{c}Last Week\end{tabular}}
 
-Last week I showed you\bigskip
-
-\begin{itemize}
-\item an algorithm for automata minimisation
-
-\item an algorithm for transforming a regular expression into an NFA
-
-\item an algorithm for transforming an NFA into a DFA (subset construction)
-
-\end{itemize}
-
-\end{frame}}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
 
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\mode<presentation>{
-\begin{frame}[c]
-\frametitle{\begin{tabular}{c}This Week\end{tabular}}
-
-Go over the algorithms again, but with two new things and \ldots\medskip
-
-\begin{itemize}
-\item with the example: what is the regular expression that accepts every string, except those ending 
-in \bl{aa}?\medskip
-
-\item Go over the proof for \bl{$L(rev(r)) = Rev(L(r))$}.\medskip
-
-\item Anything else so far.
-\end{itemize}
-
-\end{frame}}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-
+\newcommand{\qq}{\mbox{\texttt{"}}}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \mode<presentation>{
 \begin{frame}[c]
-\frametitle{\begin{tabular}{c}Proofs By Induction\end{tabular}}
+\frametitle{\begin{tabular}{c}Grammars\end{tabular}}
+
+A (context-free) Grammar \bl{$G$} consists of
 
 \begin{itemize}
-\item \bl{$P$} holds for \bl{$\varnothing$}, \bl{$\epsilon$} and \bl{c}\bigskip
-\item \bl{$P$} holds for \bl{r$_1$ + r$_2$} under the assumption that \bl{$P$} already
-holds for \bl{r$_1$} and \bl{r$_2$}.\bigskip
-\item \bl{$P$} holds for \bl{r$_1$ $\cdot$ r$_2$} under the assumption that \bl{$P$} already
-holds for \bl{r$_1$} and \bl{r$_2$}.
-\item \bl{$P$} holds for \bl{r$^*$} under the assumption that \bl{$P$} already
-holds for \bl{r}.
-\end{itemize}
-
+\item a finite set of nonterminal symbols (upper case)
+\item a finite terminal symbols or tokens (lower case)
+\item a start symbol (which must be a nonterminal)
+\item a set of rules
 \begin{center}
-\bl{$P(r):\;\;L(rev(r)) = Rev(L(r))$}
-\end{center}
-
-\end{frame}}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\mode<presentation>{
-\begin{frame}[t]
-
-What is the regular expression that accepts every string, except those ending 
-in \bl{aa}?\pause\bigskip
-
-\begin{center}
-\begin{tabular}{l}
-\bl{(a + b)$^*$ba}\\
-\bl{(a + b)$^*$ab}\\
-\bl{(a + b)$^*$bb}\\\pause
-\bl{a}\\
-\bl{\texttt{""}}
-\end{tabular}
-\end{center}\pause
-
-What are the strings to be avoided?\pause\medskip
-
-\begin{center}
-\bl{(a + b)$^*$aa}
+\bl{$A \rightarrow \text{rhs}$}
 \end{center}
 
-\end{frame}}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\mode<presentation>{
-\begin{frame}[t]
-
-An NFA for \bl{(a + b)$^*$aa}
-
-\begin{center}
-\begin{tikzpicture}[scale=2, line width=0.5mm]
-  \node[state, initial]        (q0) at ( 0,1) {$q_0$};
-  \node[state]                    (q1) at ( 1,1) {$q_1$};
-  \node[state, accepting] (q2) at ( 2,1) {$q_2$};
-  \path[->] (q0) edge node[above] {$a$} (q1)
-                  (q1) edge node[above] {$a$} (q2)
-                  (q0) edge [loop below] node {$a$} ()
-                  (q0) edge [loop above] node {$b$} ()
-            ;
-\end{tikzpicture}
-\end{center}\pause
+where \bl{rhs} are sequences involving terminals and nonterminals.\medskip\pause
 
-Minimisation for DFAs\\
-Subset Construction for NFAs
-
-\end{frame}}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\mode<presentation>{
-\begin{frame}[c]
-\frametitle{\begin{tabular}{c}DFA Minimisation\end{tabular}}
-
+We can also allow rules
+\begin{center}
+\bl{$A \rightarrow \text{rhs}_1 | \text{rhs}_2 | \ldots$}
+\end{center}
+\end{itemize}
 
-\begin{enumerate}
-\item Take all pairs \bl{(q, p)} with \bl{q $\not=$ p}
-\item Mark all pairs that accepting and non-accepting states
-\item For  all unmarked pairs \bl{(q, p)} and all characters \bl{c} tests wether
-\begin{center}
-\bl{($\delta$(q,c), $\delta$(p,c))}
-\end{center} 
-are marked. If yes, then also mark \bl{(q, p)}.
-\item Repeat last step until nothing changed.
-\item All unmarked pairs can be merged.
-\end{enumerate}
 
 \end{frame}}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
@@ -483,141 +402,46 @@
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \mode<presentation>{
 \begin{frame}[c]
-
-Minimal DFA \only<1>{\bl{(a + b)$^*$aa}}\only<2->{\alert{not} \bl{(a + b)$^*$aa}}
+\frametitle{\begin{tabular}{c}Palindromes\end{tabular}}
 
 \begin{center}
-\begin{tikzpicture}[scale=2, line width=0.5mm]
-  \only<1>{\node[state, initial]        (q0) at ( 0,1) {$q_0$};}
-  \only<2->{\node[state, initial,accepting]        (q0) at ( 0,1) {$q_0$};}
-  \only<1>{\node[state]                    (q1) at ( 1,1) {$q_1$};}
-  \only<2->{\node[state,accepting]                    (q1) at ( 1,1) {$q_1$};}
-  \only<1>{\node[state, accepting] (q2) at ( 2,1) {$q_2$};}
-  \only<2->{\node[state] (q2) at ( 2,1) {$q_2$};}
-  \path[->] (q0) edge[bend left] node[above] {$a$} (q1)
-                  (q1) edge[bend left] node[above] {$b$} (q0)
-                  (q2) edge[bend left=50] node[below] {$b$} (q0)
-                  (q1) edge node[above] {$a$} (q2)
-                  (q2) edge [loop right] node {$a$} ()
-                  (q0) edge [loop below] node {$b$} ()
-            ;
-\end{tikzpicture}
+\bl{\begin{tabular}{lcl}
+$S$ & $\rightarrow$ &  $\epsilon$ \\
+$S$ & $\rightarrow$ &  $aSa$ \\
+$S$ & $\rightarrow$ &  $bSb$ \\
+\end{tabular}}
+\end{center}\pause
+
+or
+
+\begin{center}
+\bl{\begin{tabular}{lcl}
+$S$ & $\rightarrow$ &  $\epsilon \;|\; aSa \;|\;bSb$ \\
+\end{tabular}}
 \end{center}
 
-\onslide<3>{How to get from a DFA to a regular expression?}
 
 \end{frame}}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
 
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\mode<presentation>{
-\begin{frame}[c]
 
-\begin{center}
-\begin{tikzpicture}[scale=2, line width=0.5mm]
-  \only<1->{\node[state, initial]        (q0) at ( 0,1) {$q_0$};}
-  \only<1->{\node[state]                    (q1) at ( 1,1) {$q_1$};}
-  \only<1->{\node[state] (q2) at ( 2,1) {$q_2$};}
-  \path[->] (q0) edge[bend left] node[above] {$a$} (q1)
-                  (q1) edge[bend left] node[above] {$b$} (q0)
-                  (q2) edge[bend left=50] node[below] {$b$} (q0)
-                  (q1) edge node[above] {$a$} (q2)
-                  (q2) edge [loop right] node {$a$} ()
-                  (q0) edge [loop below] node {$b$} ()
-            ;
-\end{tikzpicture}
-\end{center}\pause\bigskip
-
-\onslide<2->{
-\begin{center}
-\begin{tabular}{r@ {\hspace{2mm}}c@ {\hspace{2mm}}l}
-\bl{$q_0$} & \bl{$=$} & \bl{$2\, q_0 + 3 \,q_1 +  4\, q_2$}\\
-\bl{$q_1$} & \bl{$=$} & \bl{$2 \,q_0 + 3\, q_1 + 1\, q_2$}\\
-\bl{$q_2$} & \bl{$=$} & \bl{$1\, q_0 + 5\, q_1 + 2\, q_2$}\\
-
-\end{tabular}
-\end{center}
-}
-
-\end{frame}}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \mode<presentation>{
 \begin{frame}[c]
-
-\begin{center}
-\begin{tikzpicture}[scale=2, line width=0.5mm]
-  \only<1->{\node[state, initial]        (q0) at ( 0,1) {$q_0$};}
-  \only<1->{\node[state]                    (q1) at ( 1,1) {$q_1$};}
-  \only<1->{\node[state] (q2) at ( 2,1) {$q_2$};}
-  \path[->] (q0) edge[bend left] node[above] {$a$} (q1)
-                  (q1) edge[bend left] node[above] {$b$} (q0)
-                  (q2) edge[bend left=50] node[below] {$b$} (q0)
-                  (q1) edge node[above] {$a$} (q2)
-                  (q2) edge [loop right] node {$a$} ()
-                  (q0) edge [loop below] node {$b$} ()
-            ;
-\end{tikzpicture}
-\end{center}\bigskip
-
-\onslide<2->{
-\begin{center}
-\begin{tabular}{r@ {\hspace{2mm}}c@ {\hspace{2mm}}l}
-\bl{$q_0$} & \bl{$=$} & \bl{$\epsilon + q_0\,b + q_1\,b +  q_2\,b$}\\
-\bl{$q_1$} & \bl{$=$} & \bl{$q_0\,a$}\\
-\bl{$q_2$} & \bl{$=$} & \bl{$q_1\,a + q_2\,a$}\\
-
-\end{tabular}
-\end{center}
-}
-
-\onslide<3->{
-Arden's Lemma:
-\begin{center}
-If \bl{$q = q\,r + s$}\; then\; \bl{$q = s\, r^*$}
-\end{center}
-}
-
-\end{frame}}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\mode<presentation>{
-\begin{frame}[c]
-\frametitle{\begin{tabular}{c}Algorithms on Automata\end{tabular}}
-
-
-\begin{itemize}
-\item Reg $\rightarrow$ NFA: Thompson-McNaughton-Yamada method\medskip
-\item NFA $\rightarrow$ DFA: Subset Construction\medskip
-\item DFA $\rightarrow$ Reg: Brzozowski's Algebraic Method\medskip
-\item DFA minimisation: Hopcrofts Algorithm\medskip
-\item complement DFA
-\end{itemize}
-
-\end{frame}}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-\newcommand{\qq}{\mbox{\texttt{"}}}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\mode<presentation>{
-\begin{frame}[c]
-\frametitle{\begin{tabular}{c}Grammars\end{tabular}}
+\frametitle{\begin{tabular}{c}Arithmetic Expressions\end{tabular}}
 
 \begin{center}
 \bl{\begin{tabular}{lcl}
-$E$ & $\rightarrow$ &  $F + (F \cdot \qq*\qq \cdot F) + (F \cdot \qq\backslash\qq \cdot F)$\\
-$F$ & $\rightarrow$ & $T + (T \cdot \qq\texttt{+}\qq \cdot T) + (T \cdot \qq\texttt{-}\qq \cdot T)$\\
-$T$ & $\rightarrow$ & $num + (\qq\texttt{(}\qq \cdot E \cdot \qq\texttt{)}\qq)$\\
+$E$ & $\rightarrow$ &  $num\_token$ \\
+$E$ & $\rightarrow$ &  $E + E$ \\
+$E$ & $\rightarrow$ &  $E - E$ \\
+$E$ & $\rightarrow$ &  $E * E$ \\
+$E$ & $\rightarrow$ &  $( E )$ 
 \end{tabular}}
-\end{center}
+\end{center}\pause
 
-\bl{$E$}, \bl{$F$} and \bl{$T$} are non-terminals\\
-\bl{$E$} is start symbol\\
-\bl{$num$}, \bl{(}, \bl{)}, \bl{+} \ldots are terminals\bigskip\\
-
-
-\bl{\texttt{(2*3)+(3+4)}}
+\bl{\texttt{1 + 2 * 3 + 4}}
 
 \end{frame}}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
@@ -626,32 +450,33 @@
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \mode<presentation>{
 \begin{frame}[c]
+\frametitle{\begin{tabular}{c}Parse Trees\end{tabular}}
 
 \begin{center}
 \bl{\begin{tabular}{lcl}
-$E$ & $\rightarrow$ &  $F + (F \cdot \qq*\qq \cdot F) + (F \cdot \qq\backslash\qq \cdot F)$\\
-$F$ & $\rightarrow$ & $T + (T \cdot \qq\texttt{+}\qq \cdot T) + (T \cdot \qq\texttt{-}\qq \cdot T)$\\
-$T$ & $\rightarrow$ & $num + (\qq\texttt{(}\qq \cdot E \cdot \qq\texttt{)}\qq)$\\
+$E$ & $\rightarrow$ &  $F \;|\; F * F$\\
+$F$ & $\rightarrow$ & $T \;|\; T + T \;|\; T - T$\\
+$T$ & $\rightarrow$ & $num\_token \;|\; ( E )$\\
 \end{tabular}}
 \end{center}
 
 \begin{center}
 \begin{tikzpicture}[level distance=8mm, blue]
-  \node {E}
-    child {node {F} 
-     child {node {T} 
-                 child {node {\qq(\qq\,E\,\qq)\qq}
-                            child {node{F \qq*\qq{} F}
-                                  child {node {T} child {node {2}}}
-                                  child {node {T} child {node {3}}} 
+  \node {$E$}
+    child {node {$F$} 
+     child {node {$T$} 
+                 child {node {(\,$E$\,)}
+                            child {node{$F$ *{} $F$}
+                                  child {node {$T$} child {node {2}}}
+                                  child {node {$T$} child {node {3}}} 
                                }
                           }
               }
-     child {node {\qq+\qq}}
-     child {node {T}
-       child {node {\qq(\qq\,E\,\qq)\qq} 
-       child {node {F}
-       child {node {T \qq+\qq{} T}
+     child {node {+}}
+     child {node {$T$}
+       child {node {(\,$E$\,)} 
+       child {node {$F$}
+       child {node {$T$ +{} $T$}
                     child {node {3}}
                     child {node {4}} 
                  }
@@ -660,12 +485,59 @@
 \end{tikzpicture}
 \end{center}
 
-\begin{textblock}{5}(1, 5)
+\begin{textblock}{5}(1, 6.5)
 \bl{\texttt{(2*3)+(3+4)}}
 \end{textblock}
 
 \end{frame}}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}Ambiguous Grammars\end{tabular}}
+
+A grammar is \alert{ambiguous} if there is a string that has at least parse trees.
+
+
+\begin{center}
+\bl{\begin{tabular}{lcl}
+$E$ & $\rightarrow$ &  $num\_token$ \\
+$E$ & $\rightarrow$ &  $E + E$ \\
+$E$ & $\rightarrow$ &  $E - E$ \\
+$E$ & $\rightarrow$ &  $E * E$ \\
+$E$ & $\rightarrow$ &  $( E )$ 
+\end{tabular}}
+\end{center}
+
+\bl{\texttt{1 + 2 * 3 + 4}}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}Chomsky Normal Form\end{tabular}}
+
+All rules must be of the form
+
+\begin{center}
+\bl{$A \rightarrow a$}
+\end{center}
+
+or
+
+\begin{center}
+\bl{$A \rightarrow BC$}
+\end{center}
+
+
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+
 
 \end{document}