--- a/handouts/ho05.tex	Wed Nov 06 17:09:58 2019 +0000
+++ b/handouts/ho05.tex	Wed Nov 06 21:52:42 2019 +0000
@@ -17,7 +17,7 @@
 While regular expressions are very useful for lexing and for recognising
 many patterns in strings (like email addresses), they have their
 limitations. For example there is no regular expression that can
-recognise the language $a^nb^n$ (where you have strings with $n$ $a$'s
+recognise the language $a^nb^n$ (where you have strings starting with $n$ $a$'s
 followed by the same amount of $b$'s). Another example for which there
 exists no regular expression is the language of well-parenthesised
 expressions. In languages like Lisp, which use parentheses rather
@@ -66,7 +66,7 @@
 the ``words'' appear in. For example ambiguity issues like
 
 \begin{center}
-\tt Time flies like an arrow; fruit flies like bananas.
+\tt Time flies like an arrow. Fruit flies like bananas.
 \end{center}  
 
 \noindent
@@ -466,14 +466,14 @@
 The following grammar is in Chomsky normalform:
 
 \begin{plstx}[margin=1cm]
-  : \meta{S\/} ::= \meta{N}\cdot \meta{P}\\
-  : \meta{P\/} ::= \meta{V}\cdot \meta{N}\\
-  : \meta{N\/} ::= \meta{N}\cdot \meta{N}\\
-  : \meta{N\/} ::= \meta{A}\cdot \meta{N}\\
-  : \meta{N\/} ::= \texttt{student} | \texttt{trainer} | \texttt{team} 
-                   | \texttt{trains}\\
-  : \meta{V\/} ::= \texttt{trains} | \texttt{team}\\
-  : \meta{A\/} ::= \texttt{The} | \texttt{the}\\
+  : \meta{S} ::= \meta{N}\cdot \meta{P}\\
+  : \meta{P} ::= \meta{V}\cdot \meta{N}\\
+  : \meta{N} ::= \meta{N}\cdot \meta{N}\\
+  : \meta{N} ::= \meta{A}\cdot \meta{N}\\
+  : \meta{N} ::= \texttt{student} | \texttt{trainer} | \texttt{team} 
+                | \texttt{trains}\\
+  : \meta{V} ::= \texttt{trains} | \texttt{team}\\
+  : \meta{A} ::= \texttt{The} | \texttt{the}\\
 \end{plstx}
 
 \noindent
@@ -493,7 +493,48 @@
 is recognised by the grammar. The CYK algorithm starts with the
 following triangular data structure.
 
-TBD
+\begin{figure}[t]
+\begin{center}
+  \begin{tikzpicture}[scale=0.8,line width=0.8mm]
+  \draw (-2,0) -- (4,0);
+  \draw (-2,1) -- (4,1);
+  \draw (-2,2) -- (3,2);
+  \draw (-2,3) -- (2,3);
+  \draw (-2,4) -- (1,4);
+  \draw (-2,5) -- (0,5);
+  \draw (-2,6) -- (-1,6);
+  
+  \draw (0,0) -- (0, 5);
+  \draw (1,0) -- (1, 4);
+  \draw (2,0) -- (2, 3);
+  \draw (3,0) -- (3, 2);
+  \draw (4,0) -- (4, 1);
+  \draw (-1,0) -- (-1, 6);
+  \draw (-2,0) -- (-2, 6);
+  
+  \draw (-1.5,-0.5) node {\footnotesize{}\texttt{The}}; 
+  \draw (-0.5,-1.0) node {\footnotesize{}\texttt{trainer}}; 
+  \draw ( 0.5,-0.5) node {\footnotesize{}\texttt{trains}}; 
+  \draw ( 1.5,-1.0) node {\footnotesize{}\texttt{the}}; 
+  \draw ( 2.5,-0.5) node {\footnotesize{}\texttt{student}}; 
+  \draw ( 3.5,-1.0) node {\footnotesize{}\texttt{team}};
+  
+  \draw (-1.5,0.5) node {$A$}; 
+  \draw (-0.5,0.5) node {$N$}; 
+  \draw ( 0.5,0.5) node {$N,V$}; 
+  \draw ( 1.5,0.5) node {$A$}; 
+  \draw ( 2.5,0.5) node {$N$}; 
+  \draw ( 3.5,0.5) node {$N,V$};
+
+  \draw (-2.4, 5.5) node {$1$}; 
+  \draw (-2.4, 4.5) node {$2$}; 
+  \draw (-2.4, 3.5) node {$3$}; 
+  \draw (-2.4, 2.5) node {$4$}; 
+  \draw (-2.4, 1.5) node {$5$}; 
+  \draw (-2.4, 0.5) node {$6$}; 
+  \end{tikzpicture}
+  \end{center}
+\end{figure}
 
 \end{document}